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SageMath
E = EllipticCurve("fr1")
E.isogeny_class()
Elliptic curves in class 37440.fr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
37440.fr1 | 37440dk4 | \([0, 0, 0, -2303532, 1345672656]\) | \(261984288445803/42250\) | \(218000719872000\) | \([2]\) | \(663552\) | \(2.1535\) | |
37440.fr2 | 37440dk3 | \([0, 0, 0, -143532, 21160656]\) | \(-63378025803/812500\) | \(-4192321536000000\) | \([2]\) | \(331776\) | \(1.8070\) | |
37440.fr3 | 37440dk2 | \([0, 0, 0, -32172, 1330384]\) | \(520300455507/193072360\) | \(1366544539975680\) | \([2]\) | \(221184\) | \(1.6042\) | |
37440.fr4 | 37440dk1 | \([0, 0, 0, 6228, 147664]\) | \(3774555693/3515200\) | \(-24880191897600\) | \([2]\) | \(110592\) | \(1.2576\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 37440.fr have rank \(1\).
Complex multiplication
The elliptic curves in class 37440.fr do not have complex multiplication.Modular form 37440.2.a.fr
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.