↑

Properties

Label 76440cs
Number of curves $6$
Conductor $76440$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve("cs1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 76440cs have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1 - T\)
\(7\)\(1\)
\(13\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 8 T + 19 T^{2}\) 1.19.i
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 76440cs do not have complex multiplication.

Modular form 76440.2.a.cs

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 76440cs

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
76440.bn5 76440cs1 \([0, 1, 0, -36031, -3451246]\) \(-2748251600896/1124136195\) \(-2116055987288880\) \([2]\) \(393216\) \(1.6491\) \(\Gamma_0(N)\)-optimal
76440.bn4 76440cs2 \([0, 1, 0, -624276, -190042560]\) \(893359210685776/91298025\) \(2749727063865600\) \([2, 2]\) \(786432\) \(1.9956\)  
76440.bn3 76440cs3 \([0, 1, 0, -672296, -159156096]\) \(278944461825124/70849130625\) \(8535377273754240000\) \([2, 2]\) \(1572864\) \(2.3422\)  
76440.bn1 76440cs4 \([0, 1, 0, -9988176, -12153361200]\) \(914732517663095044/9555\) \(1151115463680\) \([2]\) \(1572864\) \(2.3422\)  
76440.bn6 76440cs5 \([0, 1, 0, 1646384, -1014285280]\) \(2048324060764798/3031899609375\) \(-730521512229600000000\) \([2]\) \(3145728\) \(2.6888\)  
76440.bn2 76440cs6 \([0, 1, 0, -3759296, 2673475104]\) \(24385137179326562/1284775885575\) \(309560521035802982400\) \([2]\) \(3145728\) \(2.6888\)