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Properties

Label 52800.cp
Number of curves $6$
Conductor $52800$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cp1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 52800.cp have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(5\)\(1\)
\(11\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 52800.cp do not have complex multiplication.

Modular form 52800.2.a.cp

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} + q^{9} + q^{11} - 2 q^{13} + 6 q^{17} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 52800.cp

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
52800.cp1 52800x4 \([0, -1, 0, -5280033, 4671611937]\) \(15897679904620804/2475\) \(2534400000000\) \([2]\) \(786432\) \(2.2267\)  
52800.cp2 52800x6 \([0, -1, 0, -2800033, -1767908063]\) \(1185450336504002/26043266205\) \(53336609187840000000\) \([2]\) \(1572864\) \(2.5733\)  
52800.cp3 52800x3 \([0, -1, 0, -380033, 49511937]\) \(5927735656804/2401490025\) \(2459125785600000000\) \([2, 2]\) \(786432\) \(2.2267\)  
52800.cp4 52800x2 \([0, -1, 0, -330033, 73061937]\) \(15529488955216/6125625\) \(1568160000000000\) \([2, 2]\) \(393216\) \(1.8801\)  
52800.cp5 52800x1 \([0, -1, 0, -17533, 1499437]\) \(-37256083456/38671875\) \(-618750000000000\) \([2]\) \(196608\) \(1.5335\) \(\Gamma_0(N)\)-optimal
52800.cp6 52800x5 \([0, -1, 0, 1239967, 358931937]\) \(102949393183198/86815346805\) \(-177797830256640000000\) \([4]\) \(1572864\) \(2.5733\)