Show commands:
SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 17640by
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 17640.u4 | 17640by1 | \([0, 0, 0, 3822, -750827]\) | \(4499456/180075\) | \(-247109347825200\) | \([2]\) | \(49152\) | \(1.4416\) | \(\Gamma_0(N)\)-optimal |
| 17640.u3 | 17640by2 | \([0, 0, 0, -104223, -12398078]\) | \(5702413264/275625\) | \(6051657497760000\) | \([2, 2]\) | \(98304\) | \(1.7882\) | |
| 17640.u1 | 17640by3 | \([0, 0, 0, -1647723, -814091978]\) | \(5633270409316/14175\) | \(1244912399539200\) | \([2]\) | \(196608\) | \(2.1347\) | |
| 17640.u2 | 17640by4 | \([0, 0, 0, -289443, 43871758]\) | \(30534944836/8203125\) | \(720435416400000000\) | \([2]\) | \(196608\) | \(2.1347\) |
Rank
sage: E.rank()
The elliptic curves in class 17640by have rank \(0\).
Complex multiplication
The elliptic curves in class 17640by do not have complex multiplication.Modular form 17640.2.a.by
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.
