Show commands for:
SageMath
sage: E = EllipticCurve("6720.h1")
sage: E.isogeny_class()
Elliptic curves in class 6720.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 6720.h1 | 6720bh5 | [0, -1, 0, -97601, 11765985] | [2] | 32768 | |
| 6720.h2 | 6720bh3 | [0, -1, 0, -6881, 135681] | [2, 2] | 16384 | |
| 6720.h3 | 6720bh2 | [0, -1, 0, -2961, -59535] | [2, 2] | 8192 | |
| 6720.h4 | 6720bh1 | [0, -1, 0, -2941, -60419] | [2] | 4096 | \(\Gamma_0(N)\)-optimal |
| 6720.h5 | 6720bh4 | [0, -1, 0, 639, -198495] | [2] | 16384 | |
| 6720.h6 | 6720bh6 | [0, -1, 0, 21119, 936481] | [2] | 32768 |
Rank
sage: E.rank()
The elliptic curves in class 6720.h have rank \(0\).
Modular form 6720.2.a.h
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
