Show commands:
SageMath
E = EllipticCurve("hh1")
E.isogeny_class()
Elliptic curves in class 162624hh
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 162624.bi4 | 162624hh1 | \([0, -1, 0, 13391, 23604913]\) | \(9148592/8301447\) | \(-240951619563798528\) | \([2]\) | \(1966080\) | \(2.0143\) | \(\Gamma_0(N)\)-optimal |
| 162624.bi3 | 162624hh2 | \([0, -1, 0, -1157889, 468925569]\) | \(1478729816932/38900169\) | \(4516349365047066624\) | \([2, 2]\) | \(3932160\) | \(2.3609\) | |
| 162624.bi1 | 162624hh3 | \([0, -1, 0, -18407649, 30404159073]\) | \(2970658109581346/2139291\) | \(496747741610115072\) | \([4]\) | \(7864320\) | \(2.7074\) | |
| 162624.bi2 | 162624hh4 | \([0, -1, 0, -2648609, -984526431]\) | \(8849350367426/3314597517\) | \(769656316469449457664\) | \([2]\) | \(7864320\) | \(2.7074\) |
Rank
sage: E.rank()
The elliptic curves in class 162624hh have rank \(1\).
Complex multiplication
The elliptic curves in class 162624hh do not have complex multiplication.Modular form 162624.2.a.hh
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
