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SageMath
E = EllipticCurve("fa1")
E.isogeny_class()
Elliptic curves in class 356928fa
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 356928.fa5 | 356928fa1 | \([0, 1, 0, -259809, -73204065]\) | \(-1532808577/938223\) | \(-1187151405490372608\) | \([2]\) | \(5505024\) | \(2.1701\) | \(\Gamma_0(N)\)-optimal |
| 356928.fa4 | 356928fa2 | \([0, 1, 0, -4640289, -3848301729]\) | \(8732907467857/1656369\) | \(2095835197347201024\) | \([2, 2]\) | \(11010048\) | \(2.5167\) | |
| 356928.fa3 | 356928fa3 | \([0, 1, 0, -5127009, -2992161249]\) | \(11779205551777/3763454409\) | \(4761970438950323748864\) | \([2, 2]\) | \(22020096\) | \(2.8633\) | |
| 356928.fa1 | 356928fa4 | \([0, 1, 0, -74241249, -246240605025]\) | \(35765103905346817/1287\) | \(1628465576804352\) | \([2]\) | \(22020096\) | \(2.8633\) | |
| 356928.fa2 | 356928fa5 | \([0, 1, 0, -32545569, 69189939807]\) | \(3013001140430737/108679952667\) | \(137514811038800177528832\) | \([2]\) | \(44040192\) | \(3.2099\) | |
| 356928.fa6 | 356928fa6 | \([0, 1, 0, 14504031, -20373484065]\) | \(266679605718863/296110251723\) | \(-374673933076749852868608\) | \([2]\) | \(44040192\) | \(3.2099\) |
Rank
sage: E.rank()
The elliptic curves in class 356928fa have rank \(0\).
Complex multiplication
The elliptic curves in class 356928fa do not have complex multiplication.Modular form 356928.2.a.fa
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.
