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SageMath
E = EllipticCurve("fe1")
E.isogeny_class()
Elliptic curves in class 155610.fe
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 155610.fe1 | 155610j7 | \([1, -1, 1, -6913872032, 221275592072939]\) | \(50137213659805457275731367898809/4113897879000\) | \(2999031553791000\) | \([6]\) | \(63700992\) | \(3.8216\) | |
| 155610.fe2 | 155610j6 | \([1, -1, 1, -432117032, 3457511648939]\) | \(12240533203187013248735018809/3506282465049000000\) | \(2556079917020721000000\) | \([2, 6]\) | \(31850496\) | \(3.4750\) | |
| 155610.fe3 | 155610j8 | \([1, -1, 1, -430362032, 3486987224939]\) | \(-12091997009671629064982138809/207252595706436249879000\) | \(-151087142269992026161791000\) | \([6]\) | \(63700992\) | \(3.8216\) | |
| 155610.fe4 | 155610j4 | \([1, -1, 1, -85365347, 303485541581]\) | \(94371532824107026279203049/40995077600666342790\) | \(29885411570885763893910\) | \([2]\) | \(21233664\) | \(3.2723\) | |
| 155610.fe5 | 155610j3 | \([1, -1, 1, -27117032, 53567648939]\) | \(3024980849878413455018809/50557689000000000000\) | \(36856555281000000000000\) | \([6]\) | \(15925248\) | \(3.1284\) | |
| 155610.fe6 | 155610j2 | \([1, -1, 1, -6183797, 3134086121]\) | \(35872512095393194378249/14944558319037792900\) | \(10894583014578551024100\) | \([2, 2]\) | \(10616832\) | \(2.9257\) | |
| 155610.fe7 | 155610j1 | \([1, -1, 1, -2903297, -1869332479]\) | \(3712533999213317890249/76090919904090000\) | \(55470280610081610000\) | \([2]\) | \(5308416\) | \(2.5791\) | \(\Gamma_0(N)\)-optimal |
| 155610.fe8 | 155610j5 | \([1, -1, 1, 20509753, 22962055061]\) | \(1308812680909424992398551/1070002284841633041990\) | \(-780031665649550487610710\) | \([2]\) | \(21233664\) | \(3.2723\) |
Rank
sage: E.rank()
The elliptic curves in class 155610.fe have rank \(1\).
Complex multiplication
The elliptic curves in class 155610.fe do not have complex multiplication.Modular form 155610.2.a.fe
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
