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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 417450.z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
417450.z1 | 417450z4 | \([1, 1, 0, -32196650, -70175950500]\) | \(133345896593725369/340006815000\) | \(9411606456065859375000\) | \([2]\) | \(58982400\) | \(3.0928\) | |
417450.z2 | 417450z2 | \([1, 1, 0, -2793650, -167407500]\) | \(87109155423289/49979073600\) | \(1383452775092025000000\) | \([2, 2]\) | \(29491200\) | \(2.7462\) | |
417450.z3 | 417450z1 | \([1, 1, 0, -1825650, 944824500]\) | \(24310870577209/114462720\) | \(3168401417280000000\) | \([2]\) | \(14745600\) | \(2.3997\) | \(\Gamma_0(N)\)-optimal* |
417450.z4 | 417450z3 | \([1, 1, 0, 11121350, -1322352500]\) | \(5495662324535111/3207841648920\) | \(-88795111865661939375000\) | \([2]\) | \(58982400\) | \(3.0928\) |
Rank
sage: E.rank()
The elliptic curves in class 417450.z have rank \(2\).
Complex multiplication
The elliptic curves in class 417450.z do not have complex multiplication.Modular form 417450.2.a.z
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.