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Frobenius Traces over Q(sqrt(5))

This module implements functionality for computing traces `a_P` of
Frobenius for an elliptic curve over Q(sqrt(5)) efficiently.

EXAMPLES::

    sage: from psage.ellcurve.lseries.aplist_sqrt5 import aplist
    sage: from psage.number_fields.sqrt5.misc import F, a
    sage: E = EllipticCurve([1,-a,a,a+2,a-5])
    sage: aplist(E, 60)
    [-3, -1, 5, -4, 5, 8, 1, 9, -10, 10, 10, -4, 6, -10, 4, 3]

The `a_P` in the above list exactly correspond to those output by the primes_of_bounded_norm function::

    sage: from psage.number_fields.sqrt5 import primes_of_bounded_norm
    sage: primes_of_bounded_norm(60)
    [2, 5a, 3, 11a, 11b, 19a, 19b, 29a, 29b, 31a, 31b, 41a, 41b, 7, 59a, 59b]
E must have base field with defining polynomial x^2-x-1psage.number_fields.sqrt5.prime/projects/68c8b2b8-03ba-44d4-a0d1-5d771c8cb465/code/alex/psage/psage/ellcurve/lseries/aplist_sqrt5.pyxcoefficients of the input curve must be algebraic integerspsage.number_fields.sqrt5.miscshort_weierstrass_invariantspsage.libs.smalljac.wrapper1__Pyx_CFunc_object____object____object____int___to_py.<locals>.wrapprimes_of_bounded_normaplist_remaining_slowglobal_minimal_modeldefining_polynomialelliptic_curve_apAttributeErrorresidue_fieldstringsource_coefficientsaplist_shorta_invariantschange_ringcfunc.to_pycardinalityResidueRingsage_ideal__pyx_vtable__base_fieldValueErrorvaluationfast_onlyconductorTypeErrorprimesparent__import__aplistrightrangeidealboundwrap__test____main__leftarg2arg1arg0opc6c4b6b4b2a6a4a3a2a1ByBxAyAxzyxvmkicPNFEBA�?;�2���������
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 AABFc�B�F�F�B�B�A�B�I�y�F�F�F�F�F�F�F�I�g�B�F�F�B�B�A�B�I�y�F�I�A�d�B�F�F�B�B�A�B�I�y�F�F�F�I�g�E�F�F�B�B�A�B�I�[�]�K�F�F�F�I�g�E�F�F�B�B�A�B�I�����.K^`A`@ A^r���:
p\��!��!���o�0
��!(�2P@	���o���o���o���o��!�:;;&;6;F;V;f;v;�;�;�;�;�;�;�;�;<<&<6<F<V<f<v<�<�<�<�<�<�<�<�<==&=6=F=V=f=v=�=�=�=�=�=�=�=�=>>&>6>F>V>f>v>�>�>�>�>�>�>�>�>??&?6?F?V?f?v?�?�?�?�?�?�?�?�?@@&@6@F@V@��!File: aplist_sqrt5.pyx (starting at line 218)

    Return list of `a_P` values that can be quickly computed for the
    elliptic curve `y^2=x^3+AX+B` and the given list of primes.  See
    the important warnings in the INPUT section below.
    
    INPUT:
        - A, B -- algebraic integers in Q(sqrt(5)), which we assume
          (without checking!) is defined by the polynomial `x^2-x-1`.
          If you violate this assumption, you will get nonsense.
        - ``primes`` -- list of Prime objects
    OUTPUT:
        - list of ints or None

    EXAMPLES::

        sage: from psage.ellcurve.lseries.aplist_sqrt5 import aplist_short
        sage: from psage.number_fields.sqrt5 import primes_of_bounded_norm
        sage: K.<a> = NumberField(x^2-x-1)
        sage: aplist_short(2*a+5, 3*a-7, primes_of_bounded_norm(100))
        [None, None, None, 2, None, 2, 0, -6, -6, -4, -8, 3, 1, 10, -10, 0, 0, -4, -3, -16, 14, -8, 8, 15]
    File: aplist_sqrt5.pyx (starting at line 160)

    Compute -- using possibly very slow methods -- the `a_P` in the
    list v that are currently set to None.

