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Properties

Label 25872cs
Number of curves $4$
Conductor $25872$
CM no
Rank $0$
Graph

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

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 25872cs have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(7\)\(1\)
\(11\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 - 4 T + 19 T^{2}\) 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 25872cs do not have complex multiplication.

Modular form 25872.2.a.cs

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

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 25872cs

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
25872.cy3 25872cs1 \([0, 1, 0, -35296, 1848692]\) \(10091699281/2737152\) \(1319007009374208\) \([2]\) \(172800\) \(1.6092\) \(\Gamma_0(N)\)-optimal
25872.cy4 25872cs2 \([0, 1, 0, 90144, 12134772]\) \(168105213359/228637728\) \(-110178304251789312\) \([2]\) \(345600\) \(1.9558\)  
25872.cy1 25872cs3 \([0, 1, 0, -7890976, -8534500108]\) \(112763292123580561/1932612\) \(931306984194048\) \([2]\) \(864000\) \(2.4140\)  
25872.cy2 25872cs4 \([0, 1, 0, -7883136, -8552296908]\) \(-112427521449300721/466873642818\) \(-224981881667153436672\) \([2]\) \(1728000\) \(2.7605\)