Show commands:
SageMath
E = EllipticCurve("hr1")
E.isogeny_class()
Elliptic curves in class 388416.hr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 388416.hr1 | 388416hr4 | \([0, 1, 0, -1470817, -687049633]\) | \(444893916104/9639\) | \(7623869320101888\) | \([2]\) | \(4128768\) | \(2.1639\) | |
| 388416.hr2 | 388416hr2 | \([0, 1, 0, -95177, -9959625]\) | \(964430272/127449\) | \(12600561792946176\) | \([2, 2]\) | \(2064384\) | \(1.8173\) | |
| 388416.hr3 | 388416hr1 | \([0, 1, 0, -24372, 1298370]\) | \(1036433728/122451\) | \(189162845543616\) | \([2]\) | \(1032192\) | \(1.4707\) | \(\Gamma_0(N)\)-optimal |
| 388416.hr4 | 388416hr3 | \([0, 1, 0, 147583, -52248417]\) | \(449455096/1753941\) | \(-1387261850728169472\) | \([2]\) | \(4128768\) | \(2.1639\) |
Rank
sage: E.rank()
The elliptic curves in class 388416.hr have rank \(1\).
Complex multiplication
The elliptic curves in class 388416.hr do not have complex multiplication.Modular form 388416.2.a.hr
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 LMFDB 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 LMFDB labels.
