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SageMath
E = EllipticCurve("dd1")
E.isogeny_class()
Elliptic curves in class 432450.dd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
432450.dd1 | 432450dd2 | \([1, -1, 0, -513646992, -7086786828584]\) | \(-2372030262025/2061298872\) | \(-13023829446665007514921875000\) | \([]\) | \(276480000\) | \(4.0921\) | |
432450.dd2 | 432450dd1 | \([1, -1, 0, -12350952, 50252074816]\) | \(-12882119799145/59982446592\) | \(-970201353106948121395200\) | \([]\) | \(55296000\) | \(3.2873\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 432450.dd have rank \(1\).
Complex multiplication
The elliptic curves in class 432450.dd do not have complex multiplication.Modular form 432450.2.a.dd
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.