Show commands:
SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 436800.q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
436800.q1 | 436800q4 | \([0, -1, 0, -1685633, -841756863]\) | \(1034529986960072/44983575\) | \(23031590400000000\) | \([2]\) | \(6291456\) | \(2.2181\) | |
436800.q2 | 436800q2 | \([0, -1, 0, -110633, -11731863]\) | \(2339923888576/419225625\) | \(26830440000000000\) | \([2, 2]\) | \(3145728\) | \(1.8715\) | |
436800.q3 | 436800q1 | \([0, -1, 0, -32508, 2096262]\) | \(3799337068864/319921875\) | \(319921875000000\) | \([2]\) | \(1572864\) | \(1.5250\) | \(\Gamma_0(N)\)-optimal* |
436800.q4 | 436800q3 | \([0, -1, 0, 214367, -67956863]\) | \(2127774087928/5119712325\) | \(-2621292710400000000\) | \([2]\) | \(6291456\) | \(2.2181\) |
Rank
sage: E.rank()
The elliptic curves in class 436800.q have rank \(1\).
Complex multiplication
The elliptic curves in class 436800.q do not have complex multiplication.Modular form 436800.2.a.q
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.