Show commands:
SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 32368q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
32368.o1 | 32368q1 | \([0, -1, 0, -32464, -2241088]\) | \(-11060825617/2744\) | \(-938727931904\) | \([]\) | \(72576\) | \(1.2849\) | \(\Gamma_0(N)\)-optimal |
32368.o2 | 32368q2 | \([0, -1, 0, 13776, -7974848]\) | \(845095823/80707214\) | \(-27610100615143424\) | \([]\) | \(217728\) | \(1.8342\) |
Rank
sage: E.rank()
The elliptic curves in class 32368q have rank \(1\).
Complex multiplication
The elliptic curves in class 32368q do not have complex multiplication.Modular form 32368.2.a.q
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.