Show commands:
SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 142912bw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 142912.bc2 | 142912bw1 | \([0, 0, 0, 820, -5136]\) | \(930433500/712327\) | \(-46683062272\) | \([2]\) | \(73728\) | \(0.73532\) | \(\Gamma_0(N)\)-optimal |
| 142912.bc1 | 142912bw2 | \([0, 0, 0, -3820, -44112]\) | \(47033129250/20804861\) | \(2726934740992\) | \([2]\) | \(147456\) | \(1.0819\) |
Rank
sage: E.rank()
The elliptic curves in class 142912bw have rank \(1\).
Complex multiplication
The elliptic curves in class 142912bw do not have complex multiplication.Modular form 142912.2.a.bw
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.
