Show commands:
SageMath
E = EllipticCurve("dz1")
E.isogeny_class()
Elliptic curves in class 232050.dz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 232050.dz1 | 232050dz3 | \([1, 1, 1, -5065438, 4385588531]\) | \(919929834039591549721/90169756021800\) | \(1408902437840625000\) | \([2]\) | \(8847360\) | \(2.5178\) | |
| 232050.dz2 | 232050dz4 | \([1, 1, 1, -1887438, -951155469]\) | \(47590666162724706841/2569185084375000\) | \(40143516943359375000\) | \([2]\) | \(8847360\) | \(2.5178\) | |
| 232050.dz3 | 232050dz2 | \([1, 1, 1, -340438, 57488531]\) | \(279265866686253721/69785974440000\) | \(1090405850625000000\) | \([2, 2]\) | \(4423680\) | \(2.1712\) | |
| 232050.dz4 | 232050dz1 | \([1, 1, 1, 51562, 5744531]\) | \(970269597296999/1467060940800\) | \(-22922827200000000\) | \([2]\) | \(2211840\) | \(1.8246\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 232050.dz have rank \(1\).
Complex multiplication
The elliptic curves in class 232050.dz do not have complex multiplication.Modular form 232050.2.a.dz
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
