Show commands:
SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 80850df
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 80850.de4 | 80850df1 | \([1, 0, 1, -143879951, 664263901298]\) | \(1433528304665250149/162339408\) | \(37302869163656250000\) | \([2]\) | \(9216000\) | \(3.1794\) | \(\Gamma_0(N)\)-optimal |
| 80850.de3 | 80850df2 | \([1, 0, 1, -144247451, 660699886298]\) | \(1444540994277943589/15251205665388\) | \(3504470889311001585937500\) | \([2]\) | \(18432000\) | \(3.5260\) | |
| 80850.de2 | 80850df3 | \([1, 0, 1, -531623076, -4021279268702]\) | \(72313087342699809269/11447096545640448\) | \(2630350510738384896000000000\) | \([2]\) | \(46080000\) | \(3.9841\) | |
| 80850.de1 | 80850df4 | \([1, 0, 1, -8152103076, -283296630308702]\) | \(260744057755293612689909/8504954620259328\) | \(1954295715075956406000000000\) | \([2]\) | \(92160000\) | \(4.3307\) |
Rank
sage: E.rank()
The elliptic curves in class 80850df have rank \(1\).
Complex multiplication
The elliptic curves in class 80850df do not have complex multiplication.Modular form 80850.2.a.df
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
