Show commands:
SageMath
E = EllipticCurve("mn1")
E.isogeny_class()
Elliptic curves in class 342720.mn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 342720.mn1 | 342720mn5 | \([0, 0, 0, -41322252, 102042501104]\) | \(40832710302042509761/91556816413125\) | \(17496780089633832960000\) | \([2]\) | \(33554432\) | \(3.1502\) | |
| 342720.mn2 | 342720mn3 | \([0, 0, 0, -3522252, 330261104]\) | \(25288177725059761/14387797265625\) | \(2749550875545600000000\) | \([2, 2]\) | \(16777216\) | \(2.8037\) | |
| 342720.mn3 | 342720mn2 | \([0, 0, 0, -2252172, -1294933264]\) | \(6610905152742241/35128130625\) | \(6713090303754240000\) | \([2, 2]\) | \(8388608\) | \(2.4571\) | |
| 342720.mn4 | 342720mn1 | \([0, 0, 0, -2249292, -1298424976]\) | \(6585576176607121/187425\) | \(35817475276800\) | \([2]\) | \(4194304\) | \(2.1105\) | \(\Gamma_0(N)\)-optimal |
| 342720.mn5 | 342720mn4 | \([0, 0, 0, -1028172, -2696658064]\) | \(-629004249876241/16074715228425\) | \(-3071925918504537292800\) | \([2]\) | \(16777216\) | \(2.8037\) | |
| 342720.mn6 | 342720mn6 | \([0, 0, 0, 13956468, 2630460656]\) | \(1573196002879828319/926055908203125\) | \(-176972040000000000000000\) | \([2]\) | \(33554432\) | \(3.1502\) |
Rank
sage: E.rank()
The elliptic curves in class 342720.mn have rank \(0\).
Complex multiplication
The elliptic curves in class 342720.mn do not have complex multiplication.Modular form 342720.2.a.mn
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
