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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 79350.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
79350.k1 | 79350e4 | \([1, 1, 0, -1460040275, -21473739057375]\) | \(148809678420065817601/20700\) | \(47880357848437500\) | \([2]\) | \(19464192\) | \(3.5255\) | |
79350.k2 | 79350e6 | \([1, 1, 0, -341602025, 2081475474375]\) | \(1905890658841300321/293666194803750\) | \(679267753390942371621093750\) | \([2]\) | \(38928384\) | \(3.8721\) | |
79350.k3 | 79350e3 | \([1, 1, 0, -93633275, -317126244375]\) | \(39248884582600321/3935264062500\) | \(9102504905342797851562500\) | \([2, 2]\) | \(19464192\) | \(3.5255\) | |
79350.k4 | 79350e2 | \([1, 1, 0, -91252775, -335553694875]\) | \(36330796409313601/428490000\) | \(991123407462656250000\) | \([2, 2]\) | \(9732096\) | \(3.1789\) | |
79350.k5 | 79350e1 | \([1, 1, 0, -5554775, -5530696875]\) | \(-8194759433281/965779200\) | \(-2233905975776700000000\) | \([2]\) | \(4866048\) | \(2.8323\) | \(\Gamma_0(N)\)-optimal |
79350.k6 | 79350e5 | \([1, 1, 0, 116247475, -1536323521125]\) | \(75108181893694559/484313964843750\) | \(-1120247630324363708496093750\) | \([2]\) | \(38928384\) | \(3.8721\) |
Rank
sage: E.rank()
The elliptic curves in class 79350.k have rank \(0\).
Complex multiplication
The elliptic curves in class 79350.k do not have complex multiplication.Modular form 79350.2.a.k
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.