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SageMath
sage: E = EllipticCurve("5445.c1")
sage: E.isogeny_class()
Elliptic curves in class 5445g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
5445.c7 | 5445g1 | [1, -1, 1, -23, 6702] | [2] | 2560 | \(\Gamma_0(N)\)-optimal |
5445.c6 | 5445g2 | [1, -1, 1, -5468, 154806] | [2, 2] | 5120 | |
5445.c5 | 5445g3 | [1, -1, 1, -10913, -200208] | [2, 2] | 10240 | |
5445.c4 | 5445g4 | [1, -1, 1, -87143, 9923136] | [2] | 10240 | |
5445.c2 | 5445g5 | [1, -1, 1, -147038, -21653508] | [2, 2] | 20480 | |
5445.c8 | 5445g6 | [1, -1, 1, 38092, -1533144] | [2] | 20480 | |
5445.c1 | 5445g7 | [1, -1, 1, -2352263, -1388010918] | [2] | 40960 | |
5445.c3 | 5445g8 | [1, -1, 1, -119813, -29940798] | [2] | 40960 |
Rank
sage: E.rank()
The elliptic curves in class 5445g have rank \(1\).
Modular form 5445.2.a.c
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 Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.