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SageMath
E = EllipticCurve("dk1")
E.isogeny_class()
Elliptic curves in class 265200.dk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 265200.dk1 | 265200dk4 | \([0, -1, 0, -27438008, -55296265488]\) | \(35694515311673154481/10400566692750\) | \(665636268336000000000\) | \([2]\) | \(19464192\) | \(2.9751\) | |
| 265200.dk2 | 265200dk3 | \([0, -1, 0, -13430008, 18498422512]\) | \(4185743240664514801/113629394531250\) | \(7272281250000000000000\) | \([2]\) | \(19464192\) | \(2.9751\) | |
| 265200.dk3 | 265200dk2 | \([0, -1, 0, -1938008, -624265488]\) | \(12577973014374481/4642947562500\) | \(297148644000000000000\) | \([2, 2]\) | \(9732096\) | \(2.6285\) | |
| 265200.dk4 | 265200dk1 | \([0, -1, 0, 373992, -69385488]\) | \(90391899763439/84690294000\) | \(-5420178816000000000\) | \([2]\) | \(4866048\) | \(2.2819\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 265200.dk have rank \(1\).
Complex multiplication
The elliptic curves in class 265200.dk do not have complex multiplication.Modular form 265200.2.a.dk
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.
