Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 32340e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
32340.e2 | 32340e1 | \([0, -1, 0, 6599, 13210]\) | \(16880451584/9823275\) | \(-18491175687600\) | \([2]\) | \(82944\) | \(1.2351\) | \(\Gamma_0(N)\)-optimal |
32340.e1 | 32340e2 | \([0, -1, 0, -26476, 132280]\) | \(68150496976/39220335\) | \(1181243697258240\) | \([2]\) | \(165888\) | \(1.5817\) |
Rank
sage: E.rank()
The elliptic curves in class 32340e have rank \(0\).
Complex multiplication
The elliptic curves in class 32340e do not have complex multiplication.Modular form 32340.2.a.e
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.