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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 249090x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 249090.x5 | 249090x1 | \([1, 1, 0, -151627, -24995651]\) | \(-8194759433281/965779200\) | \(-45435933315475200\) | \([2]\) | \(2654208\) | \(1.9321\) | \(\Gamma_0(N)\)-optimal |
| 249090.x4 | 249090x2 | \([1, 1, 0, -2490907, -1514181299]\) | \(36330796409313601/428490000\) | \(20158689549690000\) | \([2, 2]\) | \(5308416\) | \(2.2787\) | |
| 249090.x3 | 249090x3 | \([1, 1, 0, -2555887, -1431097871]\) | \(39248884582600321/3935264062500\) | \(185137964787951562500\) | \([2, 2]\) | \(10616832\) | \(2.6252\) | |
| 249090.x1 | 249090x4 | \([1, 1, 0, -39854407, -96858360599]\) | \(148809678420065817601/20700\) | \(973849736700\) | \([2]\) | \(10616832\) | \(2.6252\) | |
| 249090.x2 | 249090x5 | \([1, 1, 0, -9324637, 9384010879]\) | \(1905890658841300321/293666194803750\) | \(13815784854460040853750\) | \([2]\) | \(21233664\) | \(2.9718\) | |
| 249090.x6 | 249090x6 | \([1, 1, 0, 3173183, -6927567629]\) | \(75108181893694559/484313964843750\) | \(-22784977156677246093750\) | \([2]\) | \(21233664\) | \(2.9718\) |
Rank
sage: E.rank()
The elliptic curves in class 249090x have rank \(1\).
Complex multiplication
The elliptic curves in class 249090x do not have complex multiplication.Modular form 249090.2.a.x
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.
