Show commands:
SageMath
E = EllipticCurve("cv1")
E.isogeny_class()
Elliptic curves in class 432450cv
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
432450.cv2 | 432450cv1 | \([1, -1, 0, 101080683, -147896971659]\) | \(11298232190519/7472736000\) | \(-75543544304780413500000000\) | \([2]\) | \(176947200\) | \(3.6544\) | \(\Gamma_0(N)\)-optimal* |
432450.cv1 | 432450cv2 | \([1, -1, 0, -435157317, -1224126637659]\) | \(901456690969801/457629750000\) | \(4626280561003437621093750000\) | \([2]\) | \(353894400\) | \(4.0009\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 432450cv have rank \(1\).
Complex multiplication
The elliptic curves in class 432450cv do not have complex multiplication.Modular form 432450.2.a.cv
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.