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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 33600.ca
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
33600.ca1 | 33600eo4 | \([0, -1, 0, -10206000033, 396858041351937]\) | \(229625675762164624948320008/9568125\) | \(4898880000000000\) | \([2]\) | \(20643840\) | \(3.9113\) | |
33600.ca2 | 33600eo2 | \([0, -1, 0, -637875033, 6201065726937]\) | \(448487713888272974160064/91549016015625\) | \(5859137025000000000000\) | \([2, 2]\) | \(10321920\) | \(3.5647\) | |
33600.ca3 | 33600eo3 | \([0, -1, 0, -635688033, 6245695835937]\) | \(-55486311952875723077768/801237030029296875\) | \(-410233359375000000000000000\) | \([2]\) | \(20643840\) | \(3.9113\) | |
33600.ca4 | 33600eo1 | \([0, -1, 0, -40003908, 96203669562]\) | \(7079962908642659949376/100085966990454375\) | \(100085966990454375000000\) | \([2]\) | \(5160960\) | \(3.2181\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 33600.ca have rank \(0\).
Complex multiplication
The elliptic curves in class 33600.ca do not have complex multiplication.Modular form 33600.2.a.ca
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.