    Here v and primes should be two lists, where primes is a list of
    Cython Prime objects, as output by the primes_of_bounded_norm
    function.  This function currently does not assume anything about
    the model of E being minimal or even integral.

    INPUT:
        - `E` -- elliptic curve over Q(sqrt(5))
        - `v` -- list of ints or None
        - `primes` -- list of Prime objects
        
    OUTPUT:
        - modified v in place by changing all of the None's to ints

    EXAMPLES::

        sage: from psage.ellcurve.lseries.aplist_sqrt5 import aplist
        sage: K.<a> = NumberField(x^2-x-1); E = EllipticCurve([1,-a,a,a-1,a+3])
        sage: v = aplist(E, 60, fast_only=True); v
        [None, None, None, -4, -2, 0, -5, 0, -5, 0, 2, 11, 12, 10, -11, -6]
        sage: from psage.number_fields.sqrt5 import primes_of_bounded_norm
        sage: primes = primes_of_bounded_norm(60)
        sage: from psage.ellcurve.lseries.aplist_sqrt5 import aplist_remaining_slow
        sage: aplist_remaining_slow(E, v, primes); v
        [1, -2, 2, -4, -2, 0, -5, 0, -5, 0, 2, 11, 12, 10, -11, -6]

    We can also use this (slow) function to compute all entries of v.  This must give
    the same answer as above, of course::
    
        sage: v = [None]*len(primes)
        sage: aplist_remaining_slow(E, v, primes)
        sage: v
        [1, -2, 2, -4, -2, 0, -5, 0, -5, 0, 2, 11, 12, 10, -11, -6]
    File: aplist_sqrt5.pyx (starting at line 93)

    Compute the traces of Frobenius `a_P` of the elliptic curve E over Q(sqrt(5)) for
    all primes `P` with norm less than the given bound.

    INPUT:
        - `E` -- an elliptic curve given by an integral (but not
          necessarily minimal) model over the number field with
          defining polynomial `x^2-x-1`.
        - ``bound`` -- a nonnegative integer
        - ``fast_only`` -- (bool, default: False) -- if True, only the
          `a_P` that can be computed efficiently are computed, and the
          rest are left as None

    EXAMPLES::

        sage: from psage.ellcurve.lseries.aplist_sqrt5 import aplist
        sage: K.<a> = NumberField(x^2-x-1); E = EllipticCurve([1,-a,a,a-1,a+3])
        sage: aplist(E,60)
        [1, -2, 2, -4, -2, 0, -5, 0, -5, 0, 2, 11, 12, 10, -11, -6]
        sage: from psage.number_fields.sqrt5 import primes_of_bounded_norm
        sage: primes_of_bounded_norm(60)
        [2, 5a, 3, 11a, 11b, 19a, 19b, 29a, 29b, 31a, 31b, 41a, 41b, 7, 59a, 59b]

    If you give the fast_only option, then only the `a_P` for which
    the current implemenation can compute quickly are computed::
    
        sage: aplist(E,60,fast_only=True)
        [None, None, None, -4, -2, 0, -5, 0, -5, 0, 2, 11, 12, 10, -11, -6]

    This is fast enough that up to a million should only take a few
    seconds::
    
        sage: t = cputime(); v = aplist(E,10^6,fast_only=True)
        sage: assert cputime(t) < 15, "too slow!"

    TESTS::

    We test some edge cases::

        sage: aplist(E, 0)
        []
        sage: L.<b> = NumberField(x^2-5); F = EllipticCurve([1,-b,b,0,0])
        sage: aplist(F, 50)
        Traceback (most recent call last):
        ...
        ValueError: E must have base field with defining polynomial x^2-x-1
        sage: E = EllipticCurve([1,-a,a,a+2,a/7])
        sage: aplist(E, 50)
        Traceback (most recent call last):
        ...
        ValueError: coefficients of the input curve must be algebraic integers

        sage: K.<a> = NumberField(x^2 - x - 1)
        sage: E = EllipticCurve(K, [1, 1, 1, 0, 0])
        sage: aplist(E, 50)
        [-3, 1, 1, -4, -4, 4, 4, -2, -2, 0, 0, 10, 10, -14]
    File: aplist_sqrt5.pyx (starting at line 54)

    Compute the invariants of a short Weierstrass form of E.

