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Properties

Label 510.e
Number of curves $8$
Conductor $510$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("e1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 510.e have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 510.e do not have complex multiplication.

Modular form 510.2.a.e

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} + q^{9} + q^{10} + 4 q^{11} - q^{12} - 2 q^{13} - q^{15} + q^{16} + q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 16 & 2 & 4 & 8 & 16 & 8 & 4 \\ 16 & 1 & 8 & 4 & 2 & 4 & 8 & 16 \\ 2 & 8 & 1 & 2 & 4 & 8 & 4 & 2 \\ 4 & 4 & 2 & 1 & 2 & 4 & 2 & 4 \\ 8 & 2 & 4 & 2 & 1 & 2 & 4 & 8 \\ 16 & 4 & 8 & 4 & 2 & 1 & 8 & 16 \\ 8 & 8 & 4 & 2 & 4 & 8 & 1 & 8 \\ 4 & 16 & 2 & 4 & 8 & 16 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 510.e

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
510.e1 510e7 \([1, 1, 1, -113470, -14759143]\) \(161572377633716256481/914742821250\) \(914742821250\) \([2]\) \(2048\) \(1.4867\)  
510.e2 510e4 \([1, 1, 1, -21760, 1226417]\) \(1139466686381936641/4080\) \(4080\) \([4]\) \(512\) \(0.79359\)  
510.e3 510e5 \([1, 1, 1, -7220, -224143]\) \(41623544884956481/2962701562500\) \(2962701562500\) \([2, 2]\) \(1024\) \(1.1402\)  
510.e4 510e3 \([1, 1, 1, -1440, 16305]\) \(330240275458561/67652010000\) \(67652010000\) \([2, 4]\) \(512\) \(0.79359\)  
510.e5 510e2 \([1, 1, 1, -1360, 18737]\) \(278202094583041/16646400\) \(16646400\) \([2, 4]\) \(256\) \(0.44702\)  
510.e6 510e1 \([1, 1, 1, -80, 305]\) \(-56667352321/16711680\) \(-16711680\) \([4]\) \(128\) \(0.10045\) \(\Gamma_0(N)\)-optimal
510.e7 510e6 \([1, 1, 1, 3060, 102705]\) \(3168685387909439/6278181696900\) \(-6278181696900\) \([4]\) \(1024\) \(1.1402\)  
510.e8 510e8 \([1, 1, 1, 6550, -962215]\) \(31077313442863199/420227050781250\) \(-420227050781250\) \([2]\) \(2048\) \(1.4867\)