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SageMath
E = EllipticCurve("dq1")
E.isogeny_class()
Elliptic curves in class 163170.dq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 163170.dq1 | 163170z4 | \([1, -1, 1, -141322208, -133090737573]\) | \(3639478711331685826729/2016912141902025000\) | \(172982730808738246295025000\) | \([2]\) | \(53084160\) | \(3.7251\) | |
| 163170.dq2 | 163170z2 | \([1, -1, 1, -86197208, 306233462427]\) | \(825824067562227826729/5613755625000000\) | \(481470044198180625000000\) | \([2, 2]\) | \(26542080\) | \(3.3785\) | |
| 163170.dq3 | 163170z1 | \([1, -1, 1, -86056088, 307291749531]\) | \(821774646379511057449/38361600000\) | \(3290125627353600000\) | \([2]\) | \(13271040\) | \(3.0319\) | \(\Gamma_0(N)\)-optimal |
| 163170.dq4 | 163170z3 | \([1, -1, 1, -33330128, 677825594331]\) | \(-47744008200656797609/2286529541015625000\) | \(-196106769284820556640625000\) | \([2]\) | \(53084160\) | \(3.7251\) |
Rank
sage: E.rank()
The elliptic curves in class 163170.dq have rank \(0\).
Complex multiplication
The elliptic curves in class 163170.dq do not have complex multiplication.Modular form 163170.2.a.dq
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.
