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Properties

Label 2550b
Number of curves $6$
Conductor $2550$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("b1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 2550b have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(5\)\(1\)
\(17\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 2 T + 7 T^{2}\) 1.7.c
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 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 2550b do not have complex multiplication.

Modular form 2550.2.a.b

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} - 4 q^{11} - q^{12} + 2 q^{13} + q^{16} - q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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 2550b

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2550.c5 2550b1 \([1, 1, 0, -850, 8500]\) \(4354703137/352512\) \(5508000000\) \([2]\) \(2048\) \(0.61208\) \(\Gamma_0(N)\)-optimal
2550.c4 2550b2 \([1, 1, 0, -2850, -49500]\) \(163936758817/30338064\) \(474032250000\) \([2, 2]\) \(4096\) \(0.95865\)  
2550.c2 2550b3 \([1, 1, 0, -43350, -3492000]\) \(576615941610337/27060804\) \(422825062500\) \([2, 2]\) \(8192\) \(1.3052\)  
2550.c6 2550b4 \([1, 1, 0, 5650, -279000]\) \(1276229915423/2927177028\) \(-45737141062500\) \([2]\) \(8192\) \(1.3052\)  
2550.c1 2550b5 \([1, 1, 0, -693600, -222626250]\) \(2361739090258884097/5202\) \(81281250\) \([2]\) \(16384\) \(1.6518\)  
2550.c3 2550b6 \([1, 1, 0, -41100, -3867750]\) \(-491411892194497/125563633938\) \(-1961931780281250\) \([2]\) \(16384\) \(1.6518\)