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SageMath
E = EllipticCurve("bx1")
E.isogeny_class()
Elliptic curves in class 101640.bx
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
101640.bx1 | 101640cm4 | \([0, 1, 0, -597296, 177476880]\) | \(12990838708516/144375\) | \(261907578240000\) | \([2]\) | \(737280\) | \(1.9200\) | |
101640.bx2 | 101640cm2 | \([0, 1, 0, -38276, 2615424]\) | \(13674725584/1334025\) | \(605006505734400\) | \([2, 2]\) | \(368640\) | \(1.5735\) | |
101640.bx3 | 101640cm1 | \([0, 1, 0, -8631, -266070]\) | \(2508888064/396165\) | \(11229287417040\) | \([2]\) | \(184320\) | \(1.2269\) | \(\Gamma_0(N)\)-optimal |
101640.bx4 | 101640cm3 | \([0, 1, 0, 46424, 12643904]\) | \(6099383804/41507235\) | \(-75297381113687040\) | \([2]\) | \(737280\) | \(1.9200\) |
Rank
sage: E.rank()
The elliptic curves in class 101640.bx have rank \(1\).
Complex multiplication
The elliptic curves in class 101640.bx do not have complex multiplication.Modular form 101640.2.a.bx
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.