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SageMath
E = EllipticCurve("ds1")
E.isogeny_class()
Elliptic curves in class 487872ds
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
487872.ds4 | 487872ds1 | \([0, 0, 0, 242484, -28004240]\) | \(4657463/3696\) | \(-1251282940965421056\) | \([2]\) | \(5898240\) | \(2.1605\) | \(\Gamma_0(N)\)-optimal* |
487872.ds3 | 487872ds2 | \([0, 0, 0, -1151436, -242667920]\) | \(498677257/213444\) | \(72261589840753065984\) | \([2, 2]\) | \(11796480\) | \(2.5071\) | \(\Gamma_0(N)\)-optimal* |
487872.ds2 | 487872ds3 | \([0, 0, 0, -8817996, 9910924144]\) | \(223980311017/4278582\) | \(1448516414535095549952\) | \([2]\) | \(23592960\) | \(2.8536\) | \(\Gamma_0(N)\)-optimal* |
487872.ds1 | 487872ds4 | \([0, 0, 0, -15787596, -24134735504]\) | \(1285429208617/614922\) | \(208182199303121928192\) | \([2]\) | \(23592960\) | \(2.8536\) |
Rank
sage: E.rank()
The elliptic curves in class 487872ds have rank \(1\).
Complex multiplication
The elliptic curves in class 487872ds do not have complex multiplication.Modular form 487872.2.a.ds
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.