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Properties

Label 69360cb
Number of curves $8$
Conductor $69360$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cb1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 69360cb have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 69360cb do not have complex multiplication.

Modular form 69360.2.a.cb

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{9} + 4 q^{11} - 2 q^{13} + q^{15} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 69360cb

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
69360.q6 69360cb1 \([0, -1, 0, -370016, -106318080]\) \(-56667352321/16711680\) \(-1652241732017848320\) \([2]\) \(884736\) \(2.2102\) \(\Gamma_0(N)\)-optimal
69360.q5 69360cb2 \([0, -1, 0, -6288736, -6067652864]\) \(278202094583041/16646400\) \(1645787662752153600\) \([2, 2]\) \(1769472\) \(2.5568\)  
69360.q4 69360cb3 \([0, -1, 0, -6658656, -5313312000]\) \(330240275458561/67652010000\) \(6688583923153674240000\) \([2, 2]\) \(3538944\) \(2.9033\)  
69360.q2 69360cb4 \([0, -1, 0, -100618336, -388442119424]\) \(1139466686381936641/4080\) \(403379329105920\) \([2]\) \(3538944\) \(2.9033\)  
69360.q7 69360cb5 \([0, -1, 0, 14149344, -31897612800]\) \(3168685387909439/6278181696900\) \(-620708019828575584665600\) \([2]\) \(7077888\) \(3.2499\)  
69360.q3 69360cb6 \([0, -1, 0, -33385376, 69542885376]\) \(41623544884956481/2962701562500\) \(292914845250566400000000\) \([2, 2]\) \(7077888\) \(3.2499\)  
69360.q8 69360cb7 \([0, -1, 0, 30287104, 303399169920]\) \(31077313442863199/420227050781250\) \(-41546790641250000000000000\) \([2]\) \(14155776\) \(3.5965\)  
69360.q1 69360cb8 \([0, -1, 0, -524685376, 4626055605376]\) \(161572377633716256481/914742821250\) \(90438319985363112960000\) \([2]\) \(14155776\) \(3.5965\)