Show commands:
SageMath
E = EllipticCurve("bh1")
E.isogeny_class()
Elliptic curves in class 53958bh
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 53958.bn5 | 53958bh1 | \([1, 0, 0, -17997, -863343]\) | \(4354703137/352512\) | \(52184427303168\) | \([2]\) | \(202752\) | \(1.3751\) | \(\Gamma_0(N)\)-optimal |
| 53958.bn4 | 53958bh2 | \([1, 0, 0, -60317, 4697505]\) | \(163936758817/30338064\) | \(4491122274778896\) | \([2, 2]\) | \(405504\) | \(1.7217\) | |
| 53958.bn6 | 53958bh3 | \([1, 0, 0, 119543, 27395837]\) | \(1276229915423/2927177028\) | \(-433327253600357892\) | \([2]\) | \(811008\) | \(2.0683\) | |
| 53958.bn2 | 53958bh4 | \([1, 0, 0, -917297, 338062725]\) | \(576615941610337/27060804\) | \(4005970177194756\) | \([2, 2]\) | \(811008\) | \(2.0683\) | |
| 53958.bn3 | 53958bh5 | \([1, 0, 0, -869687, 374731947]\) | \(-491411892194497/125563633938\) | \(-18587924176082400882\) | \([2]\) | \(1622016\) | \(2.4148\) | |
| 53958.bn1 | 53958bh6 | \([1, 0, 0, -14676587, 21640195503]\) | \(2361739090258884097/5202\) | \(770082694578\) | \([2]\) | \(1622016\) | \(2.4148\) |
Rank
sage: E.rank()
The elliptic curves in class 53958bh have rank \(0\).
Complex multiplication
The elliptic curves in class 53958bh do not have complex multiplication.Modular form 53958.2.a.bh
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
