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Properties

Label 21840cg
Number of curves $6$
Conductor $21840$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cg1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 21840cg have rank \(1\).

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 \(\mathbb{F}_p\)
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 21840.2.a.cg

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} + q^{5} - q^{7} + q^{9} + 4 q^{11} + q^{13} + q^{15} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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 21840cg

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
21840.cc5 21840cg1 \([0, 1, 0, -5920, -2764300]\) \(-5602762882081/801531494400\) \(-3283073001062400\) \([2]\) \(147456\) \(1.6567\) \(\Gamma_0(N)\)-optimal
21840.cc4 21840cg2 \([0, 1, 0, -333600, -73674252]\) \(1002404925316922401/9348917760000\) \(38293167144960000\) \([2, 2]\) \(294912\) \(2.0033\)  
21840.cc3 21840cg3 \([0, 1, 0, -584480, 51866100]\) \(5391051390768345121/2833965225000000\) \(11607921561600000000\) \([2, 4]\) \(589824\) \(2.3498\)  
21840.cc2 21840cg4 \([0, 1, 0, -5325600, -4732208652]\) \(4078208988807294650401/359723582400\) \(1473427793510400\) \([2]\) \(589824\) \(2.3498\)  
21840.cc6 21840cg5 \([0, 1, 0, 2215520, 406906100]\) \(293623352309352854879/187320324116835000\) \(-767264047582556160000\) \([8]\) \(1179648\) \(2.6964\)  
21840.cc1 21840cg6 \([0, 1, 0, -7398560, 7735422708]\) \(10934663514379917006241/12996826171875000\) \(53235000000000000000\) \([4]\) \(1179648\) \(2.6964\)