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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 19650.bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19650.bc1 | 19650bf3 | \([1, 0, 0, -139738, 20094092]\) | \(19312898130234073/84888\) | \(1326375000\) | \([2]\) | \(73728\) | \(1.3788\) | |
| 19650.bc2 | 19650bf2 | \([1, 0, 0, -8738, 313092]\) | \(4722184089433/9884736\) | \(154449000000\) | \([2, 2]\) | \(36864\) | \(1.0322\) | |
| 19650.bc3 | 19650bf4 | \([1, 0, 0, -5738, 532092]\) | \(-1337180541913/7067998104\) | \(-110437470375000\) | \([2]\) | \(73728\) | \(1.3788\) | |
| 19650.bc4 | 19650bf1 | \([1, 0, 0, -738, 1092]\) | \(2845178713/1609728\) | \(25152000000\) | \([2]\) | \(18432\) | \(0.68566\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 19650.bc have rank \(0\).
Complex multiplication
The elliptic curves in class 19650.bc do not have complex multiplication.Modular form 19650.2.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 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.
