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Properties

Label 990.b
Number of curves $6$
Conductor $990$
CM no
Rank $0$
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 990.b have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(5\)\(1 + T\)
\(11\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 10 T + 29 T^{2}\) 1.29.ak
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 990.b do not have complex multiplication.

Modular form 990.2.a.b

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{5} - q^{8} + q^{10} - q^{11} + 6 q^{13} + q^{16} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 990.b

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
990.b1 990c5 \([1, -1, 0, -1539765, 735795481]\) \(553808571467029327441/12529687500\) \(9134142187500\) \([2]\) \(12288\) \(2.0111\)  
990.b2 990c3 \([1, -1, 0, -106425, -13307459]\) \(182864522286982801/463015182960\) \(337538068377840\) \([2]\) \(6144\) \(1.6645\)  
990.b3 990c4 \([1, -1, 0, -96345, 11487325]\) \(135670761487282321/643043610000\) \(468778791690000\) \([2, 2]\) \(6144\) \(1.6645\)  
990.b4 990c6 \([1, -1, 0, -46845, 23238625]\) \(-15595206456730321/310672490129100\) \(-226480245304113900\) \([2]\) \(12288\) \(2.0111\)  
990.b5 990c2 \([1, -1, 0, -9225, -29939]\) \(119102750067601/68309049600\) \(49797297158400\) \([2, 2]\) \(3072\) \(1.3179\)  
990.b6 990c1 \([1, -1, 0, 2295, -4595]\) \(1833318007919/1070530560\) \(-780416778240\) \([2]\) \(1536\) \(0.97134\) \(\Gamma_0(N)\)-optimal