    The main motivation for this function is that it doesn't require
    constructing an elliptic curve like the short_weierstrass_model
    method on E does.

    INPUT:
        - E -- an elliptic curve

    OUTPUT:
        - two elements A, B of the base field of the curve such that
          E is isomorphic to `y^2 = x^3 + Ax + B`.

    EXAMPLES::

        sage: from psage.ellcurve.lseries.aplist_sqrt5 import short_weierstrass_invariants
        sage: from psage.number_fields.sqrt5.misc import F, a
        sage: E = EllipticCurve([1,-a,a,a+2,a-5])
        sage: short_weierstrass_invariants(E)
        (1728*a + 2133, 101952*a - 206874)
        sage: E.short_weierstrass_model()
        Elliptic Curve defined by y^2 = x^3 + (1728*a+2133)*x + (101952*a-206874) over Number Field in a with defining polynomial x^2 - x - 1
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unsigned int*>kw_argsoperator NTL::UniquePtr<NTL::ZZ, NTL::DefaultDeleterPolicy>::fake_null_typeINIT_VAL_STRUCTspecial_ZN3NTL9GF2EInfoTC4ERKS0__ZN3NTL9UniquePtrINS_4LazyINS_3VecINS_4ZZ_pEEEEENS_20DefaultDeleterPolicyEEaSEMS7_KFvNS7_6Dummy1EE__pyx_vtabstruct_4sage_5rings_10polynomial_18polynomial_element_Polynomial_generic_densesq_inplace_repeat_ZNK3NTL9UniquePtrINS_3VecINS_4ZZ_pEEENS_20DefaultDeleterPolicyEEdeEv_ZNK3NTL4LazyINS_2ZZEE5builtEvnb_or__pyx_vtabstruct_4sage_9structure_7element_ElementWithCachedMethod_ZN3NTL11UniqueArrayINS_9UniquePtrINS_13FFTVectorPairENS_20DefaultDeleterPolicyEEEEC4EPS4___pyx_k_aplist_remaining_slow_ZN3NTL2ZZaSERKS0___Pyx_CyFunction_set_dict_ZN3NTL4GF2XC4Elim_class_convert_from_hash__pyx_n_s_psage_number_fields_sqrt5_prime_ZNK3NTL10WordVector4eltsEvdescrsetfuncungetwc_ZN3NTL10WrappedPtrINS_4GF2XENS_3VecIS1_E12_vec_deleterEEadEv_ZN3NTL3VecImE11AdjustAllocElcurrency_symbolnb_inplace_subtract_ZNK3NTL2ZZ4sizeEv__pyx_vtabptr_4sage_9structure_7element_ElementWithCachedMethod_ZNK3NTL11UniqueArrayImEcvMS1_KFvNS1_5DummyEEEvlong_long_ZN3NTL3VecImEC4ERKNS_16INIT_SIZE_STRUCTElRKm_ZN3NTL10ZZ_pEInfoTC4Ev_ZN3NTL9UniquePtrINS_2ZZENS_20DefaultDeleterPolicyEE5resetEPS1___wchb__pyx_t_3__pyx_t_4__pyx_t_5__pyx_t_6S_el__pyx_t_8__pyx_t_9__pyx_n_s_ValueError_ZN3NTL9UniquePtrINS_3VecINS_4ZZ_pEEENS_20DefaultDeleterPolicyEE4moveERS5_operator long