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SageMath
E = EllipticCurve("du1")
E.isogeny_class()
Elliptic curves in class 141570du
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
141570.g4 | 141570du1 | \([1, -1, 0, -650700, -2324744064]\) | \(-23592983745241/1794399750000\) | \(-2317409800706607750000\) | \([2]\) | \(6635520\) | \(2.7792\) | \(\Gamma_0(N)\)-optimal |
141570.g3 | 141570du2 | \([1, -1, 0, -30598200, -64693407564]\) | \(2453170411237305241/19353090685500\) | \(24993896721475502749500\) | \([2]\) | \(13271040\) | \(3.1258\) | |
141570.g2 | 141570du3 | \([1, -1, 0, -154608075, -739915415739]\) | \(-316472948332146183241/7074906009600\) | \(-9137014495084006502400\) | \([2]\) | \(19906560\) | \(3.3286\) | |
141570.g1 | 141570du4 | \([1, -1, 0, -2473742475, -47355908336379]\) | \(1296294060988412126189641/647824320\) | \(836644358819206080\) | \([2]\) | \(39813120\) | \(3.6751\) |
Rank
sage: E.rank()
The elliptic curves in class 141570du have rank \(1\).
Complex multiplication
The elliptic curves in class 141570du do not have complex multiplication.Modular form 141570.2.a.du
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.