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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 31433.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
31433.b1 | 31433b4 | \([1, -1, 0, -167681, 26470544]\) | \(82483294977/17\) | \(107463171833\) | \([2]\) | \(80640\) | \(1.5040\) | |
31433.b2 | 31433b2 | \([1, -1, 0, -10516, 412587]\) | \(20346417/289\) | \(1826873921161\) | \([2, 2]\) | \(40320\) | \(1.1574\) | |
31433.b3 | 31433b1 | \([1, -1, 0, -1271, -7136]\) | \(35937/17\) | \(107463171833\) | \([2]\) | \(20160\) | \(0.81082\) | \(\Gamma_0(N)\)-optimal |
31433.b4 | 31433b3 | \([1, -1, 0, -1271, 1105962]\) | \(-35937/83521\) | \(-527966563215529\) | \([2]\) | \(80640\) | \(1.5040\) |
Rank
sage: E.rank()
The elliptic curves in class 31433.b have rank \(1\).
Complex multiplication
The elliptic curves in class 31433.b do not have complex multiplication.Modular form 31433.2.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 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.