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SageMath
E = EllipticCurve("dr1")
E.isogeny_class()
Elliptic curves in class 258570.dr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 258570.dr1 | 258570dr3 | \([1, -1, 1, -282938, 57877031]\) | \(711882749089/1721250\) | \(6056637698621250\) | \([2]\) | \(2359296\) | \(1.9072\) | |
| 258570.dr2 | 258570dr4 | \([1, -1, 1, -252518, -48568633]\) | \(506071034209/2505630\) | \(8816669929874430\) | \([2]\) | \(2359296\) | \(1.9072\) | |
| 258570.dr3 | 258570dr2 | \([1, -1, 1, -24368, 164207]\) | \(454756609/260100\) | \(915225252236100\) | \([2, 2]\) | \(1179648\) | \(1.5606\) | |
| 258570.dr4 | 258570dr1 | \([1, -1, 1, 6052, 18191]\) | \(6967871/4080\) | \(-14356474544880\) | \([2]\) | \(589824\) | \(1.2140\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 258570.dr have rank \(1\).
Complex multiplication
The elliptic curves in class 258570.dr do not have complex multiplication.Modular form 258570.2.a.dr
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.
