Show commands:
SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 69312.bp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 69312.bp1 | 69312cp4 | \([0, -1, 0, -2345537, 1383413985]\) | \(115714886617/1539\) | \(18980172165021696\) | \([2]\) | \(1105920\) | \(2.2669\) | |
| 69312.bp2 | 69312cp2 | \([0, -1, 0, -150657, 20393505]\) | \(30664297/3249\) | \(40069252348379136\) | \([2, 2]\) | \(552960\) | \(1.9204\) | |
| 69312.bp3 | 69312cp1 | \([0, -1, 0, -35137, -2179103]\) | \(389017/57\) | \(702969339445248\) | \([2]\) | \(276480\) | \(1.5738\) | \(\Gamma_0(N)\)-optimal |
| 69312.bp4 | 69312cp3 | \([0, -1, 0, 195903, 100171617]\) | \(67419143/390963\) | \(-4821666699254956032\) | \([2]\) | \(1105920\) | \(2.2669\) |
Rank
sage: E.rank()
The elliptic curves in class 69312.bp have rank \(1\).
Complex multiplication
The elliptic curves in class 69312.bp do not have complex multiplication.Modular form 69312.2.a.bp
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
