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Properties

Label 16170bn
Number of curves $8$
Conductor $16170$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bn1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 16170bn have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 + T\)
\(5\)\(1 - T\)
\(7\)\(1\)
\(11\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 16170bn do not have complex multiplication.

Modular form 16170.2.a.bn

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} + q^{9} + q^{10} - q^{11} - q^{12} - 2 q^{13} - q^{15} + q^{16} - 6 q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 16170bn

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
16170.bn8 16170bn1 \([1, 1, 1, 77860, -1014595]\) \(443688652450511/260789760000\) \(-30681654474240000\) \([4]\) \(165888\) \(1.8529\) \(\Gamma_0(N)\)-optimal
16170.bn7 16170bn2 \([1, 1, 1, -314140, -8540995]\) \(29141055407581489/16604321025600\) \(1953481764340814400\) \([2, 2]\) \(331776\) \(2.1995\)  
16170.bn6 16170bn3 \([1, 1, 1, -992300, 417302717]\) \(-918468938249433649/109183593750000\) \(-12845340621093750000\) \([4]\) \(497664\) \(2.4022\)  
16170.bn4 16170bn4 \([1, 1, 1, -3675540, -2708417475]\) \(46676570542430835889/106752955783320\) \(12559378494951814680\) \([2]\) \(663552\) \(2.5461\)  
16170.bn5 16170bn5 \([1, 1, 1, -3224740, 2217485885]\) \(31522423139920199089/164434491947880\) \(19345553543176134120\) \([2]\) \(663552\) \(2.5461\)  
16170.bn3 16170bn6 \([1, 1, 1, -16304800, 25333802717]\) \(4074571110566294433649/48828650062500\) \(5744641851203062500\) \([2, 2]\) \(995328\) \(2.7488\)  
16170.bn2 16170bn7 \([1, 1, 1, -16733550, 23930589717]\) \(4404531606962679693649/444872222400201750\) \(52338772093161335685750\) \([2]\) \(1990656\) \(3.0954\)  
16170.bn1 16170bn8 \([1, 1, 1, -260876050, 1621699265717]\) \(16689299266861680229173649/2396798250\) \(281980917314250\) \([2]\) \(1990656\) \(3.0954\)