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SageMath
E = EllipticCurve("el1")
E.isogeny_class()
Elliptic curves in class 382200el
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
382200.el4 | 382200el1 | \([0, -1, 0, 53492, 56809012]\) | \(35969456/2985255\) | \(-1404849061980000000\) | \([2]\) | \(5898240\) | \(2.1617\) | \(\Gamma_0(N)\)-optimal |
382200.el3 | 382200el2 | \([0, -1, 0, -1931008, 997462012]\) | \(423026849956/16769025\) | \(31565744355600000000\) | \([2, 2]\) | \(11796480\) | \(2.5083\) | |
382200.el1 | 382200el3 | \([0, -1, 0, -30596008, 65149732012]\) | \(841356017734178/1404585\) | \(5287936661280000000\) | \([2]\) | \(23592960\) | \(2.8548\) | |
382200.el2 | 382200el4 | \([0, -1, 0, -5018008, -2984767988]\) | \(3711757787138/1124589375\) | \(4233818092140000000000\) | \([2]\) | \(23592960\) | \(2.8548\) |
Rank
sage: E.rank()
The elliptic curves in class 382200el have rank \(0\).
Complex multiplication
The elliptic curves in class 382200el do not have complex multiplication.Modular form 382200.2.a.el
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.