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SageMath
E = EllipticCurve("ep1")
E.isogeny_class()
Elliptic curves in class 116550ep
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
116550.fe4 | 116550ep1 | \([1, -1, 1, 17995, -760003]\) | \(56578878719/54390000\) | \(-619536093750000\) | \([2]\) | \(491520\) | \(1.5254\) | \(\Gamma_0(N)\)-optimal |
116550.fe3 | 116550ep2 | \([1, -1, 1, -94505, -6835003]\) | \(8194759433281/2958272100\) | \(33696568139062500\) | \([2, 2]\) | \(983040\) | \(1.8720\) | |
116550.fe2 | 116550ep3 | \([1, -1, 1, -645755, 194922497]\) | \(2614441086442081/74385450090\) | \(847296767431406250\) | \([2]\) | \(1966080\) | \(2.2186\) | |
116550.fe1 | 116550ep4 | \([1, -1, 1, -1343255, -598742503]\) | \(23531588875176481/6398929110\) | \(72887801893593750\) | \([2]\) | \(1966080\) | \(2.2186\) |
Rank
sage: E.rank()
The elliptic curves in class 116550ep have rank \(0\).
Complex multiplication
The elliptic curves in class 116550ep do not have complex multiplication.Modular form 116550.2.a.ep
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.