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SageMath
sage: E = EllipticCurve("ch1")
sage: E.isogeny_class()
Elliptic curves in class 32448.ch
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 32448.ch1 | 32448bh6 | \([0, 1, 0, -259809, -51058305]\) | \(3065617154/9\) | \(5693935583232\) | \([2]\) | \(147456\) | \(1.6768\) | |
| 32448.ch2 | 32448bh4 | \([0, 1, 0, -43489, 3475967]\) | \(28756228/3\) | \(948989263872\) | \([2]\) | \(73728\) | \(1.3303\) | |
| 32448.ch3 | 32448bh3 | \([0, 1, 0, -16449, -780129]\) | \(1556068/81\) | \(25622710124544\) | \([2, 2]\) | \(73728\) | \(1.3303\) | |
| 32448.ch4 | 32448bh2 | \([0, 1, 0, -2929, 44591]\) | \(35152/9\) | \(711741947904\) | \([2, 2]\) | \(36864\) | \(0.98370\) | |
| 32448.ch5 | 32448bh1 | \([0, 1, 0, 451, 4707]\) | \(2048/3\) | \(-14827957248\) | \([2]\) | \(18432\) | \(0.63712\) | \(\Gamma_0(N)\)-optimal |
| 32448.ch6 | 32448bh5 | \([0, 1, 0, 10591, -3067713]\) | \(207646/6561\) | \(-4150879040176128\) | \([2]\) | \(147456\) | \(1.6768\) |
Rank
sage: E.rank()
The elliptic curves in class 32448.ch have rank \(0\).
Complex multiplication
The elliptic curves in class 32448.ch do not have complex multiplication.Modular form 32448.2.a.ch
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
