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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 221067.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
221067.m1 | 221067h3 | \([1, -1, 1, -9454593479, 353846826376926]\) | \(72371679832051361738355457/1627857\) | \(2102325173612433\) | \([2]\) | \(79626240\) | \(3.8853\) | |
221067.m2 | 221067h4 | \([1, -1, 1, -592649069, 5494828926678]\) | \(17825137625614555960417/216318148151991039\) | \(279367959451692970443529791\) | \([2]\) | \(79626240\) | \(3.8853\) | |
221067.m3 | 221067h2 | \([1, -1, 1, -590912114, 5528967040248]\) | \(17668869054438249282097/2649918412449\) | \(3422284750141214346081\) | \([2, 2]\) | \(39813120\) | \(3.5387\) | |
221067.m4 | 221067h1 | \([1, -1, 1, -36823469, 86930004516]\) | \(-4275768267198290017/52843101620463\) | \(-68245173125439959449647\) | \([2]\) | \(19906560\) | \(3.1922\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 221067.m have rank \(1\).
Complex multiplication
The elliptic curves in class 221067.m do not have complex multiplication.Modular form 221067.2.a.m
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.