int~ref_GF2_ZN3NTL4ZZ_p8allocateEvco_flagsm_moduleUniqueArray<long unsigned int>TwoInvPreconTablecause_ZN3NTL3VecIlED4Ev~GF2EInfoT__pyx_k_Pfunc_globals_ZN3NTL3ZZX9normalizeEv_ZN3NTL10WrappedPtrI17_ntl_gbigint_bodyNS_2ZZ7DeleterEEaSERKPS1__ZNK3NTL9UniquePtrINS_4LazyINS_3VecINS_3GF2EEEEENS_20DefaultDeleterPolicyEEcvMS7_KFvNS7_5DummyEEEvmp_length_ZN3NTL10WrappedPtrI17_ntl_gbigint_bodyNS_2ZZ7DeleterEEadEvpy_frame__pyx_n_s_psage_modform_hilbert_sqrt5_sqrt_ZNK3NTL3VecINS_4GF2XEEixEl_ZNK3NTL10WrappedPtrIlNS_3VecIlE12_vec_deleterEEadEv_ZN3NTL9ZZ_pInfoTC4Ev__pyx_kp_s_stringsource_ZN3NTL10WordVectorC4Ev_S_app__pyx_pf_4code_4alex_5psage_5psage_8ellcurve_7lseries_12aplist_sqrt5_short_weierstrass_invariants_ZN3NTL9FFTTablesEoperator NTL::UniquePtr<NTL::FFTPrimeInfo, NTL::DefaultDeleterPolicy>::fake_null_type__pyx_vtabptr_4sage_5rings_10polynomial_19polynomial_compiled_univar_pd_ZN3NTL2ZZ8validateEv__pyx_v_primes__pyx_vtabptr_4sage_5rings_10polynomial_18polynomial_element_Polynomialappend_ZN3NTL9UniquePtrINS_10FFTMulTabsENS_20DefaultDeleterPolicyEE7releaseEv_ZNK3NTL4LazyINS_12ZZ_pFFTInfoTEEdeEvset_elementinverse_series_trunc_ZN3NTL3VecIlE4killEv__pyx_vtabstruct_4sage_5rings_10polynomial_19polynomial_compiled_coeff_pdUniquePtr<_ntl_general_rem_one_struct, NTL::sp_ZZ_reduce_struct_policy>UniquePtr<NTL::ZZ, NTL::DefaultDeleterPolicy>_Ios_Seekdir_convert_map_from_operator NTL::UniquePtr<NTL::Lazy<NTL::Vec<NTL::GF2> >, NTL::DefaultDeleterPolicy>::fake_null_type_S_in_mod__modesp_ZZ_reduce_struct_policy_ZN3NTL10WrappedPtrINS_4ZZ_pENS_3VecIS1_E12_vec_deleterEEcvRPS1_Ev__Pyx_check_binary_versionrecursion_depth_ZN3NTL3VecINS_2ZZEE4InitElco_cellvarsgetitem_as_intget_unsafe__ops_ZN3NTL3VecINS_4GF2XEE12SetMaxLengthEl__pyx_vtabstruct_4sage_9structure_10parent_old_ParentZZ_pFFTInfoT__pyx_vp_4sage_9structure_7element_coercion_model_ZN3NTL11GF2XModulusC4ERKS0___pyx_n_s_fast_only__Pyx_IternextUnpackEndCheck_ZN3NTL3GF24swapERS0__ZNK3NTL3VecINS_4GF2XEE2atElc_profilefunc17PySequenceMethods_ZN3NTL4ZZ_p7storageEvtimezone_ZN3NTL3VecINS_4GF2XEE9SetLengthEl__pyx_ptype_4sage_5rings_12number_field_20number_field_element_OrderElement_absolutelenfunc_S_ate_ZNK3NTL3VecINS_4GF2XEE8positionERKS1_~Vec12PyCodeObject_ZN3NTL3VecINS_4GF2XEEC4ERKS2__ZN3NTL3VecINS_4GF2XEE3endEvPyFrame_New_ZN15_ntl_crt_structD4Evco_nameINIT_LOOP_HOLE_TYPE__pyx_obj_4sage_9structure_7element_DedekindDomainElementoperator