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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 9075.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
9075.g1 | 9075l7 | \([1, 0, 0, -6534063, 6428154492]\) | \(1114544804970241/405\) | \(11210659453125\) | \([2]\) | \(122880\) | \(2.2945\) | |
9075.g2 | 9075l5 | \([1, 0, 0, -408438, 100383867]\) | \(272223782641/164025\) | \(4540317078515625\) | \([2, 2]\) | \(61440\) | \(1.9480\) | |
9075.g3 | 9075l8 | \([1, 0, 0, -332813, 138725742]\) | \(-147281603041/215233605\) | \(-5957804070428203125\) | \([2]\) | \(122880\) | \(2.2945\) | |
9075.g4 | 9075l3 | \([1, 0, 0, -242063, -45859758]\) | \(56667352321/15\) | \(415209609375\) | \([2]\) | \(30720\) | \(1.6014\) | |
9075.g5 | 9075l4 | \([1, 0, 0, -30313, 936992]\) | \(111284641/50625\) | \(1401332431640625\) | \([2, 2]\) | \(30720\) | \(1.6014\) | |
9075.g6 | 9075l2 | \([1, 0, 0, -15188, -711633]\) | \(13997521/225\) | \(6228144140625\) | \([2, 2]\) | \(15360\) | \(1.2548\) | |
9075.g7 | 9075l1 | \([1, 0, 0, -63, -31008]\) | \(-1/15\) | \(-415209609375\) | \([2]\) | \(7680\) | \(0.90824\) | \(\Gamma_0(N)\)-optimal |
9075.g8 | 9075l6 | \([1, 0, 0, 105812, 7062617]\) | \(4733169839/3515625\) | \(-97314752197265625\) | \([2]\) | \(61440\) | \(1.9480\) |
Rank
sage: E.rank()
The elliptic curves in class 9075.g have rank \(1\).
Complex multiplication
The elliptic curves in class 9075.g do not have complex multiplication.Modular form 9075.2.a.g
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.