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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 102850.w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
102850.w1 | 102850n2 | \([1, -1, 0, -2190667, 1248531741]\) | \(42002659053081/411400\) | \(11387815553125000\) | \([2]\) | \(1935360\) | \(2.2415\) | |
102850.w2 | 102850n1 | \([1, -1, 0, -133667, 20502741]\) | \(-9541617561/1017280\) | \(-28158962095000000\) | \([2]\) | \(967680\) | \(1.8949\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 102850.w have rank \(1\).
Complex multiplication
The elliptic curves in class 102850.w do not have complex multiplication.Modular form 102850.2.a.w
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.