Show commands:
SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 1650.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1650.k1 | 1650f3 | \([1, 0, 1, -8801, -318502]\) | \(4824238966273/66\) | \(1031250\) | \([2]\) | \(2048\) | \(0.70977\) | |
| 1650.k2 | 1650f2 | \([1, 0, 1, -551, -5002]\) | \(1180932193/4356\) | \(68062500\) | \([2, 2]\) | \(1024\) | \(0.36319\) | |
| 1650.k3 | 1650f4 | \([1, 0, 1, -301, -9502]\) | \(-192100033/2371842\) | \(-37060031250\) | \([2]\) | \(2048\) | \(0.70977\) | |
| 1650.k4 | 1650f1 | \([1, 0, 1, -51, -2]\) | \(912673/528\) | \(8250000\) | \([2]\) | \(512\) | \(0.016618\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1650.k have rank \(0\).
Complex multiplication
The elliptic curves in class 1650.k do not have complex multiplication.Modular form 1650.2.a.k
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.
