Show commands:
SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 265200.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 265200.k1 | 265200k3 | \([0, -1, 0, -2959008, 1947856512]\) | \(44769506062996441/323730468750\) | \(20718750000000000000\) | \([2]\) | \(12386304\) | \(2.5381\) | |
| 265200.k2 | 265200k2 | \([0, -1, 0, -307008, -14623488]\) | \(50002789171321/27473062500\) | \(1758276000000000000\) | \([2, 2]\) | \(6193152\) | \(2.1916\) | |
| 265200.k3 | 265200k1 | \([0, -1, 0, -235008, -43711488]\) | \(22428153804601/35802000\) | \(2291328000000000\) | \([2]\) | \(3096576\) | \(1.8450\) | \(\Gamma_0(N)\)-optimal |
| 265200.k4 | 265200k4 | \([0, -1, 0, 1192992, -116623488]\) | \(2933972022568679/1789082460750\) | \(-114501277488000000000\) | \([4]\) | \(12386304\) | \(2.5381\) |
Rank
sage: E.rank()
The elliptic curves in class 265200.k have rank \(0\).
Complex multiplication
The elliptic curves in class 265200.k do not have complex multiplication.Modular form 265200.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
