Show commands for:
SageMath
sage: E = EllipticCurve("a1")
sage: E.isogeny_class()
Elliptic curves in class 24.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
24.a1 | 24a5 | [0, -1, 0, -384, -2772] | [2] | 4 | |
24.a2 | 24a3 | [0, -1, 0, -64, 220] | [4] | 2 | |
24.a3 | 24a2 | [0, -1, 0, -24, -36] | [2, 2] | 2 | |
24.a4 | 24a1 | [0, -1, 0, -4, 4] | [2, 4] | 1 | \(\Gamma_0(N)\)-optimal |
24.a5 | 24a4 | [0, -1, 0, 1, 0] | [4] | 2 | |
24.a6 | 24a6 | [0, -1, 0, 16, -180] | [2] | 4 |
Rank
sage: E.rank()
The elliptic curves in class 24.a have rank \(0\).
Complex multiplication
The elliptic curves in class 24.a do not have complex multiplication.Modular form 24.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.