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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 458640.x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
458640.x1 | 458640x3 | \([0, 0, 0, -248283, 47583018]\) | \(9636491538/8125\) | \(1427148253440000\) | \([2]\) | \(3145728\) | \(1.8354\) | \(\Gamma_0(N)\)-optimal* |
458640.x2 | 458640x2 | \([0, 0, 0, -18963, 388962]\) | \(8586756/4225\) | \(371058545894400\) | \([2, 2]\) | \(1572864\) | \(1.4888\) | \(\Gamma_0(N)\)-optimal* |
458640.x3 | 458640x1 | \([0, 0, 0, -10143, -388962]\) | \(5256144/65\) | \(1427148253440\) | \([2]\) | \(786432\) | \(1.1422\) | \(\Gamma_0(N)\)-optimal* |
458640.x4 | 458640x4 | \([0, 0, 0, 69237, 2982042]\) | \(208974222/142805\) | \(-25083557702461440\) | \([2]\) | \(3145728\) | \(1.8354\) |
Rank
sage: E.rank()
The elliptic curves in class 458640.x have rank \(2\).
Complex multiplication
The elliptic curves in class 458640.x do not have complex multiplication.Modular form 458640.2.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 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.