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SageMath
E = EllipticCurve("cg1")
E.isogeny_class()
Elliptic curves in class 67760cg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
67760.bo4 | 67760cg1 | \([0, 0, 0, -56507, -54712086]\) | \(-2749884201/176619520\) | \(-1281606670216069120\) | \([2]\) | \(737280\) | \(2.1543\) | \(\Gamma_0(N)\)-optimal |
67760.bo3 | 67760cg2 | \([0, 0, 0, -2534587, -1543046934]\) | \(248158561089321/1859334400\) | \(13491913969657446400\) | \([2, 2]\) | \(1474560\) | \(2.5009\) | |
67760.bo2 | 67760cg3 | \([0, 0, 0, -4238267, 792016874]\) | \(1160306142246441/634128110000\) | \(4601432591072092160000\) | \([4]\) | \(2949120\) | \(2.8475\) | |
67760.bo1 | 67760cg4 | \([0, 0, 0, -40480187, -99131541014]\) | \(1010962818911303721/57392720\) | \(416459589369528320\) | \([2]\) | \(2949120\) | \(2.8475\) |
Rank
sage: E.rank()
The elliptic curves in class 67760cg have rank \(1\).
Complex multiplication
The elliptic curves in class 67760cg do not have complex multiplication.Modular form 67760.2.a.cg
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.