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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 9900.bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 9900.bc1 | 9900u4 | \([0, 0, 0, -350175, 43325750]\) | \(1628514404944/664335375\) | \(1937201953500000000\) | \([2]\) | \(165888\) | \(2.2066\) | |
| 9900.bc2 | 9900u2 | \([0, 0, 0, -161175, -24903250]\) | \(158792223184/16335\) | \(47632860000000\) | \([2]\) | \(55296\) | \(1.6573\) | |
| 9900.bc3 | 9900u1 | \([0, 0, 0, -9300, -451375]\) | \(-488095744/200475\) | \(-36536568750000\) | \([2]\) | \(27648\) | \(1.3108\) | \(\Gamma_0(N)\)-optimal |
| 9900.bc4 | 9900u3 | \([0, 0, 0, 71700, 4935125]\) | \(223673040896/187171875\) | \(-34112074218750000\) | \([2]\) | \(82944\) | \(1.8601\) |
Rank
sage: E.rank()
The elliptic curves in class 9900.bc have rank \(0\).
Complex multiplication
The elliptic curves in class 9900.bc do not have complex multiplication.Modular form 9900.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
