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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 25242d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25242.e1 | 25242d1 | \([1, 0, 1, -550, -4744]\) | \(18351695644633/1085607936\) | \(1085607936\) | \([2]\) | \(16128\) | \(0.48691\) | \(\Gamma_0(N)\)-optimal |
25242.e2 | 25242d2 | \([1, 0, 1, 410, -19336]\) | \(7648866341927/166510771392\) | \(-166510771392\) | \([2]\) | \(32256\) | \(0.83348\) |
Rank
sage: E.rank()
The elliptic curves in class 25242d have rank \(1\).
Complex multiplication
The elliptic curves in class 25242d do not have complex multiplication.Modular form 25242.2.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 Cremona 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 Cremona labels.