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SageMath
sage: E = EllipticCurve("x1")
sage: E.isogeny_class()
Elliptic curves in class 400710.x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
400710.x1 | 400710x4 | [1, 0, 1, -123102813, -525724902344] | [2] | 47775744 | |
400710.x2 | 400710x5 | [1, 0, 1, -109428133, 438672429128] | [2] | 95551488 | \(\Gamma_0(N)\)-optimal* |
400710.x3 | 400710x3 | [1, 0, 1, -10586333, -1489874632] | [2, 2] | 47775744 | \(\Gamma_0(N)\)-optimal* |
400710.x4 | 400710x2 | [1, 0, 1, -7698333, -8205052232] | [2, 2] | 23887872 | \(\Gamma_0(N)\)-optimal* |
400710.x5 | 400710x1 | [1, 0, 1, -305053, -223267144] | [2] | 11943936 | \(\Gamma_0(N)\)-optimal* |
400710.x6 | 400710x6 | [1, 0, 1, 42047467, -11869259992] | [2] | 95551488 |
Rank
sage: E.rank()
The elliptic curves in class 400710.x have rank \(0\).
Complex multiplication
The elliptic curves in class 400710.x do not have complex multiplication.Modular form 400710.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.