Show commands:
SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 6240b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6240.a3 | 6240b1 | \([0, -1, 0, -86, 240]\) | \(1111934656/342225\) | \(21902400\) | \([2, 2]\) | \(1536\) | \(0.11317\) | \(\Gamma_0(N)\)-optimal |
6240.a2 | 6240b2 | \([0, -1, 0, -536, -4440]\) | \(33324076232/1285245\) | \(658045440\) | \([2]\) | \(3072\) | \(0.45975\) | |
6240.a1 | 6240b3 | \([0, -1, 0, -1256, 17556]\) | \(428320044872/73125\) | \(37440000\) | \([2]\) | \(3072\) | \(0.45975\) | |
6240.a4 | 6240b4 | \([0, -1, 0, 239, 1345]\) | \(367061696/426465\) | \(-1746800640\) | \([2]\) | \(3072\) | \(0.45975\) |
Rank
sage: E.rank()
The elliptic curves in class 6240b have rank \(1\).
Complex multiplication
The elliptic curves in class 6240b do not have complex multiplication.Modular form 6240.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.