↑

Properties

Label 21840cl
Number of curves $8$
Conductor $21840$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve("cl1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 21840cl have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1 - T\)
\(7\)\(1 - T\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 2 T + 11 T^{2}\) 1.11.ac
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 + 4 T + 29 T^{2}\) 1.29.e
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 21840cl do not have complex multiplication.

Modular form 21840.2.a.cl

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} + q^{5} + q^{7} + q^{9} - 4 q^{11} + q^{13} + q^{15} + 2 q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 21840cl

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
21840.cg7 21840cl1 \([0, 1, 0, -348160, -80234572]\) \(-1139466686381936641/17587891077120\) \(-72040001851883520\) \([2]\) \(196608\) \(2.0368\) \(\Gamma_0(N)\)-optimal
21840.cg5 21840cl2 \([0, 1, 0, -5591040, -5090330700]\) \(4718909406724749250561/1098974822400\) \(4501400872550400\) \([2, 2]\) \(393216\) \(2.3834\)  
21840.cg4 21840cl3 \([0, 1, 0, -5611520, -5051181132]\) \(4770955732122964500481/71987251059360000\) \(294859780339138560000\) \([2, 4]\) \(786432\) \(2.7300\)  
21840.cg2 21840cl4 \([0, 1, 0, -89456640, -325691746380]\) \(19328649688935739391016961/1048320\) \(4293918720\) \([2]\) \(786432\) \(2.7300\)  
21840.cg6 21840cl5 \([0, 1, 0, -531200, -13852327500]\) \(-4047051964543660801/20235220197806250000\) \(-82883461930214400000000\) \([4]\) \(1572864\) \(3.0766\)  
21840.cg3 21840cl6 \([0, 1, 0, -11019520, 6255865268]\) \(36128658497509929012481/16775330746084419600\) \(68711754735961782681600\) \([2, 4]\) \(1572864\) \(3.0766\)  
21840.cg8 21840cl7 \([0, 1, 0, 38921280, 47267250228]\) \(1591934139020114746758719/1156766383092650262660\) \(-4738115105147495475855360\) \([4]\) \(3145728\) \(3.4231\)  
21840.cg1 21840cl8 \([0, 1, 0, -147488320, 688981977908]\) \(86623684689189325642735681/56690726941459561860\) \(232205217552218365378560\) \([4]\) \(3145728\) \(3.4231\)