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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 83790.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
83790.d1 | 83790bo4 | \([1, -1, 0, -273870, -55016550]\) | \(26487576322129/44531250\) | \(3819272575781250\) | \([2]\) | \(786432\) | \(1.8852\) | |
83790.d2 | 83790bo2 | \([1, -1, 0, -22500, -268164]\) | \(14688124849/8122500\) | \(696635317822500\) | \([2, 2]\) | \(393216\) | \(1.5386\) | |
83790.d3 | 83790bo1 | \([1, -1, 0, -13680, 615600]\) | \(3301293169/22800\) | \(1955467558800\) | \([2]\) | \(196608\) | \(1.1920\) | \(\Gamma_0(N)\)-optimal |
83790.d4 | 83790bo3 | \([1, -1, 0, 87750, -2186514]\) | \(871257511151/527800050\) | \(-45267362952106050\) | \([2]\) | \(786432\) | \(1.8852\) |
Rank
sage: E.rank()
The elliptic curves in class 83790.d have rank \(2\).
Complex multiplication
The elliptic curves in class 83790.d do not have complex multiplication.Modular form 83790.2.a.d
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.