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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 1293b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1293.b4 | 1293b1 | \([1, 0, 0, -234, 1539]\) | \(-1417383186337/229051071\) | \(-229051071\) | \([4]\) | \(498\) | \(0.33191\) | \(\Gamma_0(N)\)-optimal |
| 1293.b3 | 1293b2 | \([1, 0, 0, -3879, 92664]\) | \(6454907876131057/135419769\) | \(135419769\) | \([2, 2]\) | \(996\) | \(0.67849\) | |
| 1293.b2 | 1293b3 | \([1, 0, 0, -4014, 85833]\) | \(7152577607925217/931693026267\) | \(931693026267\) | \([2]\) | \(1992\) | \(1.0251\) | |
| 1293.b1 | 1293b4 | \([1, 0, 0, -62064, 5946075]\) | \(26438903289204662017/11637\) | \(11637\) | \([2]\) | \(1992\) | \(1.0251\) |
Rank
sage: E.rank()
The elliptic curves in class 1293b have rank \(0\).
Complex multiplication
The elliptic curves in class 1293b do not have complex multiplication.Modular form 1293.2.a.b
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
