Show commands:
SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 215985bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 215985.by4 | 215985bm1 | \([1, 0, 1, -472508, -125054107]\) | \(6585576176607121/187425\) | \(332034820425\) | \([2]\) | \(1310720\) | \(1.7204\) | \(\Gamma_0(N)\)-optimal |
| 215985.by3 | 215985bm2 | \([1, 0, 1, -473113, -124717969]\) | \(6610905152742241/35128130625\) | \(62231626218155625\) | \([2, 2]\) | \(2621440\) | \(2.0670\) | |
| 215985.by2 | 215985bm3 | \([1, 0, 1, -739918, 31736483]\) | \(25288177725059761/14387797265625\) | \(25488860511687890625\) | \([2, 2]\) | \(5242880\) | \(2.4136\) | |
| 215985.by5 | 215985bm4 | \([1, 0, 1, -215988, -259657169]\) | \(-629004249876241/16074715228425\) | \(-28477338584783821425\) | \([2]\) | \(5242880\) | \(2.4136\) | |
| 215985.by1 | 215985bm5 | \([1, 0, 1, -8680543, 9824115233]\) | \(40832710302042509761/91556816413125\) | \(162198485241652138125\) | \([2]\) | \(10485760\) | \(2.7602\) | |
| 215985.by6 | 215985bm6 | \([1, 0, 1, 2931827, 253509881]\) | \(1573196002879828319/926055908203125\) | \(-1640564530792236328125\) | \([2]\) | \(10485760\) | \(2.7602\) |
Rank
sage: E.rank()
The elliptic curves in class 215985bm have rank \(0\).
Complex multiplication
The elliptic curves in class 215985bm do not have complex multiplication.Modular form 215985.2.a.bm
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
