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SageMath
E = EllipticCurve("ct1")
E.isogeny_class()
Elliptic curves in class 8400.ct
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8400.ct1 | 8400ci4 | \([0, 1, 0, -45708, 3746088]\) | \(2640279346000/3087\) | \(12348000000\) | \([2]\) | \(20736\) | \(1.2195\) | |
8400.ct2 | 8400ci3 | \([0, 1, 0, -2833, 58838]\) | \(-10061824000/352947\) | \(-88236750000\) | \([2]\) | \(10368\) | \(0.87295\) | |
8400.ct3 | 8400ci2 | \([0, 1, 0, -708, 2088]\) | \(9826000/5103\) | \(20412000000\) | \([2]\) | \(6912\) | \(0.67022\) | |
8400.ct4 | 8400ci1 | \([0, 1, 0, 167, 338]\) | \(2048000/1323\) | \(-330750000\) | \([2]\) | \(3456\) | \(0.32365\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8400.ct have rank \(0\).
Complex multiplication
The elliptic curves in class 8400.ct do not have complex multiplication.Modular form 8400.2.a.ct
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.