Show commands:
SageMath
E = EllipticCurve("lq1")
E.isogeny_class()
Elliptic curves in class 221760.lq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 221760.lq1 | 221760iq5 | \([0, 0, 0, -1756339212, 28330922092016]\) | \(3135316978843283198764801/571725\) | \(109258348953600\) | \([2]\) | \(31457280\) | \(3.4882\) | |
| 221760.lq2 | 221760iq3 | \([0, 0, 0, -109771212, 442670562416]\) | \(765458482133960722801/326869475625\) | \(62465729555496960000\) | \([2, 2]\) | \(15728640\) | \(3.1416\) | |
| 221760.lq3 | 221760iq6 | \([0, 0, 0, -109226892, 447277904624]\) | \(-754127868744065783521/15825714261328125\) | \(-3024341092665446400000000\) | \([2]\) | \(31457280\) | \(3.4882\) | |
| 221760.lq4 | 221760iq4 | \([0, 0, 0, -14656332, -11388908176]\) | \(1821931919215868881/761147600816295\) | \(145457571691254003793920\) | \([2]\) | \(15728640\) | \(3.1416\) | |
| 221760.lq5 | 221760iq2 | \([0, 0, 0, -6894732, 6844642544]\) | \(189674274234120481/3859869269025\) | \(737632504281622118400\) | \([2, 2]\) | \(7864320\) | \(2.7950\) | |
| 221760.lq6 | 221760iq1 | \([0, 0, 0, 20148, 319761776]\) | \(4733169839/231139696095\) | \(-44171483795490078720\) | \([2]\) | \(3932160\) | \(2.4484\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 221760.lq have rank \(0\).
Complex multiplication
The elliptic curves in class 221760.lq do not have complex multiplication.Modular form 221760.2.a.lq
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
