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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 637c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 637.d1 | 637c1 | \([1, -1, 0, -5252, -145223]\) | \(-56723625/13\) | \(-3672178237\) | \([]\) | \(420\) | \(0.82691\) | \(\Gamma_0(N)\)-optimal |
| 637.d2 | 637c2 | \([1, -1, 0, 30763, 6051758]\) | \(11397810375/62748517\) | \(-17724902963955733\) | \([]\) | \(2940\) | \(1.7999\) |
Rank
sage: E.rank()
The elliptic curves in class 637c have rank \(1\).
Complex multiplication
The elliptic curves in class 637c do not have complex multiplication.Modular form 637.2.a.c
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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
