Show commands:
SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 55440.o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55440.o1 | 55440cp4 | \([0, 0, 0, -506883, -138144382]\) | \(4823468134087681/30382271150\) | \(90720975537561600\) | \([2]\) | \(663552\) | \(2.0913\) | |
| 55440.o2 | 55440cp2 | \([0, 0, 0, -38883, 2824418]\) | \(2177286259681/105875000\) | \(316141056000000\) | \([2]\) | \(221184\) | \(1.5420\) | |
| 55440.o3 | 55440cp3 | \([0, 0, 0, -12963, -4687198]\) | \(-80677568161/3131816380\) | \(-9351553601617920\) | \([2]\) | \(331776\) | \(1.7447\) | |
| 55440.o4 | 55440cp1 | \([0, 0, 0, 1437, 171362]\) | \(109902239/4312000\) | \(-12875563008000\) | \([2]\) | \(110592\) | \(1.1954\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 55440.o have rank \(1\).
Complex multiplication
The elliptic curves in class 55440.o do not have complex multiplication.Modular form 55440.2.a.o
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
