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SageMath
E = EllipticCurve("fn1")
E.isogeny_class()
Elliptic curves in class 238050.fn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 238050.fn1 | 238050fn3 | \([1, -1, 1, -13140362480, 579777814186647]\) | \(148809678420065817601/20700\) | \(34904780871510937500\) | \([2]\) | \(155713536\) | \(4.0748\) | |
| 238050.fn2 | 238050fn6 | \([1, -1, 1, -3074418230, -56202912226353]\) | \(1905890658841300321/293666194803750\) | \(495186192221996988911777343750\) | \([2]\) | \(311427072\) | \(4.4214\) | |
| 238050.fn3 | 238050fn4 | \([1, -1, 1, -842699480, 8561565898647]\) | \(39248884582600321/3935264062500\) | \(6635726075994899633789062500\) | \([2, 2]\) | \(155713536\) | \(4.0748\) | |
| 238050.fn4 | 238050fn2 | \([1, -1, 1, -821274980, 9059128486647]\) | \(36330796409313601/428490000\) | \(722528964040276406250000\) | \([2, 2]\) | \(77856768\) | \(3.7282\) | |
| 238050.fn5 | 238050fn1 | \([1, -1, 1, -49992980, 149278822647]\) | \(-8194759433281/965779200\) | \(-1628517456341214300000000\) | \([2]\) | \(38928384\) | \(3.3816\) | \(\Gamma_0(N)\)-optimal |
| 238050.fn6 | 238050fn5 | \([1, -1, 1, 1046227270, 41481781297647]\) | \(75108181893694559/484313964843750\) | \(-816660522506461143493652343750\) | \([2]\) | \(311427072\) | \(4.4214\) |
Rank
sage: E.rank()
The elliptic curves in class 238050.fn have rank \(0\).
Complex multiplication
The elliptic curves in class 238050.fn do not have complex multiplication.Modular form 238050.2.a.fn
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
