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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 4998.bp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4998.bp1 | 4998bl3 | \([1, 0, 0, -248872, -47807992]\) | \(14489843500598257/6246072\) | \(734844124728\) | \([2]\) | \(36864\) | \(1.6195\) | |
| 4998.bp2 | 4998bl4 | \([1, 0, 0, -33272, 1231992]\) | \(34623662831857/14438442312\) | \(1698668299564488\) | \([2]\) | \(36864\) | \(1.6195\) | |
| 4998.bp3 | 4998bl2 | \([1, 0, 0, -15632, -740160]\) | \(3590714269297/73410624\) | \(8636686502976\) | \([2, 2]\) | \(18432\) | \(1.2729\) | |
| 4998.bp4 | 4998bl1 | \([1, 0, 0, 48, -34560]\) | \(103823/4386816\) | \(-516104515584\) | \([2]\) | \(9216\) | \(0.92636\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4998.bp have rank \(0\).
Complex multiplication
The elliptic curves in class 4998.bp do not have complex multiplication.Modular form 4998.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
