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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 53361.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 53361.p1 | 53361bo6 | \([1, -1, 1, -41836136, 104164339920]\) | \(53297461115137/147\) | \(22335167517477507\) | \([2]\) | \(1966080\) | \(2.7954\) | |
| 53361.p2 | 53361bo4 | \([1, -1, 1, -2615801, 1626696096]\) | \(13027640977/21609\) | \(3283269625069193529\) | \([2, 2]\) | \(983040\) | \(2.4488\) | |
| 53361.p3 | 53361bo3 | \([1, -1, 1, -2082191, -1148929680]\) | \(6570725617/45927\) | \(6978144480103329687\) | \([2]\) | \(983040\) | \(2.4488\) | |
| 53361.p4 | 53361bo5 | \([1, -1, 1, -1815386, 2640341652]\) | \(-4354703137/17294403\) | \(-2627710123263711221043\) | \([2]\) | \(1966080\) | \(2.7954\) | |
| 53361.p5 | 53361bo2 | \([1, -1, 1, -214556, 8256966]\) | \(7189057/3969\) | \(603049522971892689\) | \([2, 2]\) | \(491520\) | \(2.1023\) | |
| 53361.p6 | 53361bo1 | \([1, -1, 1, 52249, 999870]\) | \(103823/63\) | \(-9572214650347503\) | \([2]\) | \(245760\) | \(1.7557\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 53361.p have rank \(0\).
Complex multiplication
The elliptic curves in class 53361.p do not have complex multiplication.Modular form 53361.2.a.p
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 8 & 4 & 4 & 8 \\ 2 & 1 & 4 & 2 & 2 & 4 \\ 8 & 4 & 1 & 8 & 2 & 4 \\ 4 & 2 & 8 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