NTL::UniquePtr<_ntl_reduce_struct>::fake_null_type_ZN3NTL10WrappedPtrINS_4GF2XENS_3VecIS1_E12_vec_deleterEEC4EvINIT_NO_ALLOC_TYPE_ZN3NTL9UniquePtrINS_3VecINS_4ZZ_pEEENS_20DefaultDeleterPolicyEE4makeEvco_freevars_ZNK3NTL9UniquePtrINS_12ZZ_pFFTInfoTENS_20DefaultDeleterPolicyEEdeEv_ZNK3NTL3VecIdE4dataEv__pyx_ptype_4sage_9structure_7element_AdditiveGroupElement_ZNK3NTL3VecIlE4eltsEv__pyx_vtabstruct_4sage_5rings_12number_field_20number_field_element_OrderElement_relative__pyx_vtabptr_4sage_5rings_10polynomial_19polynomial_compiled_dummy_pdnb_add_ZNK3NTL8SmartPtrINS_9zz_pInfoTEEptEvgetchar_ZNK3NTL19ZZ_CRTStructAdapter4evalERNS_2ZZEPKlRNS_16ZZ_TmpVecAdapterE_ZNK3NTL9UniquePtrI27_ntl_general_rem_one_structNS_26sp_ZZ_reduce_struct_policyEE18fake_null_functionENS3_5DummyE_ZN3NTL3VecImE4swapERS1_operator _ntl_gbigint_body* const&_mul_long~ZZ_pXModulus_ZN3NTL11OptionalValINS_4LazyINS_3VecINS_3GF2EEEEEEaSERKS6__ZNK3NTL3VecINS_4ZZ_pEE4dataEv__Pyx_CyFunction_CallMethodunsigned char_ZNK3NTL3VecINS_4ZZ_pEE6lengthEv_ZNK3NTL9UniquePtrI27_ntl_general_rem_one_structNS_26sp_ZZ_reduce_struct_policyEEcvMS3_KFvNS3_5DummyEEEv_ZNK3NTL22ZZ_ReduceStructAdapter6adjustERNS_2ZZEfgetcPyTypeObject__pyx_vtabptr_4sage_9structure_7element_Vectorcurexc_tracebacktp_alloctp_dealloc__gnu_debugf_builtins_ZN3NTL9UniquePtrINS_2ZZENS_20DefaultDeleterPolicyEE4swapERS3_PyErr_ClearPyCFunctionWithKeywordsthread_id_ZN3NTL10WrappedPtrI17_ntl_gbigint_bodyNS_2ZZ7DeleterEEC4Evbf_getcharbufferkw_name__pyx_k_File_aplist_sqrt5_pyx_starting_a_ZN3NTL3VecINS_4GF2XEE5beginEvfwide_ZNK3NTL3VecIdE9position1ERKd_ZN3NTL9UniquePtrINS_3VecINS_4ZZ_pEEENS_20DefaultDeleterPolicyEE7releaseEv__pyx_L4_argument_unpacking_done_ZN3NTL11UniqueArrayIPmE9SetLengthEl_ZN3NTL5ZZ_pXC4ERKNS_4ZZ_pE_ZN3NTL3VecINS_4GF2XEEC4ERKNS_16INIT_SIZE_STRUCTElRKS1___pyx_vtabptr_4sage_5rings_4ring_Field_ZN3NTL3VecINS_4ZZ_pEE4InitElRKS1__ZN3NTL5ZZ_pXC4ERKNS_16INIT_MONO_STRUCTEll_ZN3NTL3VecIdEclEl_ZN3NTL3VecImE18FixAtCurrentLengthEv_ZNK3NTL9UniquePtrINS_10FFTMulTabsENS_20DefaultDeleterPolicyEEcvMS3_KFvNS3_5DummyEEEv_ZN3NTL11UniqueArrayIPmEC4Ev_ZNK3NTL9UniquePtrI27_ntl_general_rem_one_structNS_26