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Properties

Label 159936.z
Number of curves $6$
Conductor $159936$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("z1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 159936.z have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(7\)\(1\)
\(17\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 10 T + 29 T^{2}\) 1.29.ak
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 159936.z do not have complex multiplication.

Modular form 159936.2.a.z

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{5} + q^{9} - 4 q^{11} - 2 q^{13} + 2 q^{15} - q^{17} - 4 q^{19} + O(q^{20})\) 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 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

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 159936.z

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
159936.z1 159936cy6 \([0, -1, 0, -87005249, -312338826015]\) \(2361739090258884097/5202\) \(160434775130112\) \([2]\) \(9437184\) \(2.8598\)  
159936.z2 159936cy4 \([0, -1, 0, -5437889, -4878819231]\) \(576615941610337/27060804\) \(834581700226842624\) \([2, 2]\) \(4718592\) \(2.5132\)  
159936.z3 159936cy5 \([0, -1, 0, -5155649, -5408132127]\) \(-491411892194497/125563633938\) \(-3872505454702562377728\) \([2]\) \(9437184\) \(2.8598\)  
159936.z4 159936cy2 \([0, -1, 0, -357569, -67756191]\) \(163936758817/30338064\) \(935655608558813184\) \([2, 2]\) \(2359296\) \(2.1666\)  
159936.z5 159936cy1 \([0, -1, 0, -106689, 12475233]\) \(4354703137/352512\) \(10871815349993472\) \([2]\) \(1179648\) \(1.8200\) \(\Gamma_0(N)\)-optimal
159936.z6 159936cy3 \([0, -1, 0, 708671, -395518367]\) \(1276229915423/2927177028\) \(-90277006584623136768\) \([2]\) \(4718592\) \(2.5132\)