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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 2240.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2240.k1 | 2240m3 | \([0, -1, 0, -525, -4673]\) | \(-250523582464/13671875\) | \(-875000000\) | \([]\) | \(864\) | \(0.47404\) | |
2240.k2 | 2240m1 | \([0, -1, 0, -5, 7]\) | \(-262144/35\) | \(-2240\) | \([]\) | \(96\) | \(-0.62458\) | \(\Gamma_0(N)\)-optimal |
2240.k3 | 2240m2 | \([0, -1, 0, 35, -25]\) | \(71991296/42875\) | \(-2744000\) | \([]\) | \(288\) | \(-0.075270\) |
Rank
sage: E.rank()
The elliptic curves in class 2240.k have rank \(1\).
Complex multiplication
The elliptic curves in class 2240.k do not have complex multiplication.Modular form 2240.2.a.k
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.
\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.