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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 2240k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2240.p3 | 2240k1 | \([0, 0, 0, -47, 124]\) | \(179406144/35\) | \(2240\) | \([2]\) | \(192\) | \(-0.35741\) | \(\Gamma_0(N)\)-optimal |
2240.p2 | 2240k2 | \([0, 0, 0, -52, 96]\) | \(3796416/1225\) | \(5017600\) | \([2, 2]\) | \(384\) | \(-0.010833\) | |
2240.p1 | 2240k3 | \([0, 0, 0, -332, -2256]\) | \(123505992/4375\) | \(143360000\) | \([2]\) | \(768\) | \(0.33574\) | |
2240.p4 | 2240k4 | \([0, 0, 0, 148, 656]\) | \(10941048/12005\) | \(-393379840\) | \([4]\) | \(768\) | \(0.33574\) |
Rank
sage: E.rank()
The elliptic curves in class 2240k have rank \(1\).
Complex multiplication
The elliptic curves in class 2240k 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 Cremona 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 Cremona labels.