Show commands:
SageMath
E = EllipticCurve("dq1")
E.isogeny_class()
Elliptic curves in class 68544dq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 68544.t5 | 68544dq1 | \([0, 0, 0, -40460556, 98935513616]\) | \(38331145780597164097/55468445663232\) | \(10600185040337928978432\) | \([2]\) | \(5898240\) | \(3.1284\) | \(\Gamma_0(N)\)-optimal |
| 68544.t4 | 68544dq2 | \([0, 0, 0, -52257036, 36541571600]\) | \(82582985847542515777/44772582831427584\) | \(8556173822292317630889984\) | \([2, 2]\) | \(11796480\) | \(3.4750\) | |
| 68544.t6 | 68544dq3 | \([0, 0, 0, 201551604, 287406031376]\) | \(4738217997934888496063/2928751705237796928\) | \(-559693166836017780624457728\) | \([2]\) | \(23592960\) | \(3.8216\) | |
| 68544.t2 | 68544dq4 | \([0, 0, 0, -494809356, -4207535177200]\) | \(70108386184777836280897/552468975892674624\) | \(105578465440762377246081024\) | \([2, 2]\) | \(23592960\) | \(3.8216\) | |
| 68544.t3 | 68544dq5 | \([0, 0, 0, -168539916, -9673331343856]\) | \(-2770540998624539614657/209924951154647363208\) | \(-40117282902307747339601707008\) | \([2]\) | \(47185920\) | \(4.1682\) | |
| 68544.t1 | 68544dq6 | \([0, 0, 0, -7901915916, -270362650933744]\) | \(285531136548675601769470657/17941034271597192\) | \(3428585041820215664443392\) | \([2]\) | \(47185920\) | \(4.1682\) |
Rank
sage: E.rank()
The elliptic curves in class 68544dq have rank \(1\).
Complex multiplication
The elliptic curves in class 68544dq do not have complex multiplication.Modular form 68544.2.a.dq
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
