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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 23100b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
23100.b3 | 23100b1 | \([0, -1, 0, -66533, 6725562]\) | \(-130287139815424/2250652635\) | \(-562663158750000\) | \([2]\) | \(124416\) | \(1.6283\) | \(\Gamma_0(N)\)-optimal |
23100.b2 | 23100b2 | \([0, -1, 0, -1068908, 425718312]\) | \(33766427105425744/9823275\) | \(39293100000000\) | \([2]\) | \(248832\) | \(1.9748\) | |
23100.b4 | 23100b3 | \([0, -1, 0, 257467, 32038062]\) | \(7549996227362816/6152409907875\) | \(-1538102476968750000\) | \([2]\) | \(373248\) | \(2.1776\) | |
23100.b1 | 23100b4 | \([0, -1, 0, -1239908, 280602312]\) | \(52702650535889104/22020583921875\) | \(88082335687500000000\) | \([2]\) | \(746496\) | \(2.5241\) |
Rank
sage: E.rank()
The elliptic curves in class 23100b have rank \(0\).
Complex multiplication
The elliptic curves in class 23100b do not have complex multiplication.Modular form 23100.2.a.b
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.