Show commands:
SageMath
E = EllipticCurve("ec1")
E.isogeny_class()
Elliptic curves in class 44352ec
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 44352.et1 | 44352ec1 | \([0, 0, 0, -3216, 70198]\) | \(-78843215872/539\) | \(-25147584\) | \([]\) | \(28800\) | \(0.60088\) | \(\Gamma_0(N)\)-optimal |
| 44352.et2 | 44352ec2 | \([0, 0, 0, -1776, 133198]\) | \(-13278380032/156590819\) | \(-7305901251264\) | \([]\) | \(86400\) | \(1.1502\) | |
| 44352.et3 | 44352ec3 | \([0, 0, 0, 15864, -3447722]\) | \(9463555063808/115539436859\) | \(-5390607966093504\) | \([]\) | \(259200\) | \(1.6995\) |
Rank
sage: E.rank()
The elliptic curves in class 44352ec have rank \(0\).
Complex multiplication
The elliptic curves in class 44352ec do not have complex multiplication.Modular form 44352.2.a.ec
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
