Show commands:
SageMath
E = EllipticCurve("hw1")
E.isogeny_class()
Elliptic curves in class 81600.hw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 81600.hw1 | 81600hu8 | \([0, 1, 0, -181552033, 941498756063]\) | \(161572377633716256481/914742821250\) | \(3746786595840000000000\) | \([2]\) | \(9437184\) | \(3.3312\) | |
| 81600.hw2 | 81600hu4 | \([0, 1, 0, -34816033, -79082571937]\) | \(1139466686381936641/4080\) | \(16711680000000\) | \([2]\) | \(2359296\) | \(2.6380\) | |
| 81600.hw3 | 81600hu6 | \([0, 1, 0, -11552033, 14148756063]\) | \(41623544884956481/2962701562500\) | \(12135225600000000000000\) | \([2, 2]\) | \(4718592\) | \(2.9846\) | |
| 81600.hw4 | 81600hu3 | \([0, 1, 0, -2304033, -1082699937]\) | \(330240275458561/67652010000\) | \(277102632960000000000\) | \([2, 2]\) | \(2359296\) | \(2.6380\) | |
| 81600.hw5 | 81600hu2 | \([0, 1, 0, -2176033, -1236171937]\) | \(278202094583041/16646400\) | \(68183654400000000\) | \([2, 2]\) | \(1179648\) | \(2.2915\) | |
| 81600.hw6 | 81600hu1 | \([0, 1, 0, -128033, -21707937]\) | \(-56667352321/16711680\) | \(-68451041280000000\) | \([2]\) | \(589824\) | \(1.9449\) | \(\Gamma_0(N)\)-optimal |
| 81600.hw7 | 81600hu5 | \([0, 1, 0, 4895967, -6489899937]\) | \(3168685387909439/6278181696900\) | \(-25715432230502400000000\) | \([2]\) | \(4718592\) | \(2.9846\) | |
| 81600.hw8 | 81600hu7 | \([0, 1, 0, 10479967, 61759908063]\) | \(31077313442863199/420227050781250\) | \(-1721250000000000000000000\) | \([2]\) | \(9437184\) | \(3.3312\) |
Rank
sage: E.rank()
The elliptic curves in class 81600.hw have rank \(1\).
Complex multiplication
The elliptic curves in class 81600.hw do not have complex multiplication.Modular form 81600.2.a.hw
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 16 & 2 & 4 & 8 & 16 & 8 & 4 \\ 16 & 1 & 8 & 4 & 2 & 4 & 8 & 16 \\ 2 & 8 & 1 & 2 & 4 & 8 & 4 & 2 \\ 4 & 4 & 2 & 1 & 2 & 4 & 2 & 4 \\ 8 & 2 & 4 & 2 & 1 & 2 & 4 & 8 \\ 16 & 4 & 8 & 4 & 2 & 1 & 8 & 16 \\ 8 & 8 & 4 & 2 & 4 & 8 & 1 & 8 \\ 4 & 16 & 2 & 4 & 8 & 16 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
