Show commands:
SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 259182x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
259182.x2 | 259182x1 | \([1, -1, 0, -49803, 2630645]\) | \(521465741649/188737808\) | \(4944566871843984\) | \([2]\) | \(1548288\) | \(1.7120\) | \(\Gamma_0(N)\)-optimal |
259182.x1 | 259182x2 | \([1, -1, 0, -340863, -74616679]\) | \(167183982669969/4730963524\) | \(123942127762089252\) | \([2]\) | \(3096576\) | \(2.0586\) |
Rank
sage: E.rank()
The elliptic curves in class 259182x have rank \(1\).
Complex multiplication
The elliptic curves in class 259182x do not have complex multiplication.Modular form 259182.2.a.x
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.