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 <div class="section" id="isomorphisms-between-weierstrass-models-of-elliptic-curves">
<h1>Isomorphisms between Weierstrass models of elliptic curves<a class="headerlink" href="#isomorphisms-between-weierstrass-models-of-elliptic-curves" title="Permalink to this headline">¶</a></h1>
AUTHORS:
<ul class="simple">
<li>Robert Bradshaw (2007): initial version</li>
<li>John Cremona (Jan 2008): isomorphisms, automorphisms and twists
in all characteristics</li>
</ul>
<dl class="class">
<dt id="sage.schemes.elliptic_curves.weierstrass_morphism.WeierstrassIsomorphism">
class <tt class="descclassname">sage.schemes.elliptic_curves.weierstrass_morphism.</tt><tt class="descname">WeierstrassIsomorphism</tt><big>(</big>E=None, urst=None, F=None<big>)</big><a class="headerlink" href="#sage.schemes.elliptic_curves.weierstrass_morphism.WeierstrassIsomorphism" title="Permalink to this definition">¶</a></dt>
<dd>Bases: <a class="reference internal" href="#sage.schemes.elliptic_curves.weierstrass_morphism.baseWI" title="sage.schemes.elliptic_curves.weierstrass_morphism.baseWI"><tt class="xref py py-class docutils literal">sage.schemes.elliptic_curves.weierstrass_morphism.baseWI</tt></a>, <a class="reference external" href="../../../../categories/sage/categories/morphism.html#sage.categories.morphism.Morphism" title="(in Sage Reference Manual: Category Theory v6.8)"><tt class="xref py py-class docutils literal">sage.categories.morphism.Morphism</tt></a>
Class representing a Weierstrass isomorphism between two elliptic curves.
</dd></dl>

<dl class="class">
<dt id="sage.schemes.elliptic_curves.weierstrass_morphism.baseWI">
class <tt class="descclassname">sage.schemes.elliptic_curves.weierstrass_morphism.</tt><tt class="descname">baseWI</tt><big>(</big>u=1, r=0, s=0, t=0<big>)</big><a class="headerlink" href="#sage.schemes.elliptic_curves.weierstrass_morphism.baseWI" title="Permalink to this definition">¶</a></dt>
<dd>This class implements the basic arithmetic of isomorphisms between
Weierstrass models of elliptic curves. These are specified by
lists of the form $[u,r,s,t]$ (with $u\not=0$) which specifies a
transformation $(x,y) \mapsto (x',y')$ where
<blockquote>
<div>$(x,y) = (u^2x'+r , u^3y' + su^2x' + t).$</div></blockquote>
INPUT:
<ul class="simple">
<li><tt class="docutils literal">u,r,s,t</tt> (default (1,0,0,0)) &#8211; standard parameters of an isomorphism between Weierstrass models.</li>
</ul>
EXAMPLES:
<div class="highlight-python"><div class="highlight"><pre>sage: from sage.schemes.elliptic_curves.weierstrass_morphism import *
sage: baseWI()
(1, 0, 0, 0)
sage: baseWI(2,3,4,5)
(2, 3, 4, 5)
sage: R.&lt;u,r,s,t&gt;=QQ[]; baseWI(u,r,s,t)
(u, r, s, t)
</pre></div>
</div>
<dl class="method">
<dt id="sage.schemes.elliptic_curves.weierstrass_morphism.baseWI.is_identity">
<tt class="descname">is_identity</tt><big>(</big><big>)</big><a class="headerlink" href="#sage.schemes.elliptic_curves.weierstrass_morphism.baseWI.is_identity" title="Permalink to this definition">¶</a></dt>
<dd>Returns True if this is the identity isomorphism.
EXAMPLES:
<div class="highlight-python"><div class="highlight"><pre>sage: from sage.schemes.elliptic_curves.weierstrass_morphism import *
sage: w=baseWI(); w.is_identity()
True
sage: w=baseWI(2,3,4,5); w.is_identity()
False
</pre></div>
</div>
</dd></dl>

<dl class="method">
<dt id="sage.schemes.elliptic_curves.weierstrass_morphism.baseWI.tuple">
<tt class="descname">tuple</tt><big>(</big><big>)</big><a class="headerlink" href="#sage.schemes.elliptic_curves.weierstrass_morphism.baseWI.tuple" title="Permalink to this definition">¶</a></dt>
<dd>Returns the parameters $u,r,s,t$ as a tuple.
EXAMPLES:
<div class="highlight-python"><div class="highlight"><pre>sage: from sage.schemes.elliptic_curves.weierstrass_morphism import *
sage: u,r,s,t=baseWI(2,3,4,5).tuple()
sage: w=baseWI(2,3,4,5)
sage: u,r,s,t=w.tuple()
sage: u
2
</pre></div>
</div>
</dd></dl>

</dd></dl>

<dl class="function">
<dt id="sage.schemes.elliptic_curves.weierstrass_morphism.isomorphisms">
<tt class="descclassname">sage.schemes.elliptic_curves.weierstrass_morphism.</tt><tt class="descname">isomorphisms</tt><big>(</big>E, F, JustOne=False<big>)</big><a class="headerlink" href="#sage.schemes.elliptic_curves.weierstrass_morphism.isomorphisms" title="Permalink to this definition">¶</a></dt>
<dd>Returns one or all isomorphisms between two elliptic curves.
INPUT:
<ul class="simple">
<li><tt class="docutils literal">E</tt>, <tt class="docutils literal">F</tt> (EllipticCurve) &#8211; Two elliptic curves.</li>
<li><tt class="docutils literal">JustOne</tt> (bool) If True, returns one isomorphism, or None if
the curves are not isomorphic. If False, returns a (possibly
empty) list of isomorphisms.</li>
</ul>
OUTPUT:
Either None, or a 4-tuple $(u,r,s,t)$ representing an isomorphism,
or a list of these.
<div class="admonition note">
Note
This function is not intended for users, who should use the
interface provided by <tt class="docutils literal">ell_generic</tt>.
</div>
EXAMPLES:
<div class="highlight-python"><div class="highlight"><pre>sage: from sage.schemes.elliptic_curves.weierstrass_morphism import *
sage: isomorphisms(EllipticCurve_from_j(0),EllipticCurve(&#39;27a3&#39;))
[(-1, 0, 0, -1), (1, 0, 0, 0)]
sage: isomorphisms(EllipticCurve_from_j(0),EllipticCurve(&#39;27a3&#39;),JustOne=True)
(1, 0, 0, 0)
sage: isomorphisms(EllipticCurve_from_j(0),EllipticCurve(&#39;27a1&#39;))
[]
sage: isomorphisms(EllipticCurve_from_j(0),EllipticCurve(&#39;27a1&#39;),JustOne=True)
</pre></div>
</div>
</dd></dl>

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