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SageMath
E = EllipticCurve("fh1")
E.isogeny_class()
Elliptic curves in class 80850fh
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 80850.el4 | 80850fh1 | \([1, 1, 1, -5755198, 5311809131]\) | \(1433528304665250149/162339408\) | \(2387383626474000\) | \([2]\) | \(1843200\) | \(2.3747\) | \(\Gamma_0(N)\)-optimal |
| 80850.el3 | 80850fh2 | \([1, 1, 1, -5769898, 5283291131]\) | \(1444540994277943589/15251205665388\) | \(224286136915904101500\) | \([2]\) | \(3686400\) | \(2.7212\) | |
| 80850.el2 | 80850fh3 | \([1, 1, 1, -21264923, -32178740119]\) | \(72313087342699809269/11447096545640448\) | \(168342432687256633344000\) | \([2]\) | \(9216000\) | \(3.1794\) | |
| 80850.el1 | 80850fh4 | \([1, 1, 1, -326084123, -2266503476119]\) | \(260744057755293612689909/8504954620259328\) | \(125074925764861209984000\) | \([2]\) | \(18432000\) | \(3.5260\) |
Rank
sage: E.rank()
The elliptic curves in class 80850fh have rank \(1\).
Complex multiplication
The elliptic curves in class 80850fh do not have complex multiplication.Modular form 80850.2.a.fh
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.
