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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 17136.bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 17136.bg1 | 17136y5 | \([0, 0, 0, -1975478979, -33795331366718]\) | \(285531136548675601769470657/17941034271597192\) | \(53571641278440869756928\) | \([2]\) | \(5898240\) | \(3.8216\) | |
| 17136.bg2 | 17136y3 | \([0, 0, 0, -123702339, -525941897150]\) | \(70108386184777836280897/552468975892674624\) | \(1649663522511912144470016\) | \([2, 2]\) | \(2949120\) | \(3.4750\) | |
| 17136.bg3 | 17136y6 | \([0, 0, 0, -42134979, -1209166417982]\) | \(-2770540998624539614657/209924951154647363208\) | \(-626832545348558552181276672\) | \([2]\) | \(5898240\) | \(3.8216\) | |
| 17136.bg4 | 17136y2 | \([0, 0, 0, -13064259, 4567696450]\) | \(82582985847542515777/44772582831427584\) | \(133690215973317462982656\) | \([2, 2]\) | \(1474560\) | \(3.1284\) | |
| 17136.bg5 | 17136y1 | \([0, 0, 0, -10115139, 12366939202]\) | \(38331145780597164097/55468445663232\) | \(165627891255280140288\) | \([2]\) | \(737280\) | \(2.7819\) | \(\Gamma_0(N)\)-optimal |
| 17136.bg6 | 17136y4 | \([0, 0, 0, 50387901, 35925753922]\) | \(4738217997934888496063/2928751705237796928\) | \(-8745205731812777822257152\) | \([2]\) | \(2949120\) | \(3.4750\) |
Rank
sage: E.rank()
The elliptic curves in class 17136.bg have rank \(1\).
Complex multiplication
The elliptic curves in class 17136.bg do not have complex multiplication.Modular form 17136.2.a.bg
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
