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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 271440k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
271440.k2 | 271440k1 | \([0, 0, 0, -14283, 1781882]\) | \(-2913790403187/10716526600\) | \(-1185162109747200\) | \([]\) | \(1078272\) | \(1.5781\) | \(\Gamma_0(N)\)-optimal |
271440.k1 | 271440k2 | \([0, 0, 0, -1642923, 810540378]\) | \(-6083088015781323/11781250\) | \(-949822848000000\) | \([]\) | \(3234816\) | \(2.1274\) |
Rank
sage: E.rank()
The elliptic curves in class 271440k have rank \(2\).
Complex multiplication
The elliptic curves in class 271440k do not have complex multiplication.Modular form 271440.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.