Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 87360.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 87360.e1 | 87360ea4 | \([0, -1, 0, -815361, -283110879]\) | \(914732517663095044/9555\) | \(626196480\) | \([2]\) | \(524288\) | \(1.7158\) | |
| 87360.e2 | 87360ea6 | \([0, -1, 0, -306881, 62486625]\) | \(24385137179326562/1284775885575\) | \(168398144874086400\) | \([2]\) | \(1048576\) | \(2.0624\) | |
| 87360.e3 | 87360ea3 | \([0, -1, 0, -54881, -3688575]\) | \(278944461825124/70849130625\) | \(4643168624640000\) | \([2, 2]\) | \(524288\) | \(1.7158\) | |
| 87360.e4 | 87360ea2 | \([0, -1, 0, -50961, -4410639]\) | \(893359210685776/91298025\) | \(1495826841600\) | \([2, 2]\) | \(262144\) | \(1.3693\) | |
| 87360.e5 | 87360ea1 | \([0, -1, 0, -2941, -79235]\) | \(-2748251600896/1124136195\) | \(-1151115463680\) | \([2]\) | \(131072\) | \(1.0227\) | \(\Gamma_0(N)\)-optimal |
| 87360.e6 | 87360ea5 | \([0, -1, 0, 134399, -23714399]\) | \(2048324060764798/3031899609375\) | \(-397397145600000000\) | \([2]\) | \(1048576\) | \(2.0624\) |
Rank
sage: E.rank()
The elliptic curves in class 87360.e have rank \(0\).
Complex multiplication
The elliptic curves in class 87360.e do not have complex multiplication.Modular form 87360.2.a.e
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
