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SageMath
E = EllipticCurve("ex1")
E.isogeny_class()
Elliptic curves in class 380880ex
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 380880.ex5 | 380880ex1 | \([0, 0, 0, -31995507, -76424358094]\) | \(-8194759433281/965779200\) | \(-426906080075111281459200\) | \([2]\) | \(38928384\) | \(3.2701\) | \(\Gamma_0(N)\)-optimal |
| 380880.ex4 | 380880ex2 | \([0, 0, 0, -525615987, -4638168661966]\) | \(36330796409313601/428490000\) | \(189406632749374218240000\) | \([2, 2]\) | \(77856768\) | \(3.6166\) | |
| 380880.ex3 | 380880ex3 | \([0, 0, 0, -539327667, -4383413874574]\) | \(39248884582600321/3935264062500\) | \(1739515776465606969600000000\) | \([2, 2]\) | \(155713536\) | \(3.9632\) | |
| 380880.ex1 | 380880ex4 | \([0, 0, 0, -8409831987, -296844558897166]\) | \(148809678420065817601/20700\) | \(9150078876781363200\) | \([2]\) | \(155713536\) | \(3.9632\) | |
| 380880.ex2 | 380880ex5 | \([0, 0, 0, -1967627667, 28776284585426]\) | \(1905890658841300321/293666194803750\) | \(129810089173843178661288960000\) | \([2]\) | \(311427072\) | \(4.3098\) | |
| 380880.ex6 | 380880ex6 | \([0, 0, 0, 669585453, -21238805941486]\) | \(75108181893694559/484313964843750\) | \(-214082656011933750000000000000\) | \([2]\) | \(311427072\) | \(4.3098\) |
Rank
sage: E.rank()
The elliptic curves in class 380880ex have rank \(1\).
Complex multiplication
The elliptic curves in class 380880ex do not have complex multiplication.Modular form 380880.2.a.ex
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.
