Show commands:
SageMath
E = EllipticCurve("hz1")
E.isogeny_class()
Elliptic curves in class 169050hz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
169050.d2 | 169050hz1 | \([1, 1, 0, -3191150, -1448815500]\) | \(5699846954647/1863000000\) | \(1174668278765625000000\) | \([2]\) | \(9289728\) | \(2.7461\) | \(\Gamma_0(N)\)-optimal |
169050.d1 | 169050hz2 | \([1, 1, 0, -46066150, -120341190500]\) | \(17146168720634647/3470769000\) | \(2188407003340359375000\) | \([2]\) | \(18579456\) | \(3.0926\) |
Rank
sage: E.rank()
The elliptic curves in class 169050hz have rank \(2\).
Complex multiplication
The elliptic curves in class 169050hz do not have complex multiplication.Modular form 169050.2.a.hz
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.