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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 5445g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5445.c7 | 5445g1 | \([1, -1, 1, -23, 6702]\) | \(-1/15\) | \(-19372019535\) | \([2]\) | \(2560\) | \(0.65283\) | \(\Gamma_0(N)\)-optimal |
5445.c6 | 5445g2 | \([1, -1, 1, -5468, 154806]\) | \(13997521/225\) | \(290580293025\) | \([2, 2]\) | \(5120\) | \(0.99940\) | |
5445.c5 | 5445g3 | \([1, -1, 1, -10913, -200208]\) | \(111284641/50625\) | \(65380565930625\) | \([2, 2]\) | \(10240\) | \(1.3460\) | |
5445.c4 | 5445g4 | \([1, -1, 1, -87143, 9923136]\) | \(56667352321/15\) | \(19372019535\) | \([2]\) | \(10240\) | \(1.3460\) | |
5445.c2 | 5445g5 | \([1, -1, 1, -147038, -21653508]\) | \(272223782641/164025\) | \(211833033615225\) | \([2, 2]\) | \(20480\) | \(1.6926\) | |
5445.c8 | 5445g6 | \([1, -1, 1, 38092, -1533144]\) | \(4733169839/3515625\) | \(-4540317078515625\) | \([2]\) | \(20480\) | \(1.6926\) | |
5445.c1 | 5445g7 | \([1, -1, 1, -2352263, -1388010918]\) | \(1114544804970241/405\) | \(523044527445\) | \([2]\) | \(40960\) | \(2.0391\) | |
5445.c3 | 5445g8 | \([1, -1, 1, -119813, -29940798]\) | \(-147281603041/215233605\) | \(-277967306709898245\) | \([2]\) | \(40960\) | \(2.0391\) |
Rank
sage: E.rank()
The elliptic curves in class 5445g have rank \(1\).
Complex multiplication
The elliptic curves in class 5445g do not have complex multiplication.Modular form 5445.2.a.g
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.