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Properties

Label 52800fb
Number of curves $4$
Conductor $52800$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("fb1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 52800fb have rank \(1\).

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 \(\mathbb{F}_p\)
\(7\) \( 1 + 2 T + 7 T^{2}\) 1.7.c
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 52800.2.a.fb

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{7} + q^{9} + q^{11} + 4 q^{13} + 2 q^{17} + O(q^{20})\) 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 52800fb

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
52800.bm3 52800fb1 \([0, -1, 0, -72033, -5400063]\) \(10091699281/2737152\) \(11211374592000000\) \([2]\) \(307200\) \(1.7876\) \(\Gamma_0(N)\)-optimal
52800.bm4 52800fb2 \([0, -1, 0, 183967, -35352063]\) \(168105213359/228637728\) \(-936500133888000000\) \([2]\) \(614400\) \(2.1341\)  
52800.bm1 52800fb3 \([0, -1, 0, -16104033, 24879623937]\) \(112763292123580561/1932612\) \(7915978752000000\) \([2]\) \(1536000\) \(2.5923\)  
52800.bm2 52800fb4 \([0, -1, 0, -16088033, 24931511937]\) \(-112427521449300721/466873642818\) \(-1912314440982528000000\) \([2]\) \(3072000\) \(2.9389\)