Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 2016.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2016.a1 | 2016d2 | \([0, 0, 0, -3036, -64384]\) | \(1036433728/63\) | \(188116992\) | \([2]\) | \(1024\) | \(0.64825\) | |
2016.a2 | 2016d3 | \([0, 0, 0, -1011, 11594]\) | \(306182024/21609\) | \(8065516032\) | \([2]\) | \(1024\) | \(0.64825\) | |
2016.a3 | 2016d1 | \([0, 0, 0, -201, -880]\) | \(19248832/3969\) | \(185177664\) | \([2, 2]\) | \(512\) | \(0.30168\) | \(\Gamma_0(N)\)-optimal |
2016.a4 | 2016d4 | \([0, 0, 0, 429, -5290]\) | \(23393656/45927\) | \(-17142160896\) | \([2]\) | \(1024\) | \(0.64825\) |
Rank
sage: E.rank()
The elliptic curves in class 2016.a have rank \(0\).
Complex multiplication
The elliptic curves in class 2016.a do not have complex multiplication.Modular form 2016.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.