Show commands:
SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 189618.l
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 189618.l1 | 189618bq6 | \([1, 1, 0, -66745539, -209912851143]\) | \(6812873765474836663297/74052\) | \(357434860068\) | \([2]\) | \(9437184\) | \(2.7204\) | |
| 189618.l2 | 189618bq4 | \([1, 1, 0, -4171599, -3281186475]\) | \(1663303207415737537/5483698704\) | \(26468766257755536\) | \([2, 2]\) | \(4718592\) | \(2.3738\) | |
| 189618.l3 | 189618bq5 | \([1, 1, 0, -4114139, -3375892047]\) | \(-1595514095015181697/95635786040388\) | \(-461615672781819161892\) | \([2]\) | \(9437184\) | \(2.7204\) | |
| 189618.l4 | 189618bq2 | \([1, 1, 0, -264319, -49865915]\) | \(423108074414017/23284318464\) | \(112388957920901376\) | \([2, 2]\) | \(2359296\) | \(2.0273\) | |
| 189618.l5 | 189618bq1 | \([1, 1, 0, -47999, 3045957]\) | \(2533811507137/625016832\) | \(3016836869849088\) | \([2]\) | \(1179648\) | \(1.6807\) | \(\Gamma_0(N)\)-optimal |
| 189618.l6 | 189618bq3 | \([1, 1, 0, 181841, -200578763]\) | \(137763859017023/3683199928848\) | \(-17778102565362886032\) | \([2]\) | \(4718592\) | \(2.3738\) |
Rank
sage: E.rank()
The elliptic curves in class 189618.l have rank \(1\).
Complex multiplication
The elliptic curves in class 189618.l do not have complex multiplication.Modular form 189618.2.a.l
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
