Show commands:
SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 13104.by
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 13104.by1 | 13104ck3 | \([0, 0, 0, -281739, 57548090]\) | \(828279937799497/193444524\) | \(577622253551616\) | \([2]\) | \(73728\) | \(1.8231\) | |
| 13104.by2 | 13104ck2 | \([0, 0, 0, -19659, 676730]\) | \(281397674377/96589584\) | \(288414952390656\) | \([2, 2]\) | \(36864\) | \(1.4765\) | |
| 13104.by3 | 13104ck1 | \([0, 0, 0, -8139, -274822]\) | \(19968681097/628992\) | \(1878160048128\) | \([2]\) | \(18432\) | \(1.1299\) | \(\Gamma_0(N)\)-optimal |
| 13104.by4 | 13104ck4 | \([0, 0, 0, 58101, 4704698]\) | \(7264187703863/7406095788\) | \(-22114483525435392\) | \([4]\) | \(73728\) | \(1.8231\) |
Rank
sage: E.rank()
The elliptic curves in class 13104.by have rank \(1\).
Complex multiplication
The elliptic curves in class 13104.by do not have complex multiplication.Modular form 13104.2.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 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.
