Show commands:
SageMath
E = EllipticCurve("cj1")
E.isogeny_class()
Elliptic curves in class 69696cj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 69696.fn2 | 69696cj1 | \([0, 0, 0, -336864, 87473320]\) | \(-3196715008/649539\) | \(-858991424631081984\) | \([2]\) | \(921600\) | \(2.1636\) | \(\Gamma_0(N)\)-optimal |
| 69696.fn1 | 69696cj2 | \([0, 0, 0, -5629404, 5140790512]\) | \(932410994128/29403\) | \(622150167634034688\) | \([2]\) | \(1843200\) | \(2.5102\) |
Rank
sage: E.rank()
The elliptic curves in class 69696cj have rank \(1\).
Complex multiplication
The elliptic curves in class 69696cj do not have complex multiplication.Modular form 69696.2.a.cj
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
