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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 405720k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
405720.k1 | 405720k1 | \([0, 0, 0, -4263, 29498]\) | \(10536048/5635\) | \(4582325018880\) | \([2]\) | \(737280\) | \(1.1207\) | \(\Gamma_0(N)\)-optimal |
405720.k2 | 405720k2 | \([0, 0, 0, 16317, 231182]\) | \(147704148/92575\) | \(-301124215526400\) | \([2]\) | \(1474560\) | \(1.4673\) |
Rank
sage: E.rank()
The elliptic curves in class 405720k have rank \(1\).
Complex multiplication
The elliptic curves in class 405720k do not have complex multiplication.Modular form 405720.2.a.k
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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.