Show commands:
SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 7770r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7770.q4 | 7770r1 | \([1, 1, 1, 80, 257]\) | \(56578878719/54390000\) | \(-54390000\) | \([4]\) | \(2560\) | \(0.17140\) | \(\Gamma_0(N)\)-optimal |
| 7770.q3 | 7770r2 | \([1, 1, 1, -420, 1857]\) | \(8194759433281/2958272100\) | \(2958272100\) | \([2, 2]\) | \(5120\) | \(0.51797\) | |
| 7770.q2 | 7770r3 | \([1, 1, 1, -2870, -58903]\) | \(2614441086442081/74385450090\) | \(74385450090\) | \([2]\) | \(10240\) | \(0.86454\) | |
| 7770.q1 | 7770r4 | \([1, 1, 1, -5970, 175017]\) | \(23531588875176481/6398929110\) | \(6398929110\) | \([2]\) | \(10240\) | \(0.86454\) |
Rank
sage: E.rank()
The elliptic curves in class 7770r have rank \(0\).
Complex multiplication
The elliptic curves in class 7770r do not have complex multiplication.Modular form 7770.2.a.r
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.
