Show commands for:
SageMath
sage: E = EllipticCurve("hd1")
sage: E.isogeny_class()
Elliptic curves in class 177600.hd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 177600.hd1 | 177600bp4 | [0, 1, 0, -545608033, -4905523347937] | [2] | 31850496 | |
| 177600.hd2 | 177600bp5 | [0, 1, 0, -485000033, 4093090860063] | [4] | 63700992 | |
| 177600.hd3 | 177600bp3 | [0, 1, 0, -46920033, -13909139937] | [2, 2] | 31850496 | |
| 177600.hd4 | 177600bp2 | [0, 1, 0, -34120033, -76565139937] | [2, 2] | 15925248 | |
| 177600.hd5 | 177600bp1 | [0, 1, 0, -1352033, -2083475937] | [2] | 7962624 | \(\Gamma_0(N)\)-optimal |
| 177600.hd6 | 177600bp6 | [0, 1, 0, 186359967, -110720339937] | [2] | 63700992 |
Rank
sage: E.rank()
The elliptic curves in class 177600.hd have rank \(0\).
Complex multiplication
The elliptic curves in class 177600.hd do not have complex multiplication.Modular form 177600.2.a.hd
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
