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SageMath
sage: E = EllipticCurve("f1")
sage: E.isogeny_class()
Elliptic curves in class 630.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
630.f1 | 630f7 | [1, -1, 0, -58059, -3373137] | [6] | 4608 | |
630.f2 | 630f4 | [1, -1, 0, -51849, -4531275] | [2] | 1536 | |
630.f3 | 630f6 | [1, -1, 0, -24309, 1426113] | [2, 6] | 2304 | |
630.f4 | 630f3 | [1, -1, 0, -24129, 1448685] | [6] | 1152 | |
630.f5 | 630f2 | [1, -1, 0, -3249, -69795] | [2, 2] | 768 | |
630.f6 | 630f5 | [1, -1, 0, -729, -177147] | [2] | 1536 | |
630.f7 | 630f1 | [1, -1, 0, -369, 1053] | [2] | 384 | \(\Gamma_0(N)\)-optimal |
630.f8 | 630f8 | [1, -1, 0, 6561, 4778595] | [6] | 4608 |
Rank
sage: E.rank()
The elliptic curves in class 630.f have rank \(0\).
Complex multiplication
The elliptic curves in class 630.f do not have complex multiplication.Modular form 630.2.a.f
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.