Show commands:
SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 15680ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 15680.p2 | 15680ca1 | \([0, 1, 0, 51679, -5942945]\) | \(10100279/16000\) | \(-24179327893504000\) | \([]\) | \(112896\) | \(1.8289\) | \(\Gamma_0(N)\)-optimal |
| 15680.p1 | 15680ca2 | \([0, 1, 0, -497121, 234102175]\) | \(-8990558521/10485760\) | \(-15846164328286781440\) | \([]\) | \(338688\) | \(2.3782\) |
Rank
sage: E.rank()
The elliptic curves in class 15680ca have rank \(0\).
Complex multiplication
The elliptic curves in class 15680ca do not have complex multiplication.Modular form 15680.2.a.ca
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
