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SageMath
E = EllipticCurve("ch1")
E.isogeny_class()
Elliptic curves in class 27600ch
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 27600.cn5 | 27600ch1 | \([0, 1, 0, -168008, -29136012]\) | \(-8194759433281/965779200\) | \(-61809868800000000\) | \([2]\) | \(221184\) | \(1.9577\) | \(\Gamma_0(N)\)-optimal |
| 27600.cn4 | 27600ch2 | \([0, 1, 0, -2760008, -1765776012]\) | \(36330796409313601/428490000\) | \(27423360000000000\) | \([2, 2]\) | \(442368\) | \(2.3043\) | |
| 27600.cn3 | 27600ch3 | \([0, 1, 0, -2832008, -1668864012]\) | \(39248884582600321/3935264062500\) | \(251856900000000000000\) | \([2, 2]\) | \(884736\) | \(2.6509\) | |
| 27600.cn1 | 27600ch4 | \([0, 1, 0, -44160008, -112966176012]\) | \(148809678420065817601/20700\) | \(1324800000000\) | \([2]\) | \(884736\) | \(2.6509\) | |
| 27600.cn6 | 27600ch5 | \([0, 1, 0, 3515992, -8080344012]\) | \(75108181893694559/484313964843750\) | \(-30996093750000000000000\) | \([2]\) | \(1769472\) | \(2.9975\) | |
| 27600.cn2 | 27600ch6 | \([0, 1, 0, -10332008, 10946135988]\) | \(1905890658841300321/293666194803750\) | \(18794636467440000000000\) | \([2]\) | \(1769472\) | \(2.9975\) |
Rank
sage: E.rank()
The elliptic curves in class 27600ch have rank \(1\).
Complex multiplication
The elliptic curves in class 27600ch do not have complex multiplication.Modular form 27600.2.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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
