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SageMath
E = EllipticCurve("dk1")
E.isogeny_class()
Elliptic curves in class 259920dk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 259920.dk4 | 259920dk1 | \([0, 0, 0, 648717, 175302322]\) | \(214921799/218880\) | \(-30747878907335147520\) | \([2]\) | \(8847360\) | \(2.4262\) | \(\Gamma_0(N)\)-optimal |
| 259920.dk3 | 259920dk2 | \([0, 0, 0, -3510003, 1613387698]\) | \(34043726521/11696400\) | \(1643089779110721945600\) | \([2, 2]\) | \(17694720\) | \(2.7728\) | |
| 259920.dk1 | 259920dk3 | \([0, 0, 0, -50295603, 137263556338]\) | \(100162392144121/23457780\) | \(3295307834772059013120\) | \([2]\) | \(35389440\) | \(3.1194\) | |
| 259920.dk2 | 259920dk4 | \([0, 0, 0, -23263923, -41999316878]\) | \(9912050027641/311647500\) | \(43779694772358051840000\) | \([2]\) | \(35389440\) | \(3.1194\) |
Rank
sage: E.rank()
The elliptic curves in class 259920dk have rank \(0\).
Complex multiplication
The elliptic curves in class 259920dk do not have complex multiplication.Modular form 259920.2.a.dk
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.
