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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 66654j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 66654.h5 | 66654j1 | \([1, -1, 0, 599787, 366034549]\) | \(221115865823/664731648\) | \(-71736618393965887488\) | \([2]\) | \(2162688\) | \(2.4933\) | \(\Gamma_0(N)\)-optimal |
| 66654.h4 | 66654j2 | \([1, -1, 0, -5494293, 4243088245]\) | \(169967019783457/26337394944\) | \(2842283282695296862464\) | \([2, 2]\) | \(4325376\) | \(2.8399\) | |
| 66654.h3 | 66654j3 | \([1, -1, 0, -24157413, -41574871355]\) | \(14447092394873377/1439452851984\) | \(155343107627819866402704\) | \([2, 2]\) | \(8650752\) | \(3.1865\) | |
| 66654.h2 | 66654j4 | \([1, -1, 0, -84336453, 298119355429]\) | \(614716917569296417/19093020912\) | \(2060483744490159349872\) | \([2]\) | \(8650752\) | \(3.1865\) | |
| 66654.h6 | 66654j5 | \([1, -1, 0, 29832327, -201092957159]\) | \(27207619911317663/177609314617308\) | \(-19167270979575281849205948\) | \([2]\) | \(17301504\) | \(3.5330\) | |
| 66654.h1 | 66654j6 | \([1, -1, 0, -376757073, -2814630157391]\) | \(54804145548726848737/637608031452\) | \(68809487519992313223612\) | \([2]\) | \(17301504\) | \(3.5330\) |
Rank
sage: E.rank()
The elliptic curves in class 66654j have rank \(1\).
Complex multiplication
The elliptic curves in class 66654j do not have complex multiplication.Modular form 66654.2.a.j
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
