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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 3360x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3360.x3 | 3360x1 | \([0, 1, 0, -6378750, -6202979352]\) | \(448487713888272974160064/91549016015625\) | \(5859137025000000\) | \([2, 2]\) | \(107520\) | \(2.4134\) | \(\Gamma_0(N)\)-optimal |
3360.x1 | 3360x2 | \([0, 1, 0, -102060000, -396888659352]\) | \(229625675762164624948320008/9568125\) | \(4898880000\) | \([2]\) | \(215040\) | \(2.7600\) | |
3360.x2 | 3360x3 | \([0, 1, 0, -6400625, -6158314977]\) | \(7079962908642659949376/100085966990454375\) | \(409952120792901120000\) | \([4]\) | \(215040\) | \(2.7600\) | |
3360.x4 | 3360x4 | \([0, 1, 0, -6356880, -6247602900]\) | \(-55486311952875723077768/801237030029296875\) | \(-410233359375000000000\) | \([2]\) | \(215040\) | \(2.7600\) |
Rank
sage: E.rank()
The elliptic curves in class 3360x have rank \(0\).
Complex multiplication
The elliptic curves in class 3360x do not have complex multiplication.Modular form 3360.2.a.x
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.