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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 458640.cp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
458640.cp1 | 458640cp4 | \([0, 0, 0, -3546963, 1540085778]\) | \(520300455507/193072360\) | \(1831300380741304811520\) | \([2]\) | \(23887872\) | \(2.7799\) | |
458640.cp2 | 458640cp2 | \([0, 0, 0, -3135363, 2136878338]\) | \(261984288445803/42250\) | \(549716364288000\) | \([2]\) | \(7962624\) | \(2.2306\) | \(\Gamma_0(N)\)-optimal* |
458640.cp3 | 458640cp1 | \([0, 0, 0, -195363, 33602338]\) | \(-63378025803/812500\) | \(-10571468544000000\) | \([2]\) | \(3981312\) | \(1.8840\) | \(\Gamma_0(N)\)-optimal* |
458640.cp4 | 458640cp3 | \([0, 0, 0, 686637, 170939538]\) | \(3774555693/3515200\) | \(-33341836699887206400\) | \([2]\) | \(11943936\) | \(2.4333\) |
Rank
sage: E.rank()
The elliptic curves in class 458640.cp have rank \(1\).
Complex multiplication
The elliptic curves in class 458640.cp do not have complex multiplication.Modular form 458640.2.a.cp
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.