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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 53040bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 53040.cv1 | 53040bc1 | \([0, 1, 0, -97700, 11369820]\) | \(402876451435348816/13746755117745\) | \(3519169310142720\) | \([2]\) | \(368640\) | \(1.7551\) | \(\Gamma_0(N)\)-optimal |
| 53040.cv2 | 53040bc2 | \([0, 1, 0, 33520, 39765828]\) | \(4067455675907516/669098843633025\) | \(-685157215880217600\) | \([2]\) | \(737280\) | \(2.1017\) |
Rank
sage: E.rank()
The elliptic curves in class 53040bc have rank \(1\).
Complex multiplication
The elliptic curves in class 53040bc do not have complex multiplication.Modular form 53040.2.a.bc
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.
