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SageMath
E = EllipticCurve("cn1")
E.isogeny_class()
Elliptic curves in class 134640cn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 134640.cq8 | 134640cn1 | \([0, 0, 0, 6245637, -2542088662]\) | \(9023321954633914439/6156756739584000\) | \(-18383977116289990656000\) | \([2]\) | \(10616832\) | \(2.9608\) | \(\Gamma_0(N)\)-optimal |
| 134640.cq7 | 134640cn2 | \([0, 0, 0, -27346683, -21212700118]\) | \(757443433548897303481/373234243041000000\) | \(1114471477972537344000000\) | \([2, 2]\) | \(21233664\) | \(3.3074\) | |
| 134640.cq6 | 134640cn3 | \([0, 0, 0, -112435563, -470076090982]\) | \(-52643812360427830814761/1504091705903677440\) | \(-4491193768361086377000960\) | \([2]\) | \(31850496\) | \(3.5101\) | |
| 134640.cq5 | 134640cn4 | \([0, 0, 0, -234343803, 1366040598698]\) | \(476646772170172569823801/5862293314453125000\) | \(17504714040264000000000000\) | \([2]\) | \(42467328\) | \(3.6539\) | |
| 134640.cq4 | 134640cn5 | \([0, 0, 0, -357826683, -2603385132118]\) | \(1696892787277117093383481/1440538624914939000\) | \(4301425285378009214976000\) | \([2]\) | \(42467328\) | \(3.6539\) | |
| 134640.cq3 | 134640cn6 | \([0, 0, 0, -1811128683, -29666873698918]\) | \(220031146443748723000125481/172266701724057600\) | \(514385615080808408678400\) | \([2, 2]\) | \(63700992\) | \(3.8567\) | |
| 134640.cq2 | 134640cn7 | \([0, 0, 0, -1823293803, -29248128371302]\) | \(224494757451893010998773801/6152490825146276160000\) | \(18371239164033578273341440000\) | \([2]\) | \(127401984\) | \(4.2032\) | |
| 134640.cq1 | 134640cn8 | \([0, 0, 0, -28978053483, -1898680665934438]\) | \(901247067798311192691198986281/552431869440\) | \(1649552723237928960\) | \([2]\) | \(127401984\) | \(4.2032\) |
Rank
sage: E.rank()
The elliptic curves in class 134640cn have rank \(1\).
Complex multiplication
The elliptic curves in class 134640cn do not have complex multiplication.Modular form 134640.2.a.cn
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
