Show commands:
SageMath
E = EllipticCurve("ep1")
E.isogeny_class()
Elliptic curves in class 258570ep
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 258570.ep2 | 258570ep1 | \([1, -1, 1, 577948, 4723526279]\) | \(2761677827/1248480000\) | \(-9651599419981016160000\) | \([2]\) | \(21086208\) | \(2.8975\) | \(\Gamma_0(N)\)-optimal |
| 258570.ep1 | 258570ep2 | \([1, -1, 1, -38968052, 91281811079]\) | \(846509996114173/24354723600\) | \(188278575685279672741200\) | \([2]\) | \(42172416\) | \(3.2441\) |
Rank
sage: E.rank()
The elliptic curves in class 258570ep have rank \(0\).
Complex multiplication
The elliptic curves in class 258570ep do not have complex multiplication.Modular form 258570.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
