Show commands:
SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 155526.cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 155526.cp1 | 155526h6 | \([1, 0, 0, -2051232954, -35757595043640]\) | \(54804145548726848737/637608031452\) | \(11104756374814232727633372\) | \([2]\) | \(103809024\) | \(3.9567\) | |
| 155526.cp2 | 155526h3 | \([1, 0, 0, -459165134, 3786913849620]\) | \(614716917569296417/19093020912\) | \(332529289513748638344432\) | \([2]\) | \(51904512\) | \(3.6101\) | |
| 155526.cp3 | 155526h4 | \([1, 0, 0, -131523694, -528242529676]\) | \(14447092394873377/1439452851984\) | \(25069905719211768809892624\) | \([2, 2]\) | \(51904512\) | \(3.6101\) | |
| 155526.cp4 | 155526h2 | \([1, 0, 0, -29913374, 53882993604]\) | \(169967019783457/26337394944\) | \(458699294822796955517184\) | \([2, 2]\) | \(25952256\) | \(3.2635\) | |
| 155526.cp5 | 155526h1 | \([1, 0, 0, 3265506, 4652171460]\) | \(221115865823/664731648\) | \(-11577148720756780105728\) | \([2]\) | \(12976128\) | \(2.9170\) | \(\Gamma_0(N)\)-optimal |
| 155526.cp6 | 155526h5 | \([1, 0, 0, 162420446, -2554517064352]\) | \(27207619911317663/177609314617308\) | \(-3093292542491155465400864988\) | \([2]\) | \(103809024\) | \(3.9567\) |
Rank
sage: E.rank()
The elliptic curves in class 155526.cp have rank \(1\).
Complex multiplication
The elliptic curves in class 155526.cp do not have complex multiplication.Modular form 155526.2.a.cp
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
