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SageMath
E = EllipticCurve("co1")
E.isogeny_class()
Elliptic curves in class 286110co
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 286110.co4 | 286110co1 | \([1, -1, 0, -19013364, -31905906480]\) | \(43199583152847841/89760000\) | \(1579442793017760000\) | \([2]\) | \(14155776\) | \(2.7414\) | \(\Gamma_0(N)\)-optimal |
| 286110.co3 | 286110co2 | \([1, -1, 0, -19221444, -31171675392]\) | \(44633474953947361/1967006250000\) | \(34612008081365756250000\) | \([2, 2]\) | \(28311552\) | \(3.0879\) | |
| 286110.co2 | 286110co3 | \([1, -1, 0, -51733944, 102123072108]\) | \(870220733067747361/247623269602500\) | \(4357250318150204869102500\) | \([2, 2]\) | \(56623104\) | \(3.4345\) | |
| 286110.co5 | 286110co4 | \([1, -1, 0, 9961776, -117478130220]\) | \(6213165856218719/342407226562500\) | \(-6025096103735961914062500\) | \([2]\) | \(56623104\) | \(3.4345\) | |
| 286110.co1 | 286110co5 | \([1, -1, 0, -759856194, 8061275537658]\) | \(2757381641970898311361/379829992662450\) | \(6683597866340188445772450\) | \([2]\) | \(113246208\) | \(3.7811\) | |
| 286110.co6 | 286110co6 | \([1, -1, 0, 136188306, 673444296558]\) | \(15875306080318016639/20322604533582450\) | \(-357602398238824159762692450\) | \([2]\) | \(113246208\) | \(3.7811\) |
Rank
sage: E.rank()
The elliptic curves in class 286110co have rank \(1\).
Complex multiplication
The elliptic curves in class 286110co do not have complex multiplication.Modular form 286110.2.a.co
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.
