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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 53312bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 53312.q1 | 53312bc1 | \([0, 1, 0, -289, -2241]\) | \(-208537/34\) | \(-436731904\) | \([]\) | \(23040\) | \(0.38548\) | \(\Gamma_0(N)\)-optimal |
| 53312.q2 | 53312bc2 | \([0, 1, 0, 1951, 8959]\) | \(63905303/39304\) | \(-504862081024\) | \([]\) | \(69120\) | \(0.93478\) |
Rank
sage: E.rank()
The elliptic curves in class 53312bc have rank \(1\).
Complex multiplication
The elliptic curves in class 53312bc do not have complex multiplication.Modular form 53312.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