sp_ZZ_reduce_struct_policyEE3getEv_convert_method_namebufferinfo_ZNK3NTL3VecINS_2ZZEE4dataEv_ZN3NTL4GF2X9normalizeEvtmpfilewcsspn_ZN3NTL3VecINS_4GF2XEE2atElPyMemberDef_ZN3NTL5ZZ_pXC4El_hash_valuetp_richcompare_ZN3NTL3VecINS_2ZZEE9SetLengthElRKS1__ZN3NTL5ZZ_pXC4Ev_ZN3NTL3ZZXC4ERKNS_16INIT_SIZE_STRUCTEl__pyx_k_import__pyx_ptype_4sage_5rings_10polynomial_19polynomial_compiled_binary_pdPyLong_Type_ZN3NTL5ZZ_pX9normalizeEvasync_exc_ZN3NTL10AtomicBoolC4ERKS0_im_weakreflistsetattrfunc_ZN3NTL4ZZ_pC4ERKNS_20INIT_NO_ALLOC_STRUCTE_ZNK3NTL9UniquePtrI12_ntl_tmp_vecNS_20DefaultDeleterPolicyEEcvMS3_KFvNS3_5DummyEEEv_ZN3NTL4LazyINS_3VecINS_3GF2EEEE4killEv_Ios_Iostate__pyx_vtabstruct_4sage_5rings_4ring_CommutativeAlgebrakwargs__pyx_codeobj__6_ZNK3NTL3VecImE4dataEv__pyx_codeobj__8__pyx_ptype_4sage_5rings_10polynomial_19polynomial_compiled_dummy_pd__pyx_L1_error_ZN3NTL3GF2C4El_ZN3NTL3GF2C4Ev_ZN3NTL9UniquePtrINS_10FFTMulTabsENS_20DefaultDeleterPolicyEEC4Ev_ZN3NTL10WordVectorC4ERKS0_srand_ZN3NTL9UniquePtrINS_12ZZ_pFFTInfoTENS_20DefaultDeleterPolicyEE7releaseEvfgetpos__pyx_k_valuation_ZN3NTL10WrappedPtrIlNS_3VecIlE12_vec_deleterEEaSERKPl_ZN3NTL11UniqueArrayImEC4Ev__pyx_ptype_4sage_9structure_7element_Matrix~Init_ZN3NTL11OptionalValINS_4LazyINS_3VecINS_4ZZ_pEEEEEE4swapERS6_Vec<NTL::ZZ_p>_new_cis_inert_exact_log_mpfi_log__Pyx_CodeObjectCache17PyCFunctionObject_ZNK3NTL9UniquePtrINS_3VecINS_3GF2EEENS_20DefaultDeleterPolicyEEcvMS5_KFvNS5_5DummyEEEv_ZN3NTL3VecINS_4ZZ_pEEC4ERKNS_16INIT_SIZE_STRUCTElRKS1_DoSetLength_ZNK3NTL4LazyINS_2ZZEEdeEvoperator NTL::UniquePtr<NTL::Vec<NTL::ZZ_p>, NTL::DefaultDeleterPolicy>::fake_null_type__pyx_vtabptr_4sage_5rings_10polynomial_18polynomial_element_Polynomial_generic_dense_ZN3NTL9UniquePtrINS_3VecINS_4ZZ_pEEENS_20DefaultDeleterPolicyEEC4EPS3_wcsstrPyString_InternFromString__pyx_vtabptr_4sage_5rings_10polynomial_19polynomial_compiled_var_pdfsetpos64_ZN3NTL10ZZ_pEInfoTC4ERKS0___gmpz_set_si_ZN3NTL4GF2XC4ERKNS_16INIT_SIZE_STRUCTElINIT_TRANS_TYPEPyExc_OverflowError_ZNK3NTL19ZZ_RemStructAdapter4evalEPlRKNS_2ZZERNS_16ZZ_TmpVecAdapterE_ZN3NTL11UniqueArrayImEaSEMS1_KFvNS1_6Dummy1EE_ZNK3NTL3VecINS_3GF2EE9MaxLengthEv_ZN3NTL10WrappedPtrIdNS_3VecIdE12_vec_deleterEEaSERKPdb_levelKillBig__Pyx_IterFinish_ZN3NTL3VecImE4dataEv_ZNK3NTL9UniquePtrINS_10FFTMulTabsENS_20DefaultDeleterPolicyEE26cannot_compare_these_typesEv__pyx_ptype_5psage_7modform_7hilbert_5sqrt5_10sqrt5_fast_ResidueRing_abstractPyLong_FromLong_ZN3NTL3VecINS_4ZZ_pEEC4ERKNS_16INIT_SIZE_STRUCTEl__destnb_and_ZN3NTL3VecINS_3GF2EEaSERKS2__old_offset_shift_helperLazy<NTL::ZZ_pFFTInfoT>__pyx_pf_4code_4alex_5psage_5psage_8ellcurve_7lseries_12aplist_sqrt5_2aplist~ZZX__pyx_vtabstruct_4sage_9structure_7element_AlgebraElement_ZNK3NTL13Unique2DArrayIlE26cannot_compare_these_typesEvUniquePtr<NTL::Lazy<NTL::Vec<NTL::ZZ_p> >, NTL::DefaultDeleterPolicy>UniquePtr<_ntl_tmp_vec, NTL::DefaultDeleterPolicy>OptionalVal<NTL::Lazy<NTL::Vec<NTL::ZZ_p> > >__pyx_vtabptr_4sage_10categories_3map_FormalCompositeMap_ZN3NTL3VecIdE12AdjustLengthEl_ZN3NTL3VecIlE15AdjustMaxLengthEl_ZNK3NTL3VecImEclEl_ZN3NTL9UniquePtrINS_4LazyINS_3VecINS_3GF2EEEEENS_20DefaultDeleterPolicyEE4makeEv_ZN3NTL26sp_ZZ_reduce_struct_policy7deleterEP27_ntl_general_rem_one_structget_as_intdefaults_kwdictfopen__pyx_f_4code_4alex_5psage_5psage_8ellcurve_7lseries_12aplist_sqrt5_compute_inert_trace_ZN3NTL10WordVectorC4ERKNS_16INIT_SIZE_STRUCTEl_ZNK3NTL11UniqueArrayIPmEcvMS2_KFvNS2_5DummyEEEv_ZN3NTL11UniqueArrayINS_9UniquePtrINS_13FFTVectorPairENS_20DefaultDeleterPolicyEEEEC4Ev_S_badbit_ZN3NTL3VecINS_4ZZ_pEE10ReAllocateElNS_21NotRelocatableTypeTagE__pyx_vtabstruct_4sage_9structure_6parent_Parent_ZN3NTL9GF2EInfoTaSERKS0_wcslen__pyx_k_bound__pyx_vtabptr_4sage_5rings_4ring_CommutativeAlgebra_ZN3NTL9UniquePtrI27_ntl_general_rem_one_structNS_26sp_ZZ_reduce_struct_policyEE4makeEv__pyx_vtabptr_4sage_9structure_7element_Element__Pyx__PyObject_CallOneArg5div_t_ZNK3NTL7ref_GF2cvKNS_3GF2EEv_ZN3NTL9UniquePtrI18_ntl_reduce_structNS_20DefaultDeleterPolicyEE4moveERS3_tp_iter__pyx_dwcstoull__pyx_f_ZNK3NTL9UniquePtrI15_ntl_rem_structNS_20DefaultDeleterPolicyEEdeEv__pyx_m__pyx_n__pyx_r_ZNK3NTL11OptionalValINS_4LazyINS_3VecINS_4ZZ_pEEEEEE6existsEv_ZNK3NTL3VecImE6lengthEv__Pyx_PyObject_CallNoArg__pyx_ptype_4sage_5rings_4ring_Algebra__pyx_ptype_7cpython_7complex_complex_ZN3NTL9UniquePtrINS_12ZZ_pFFTInfoTENS_20DefaultDeleterPolicyEEC4ERKS3_swap__pyx_ptype_4sage_5rings_4ring_DedekindDomainoperator NTL::UniqueArray<long 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