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Properties

Label 17136y
Number of curves $6$
Conductor $17136$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("y1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 17136y have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(7\)\(1 + T\)
\(17\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - T + 5 T^{2}\) 1.5.ab
\(11\) \( 1 + 3 T + 11 T^{2}\) 1.11.d
\(13\) \( 1 + 5 T + 13 T^{2}\) 1.13.f
\(19\) \( 1 - 6 T + 19 T^{2}\) 1.19.ag
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 - 4 T + 29 T^{2}\) 1.29.ae
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 17136y do not have complex multiplication.

Modular form 17136.2.a.y

Copy content sage:E.q_eigenform(10)
 
\(q + 2 q^{5} - q^{7} + 4 q^{11} - 2 q^{13} - q^{17} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.

Elliptic curves in class 17136y

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
17136.bg5 17136y1 \([0, 0, 0, -10115139, 12366939202]\) \(38331145780597164097/55468445663232\) \(165627891255280140288\) \([2]\) \(737280\) \(2.7819\) \(\Gamma_0(N)\)-optimal
17136.bg4 17136y2 \([0, 0, 0, -13064259, 4567696450]\) \(82582985847542515777/44772582831427584\) \(133690215973317462982656\) \([2, 2]\) \(1474560\) \(3.1284\)  
17136.bg2 17136y3 \([0, 0, 0, -123702339, -525941897150]\) \(70108386184777836280897/552468975892674624\) \(1649663522511912144470016\) \([2, 2]\) \(2949120\) \(3.4750\)  
17136.bg6 17136y4 \([0, 0, 0, 50387901, 35925753922]\) \(4738217997934888496063/2928751705237796928\) \(-8745205731812777822257152\) \([2]\) \(2949120\) \(3.4750\)  
17136.bg1 17136y5 \([0, 0, 0, -1975478979, -33795331366718]\) \(285531136548675601769470657/17941034271597192\) \(53571641278440869756928\) \([2]\) \(5898240\) \(3.8216\)  
17136.bg3 17136y6 \([0, 0, 0, -42134979, -1209166417982]\) \(-2770540998624539614657/209924951154647363208\) \(-626832545348558552181276672\) \([2]\) \(5898240\) \(3.8216\)