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Properties

Label 5202b
Number of curves $6$
Conductor $5202$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve([1, -1, 0, -88488, -9374144]) E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 5202b have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 3 T + 5 T^{2}\) 1.5.d
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(11\) \( 1 - 3 T + 11 T^{2}\) 1.11.ad
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(19\) \( 1 - 8 T + 19 T^{2}\) 1.19.ai
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
\(29\) \( 1 - 3 T + 29 T^{2}\) 1.29.ad
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 5202b do not have complex multiplication.

Modular form 5202.2.a.b

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 2 q^{5} - q^{8} + 2 q^{10} - 4 q^{11} - 2 q^{13} + q^{16} + 4 q^{19} + O(q^{20})\) 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 5202b

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5202.c5 5202b1 \([1, -1, 0, -88488, -9374144]\) \(4354703137/352512\) \(6202902605306112\) \([2]\) \(36864\) \(1.7733\) \(\Gamma_0(N)\)-optimal
5202.c4 5202b2 \([1, -1, 0, -296568, 51343600]\) \(163936758817/30338064\) \(533837305469157264\) \([2, 2]\) \(73728\) \(2.1198\)  
5202.c2 5202b3 \([1, -1, 0, -4510188, 3687697660]\) \(576615941610337/27060804\) \(476169695310452004\) \([2, 2]\) \(147456\) \(2.4664\)  
5202.c6 5202b4 \([1, -1, 0, 587772, 298428196]\) \(1276229915423/2927177028\) \(-51507449429163835428\) \([2]\) \(147456\) \(2.4664\)  
5202.c1 5202b5 \([1, -1, 0, -72162198, 235964108794]\) \(2361739090258884097/5202\) \(91535889140802\) \([2]\) \(294912\) \(2.8130\)  
5202.c3 5202b6 \([1, -1, 0, -4276098, 4087382926]\) \(-491411892194497/125563633938\) \(-2209453840112458990338\) \([2]\) \(294912\) \(2.8130\)