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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("du1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 69360du 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.ad
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 - T + 13 T^{2}\) 1.13.ab
\(19\) \( 1 + T + 19 T^{2}\) 1.19.b
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 - 4 T + 29 T^{2}\) 1.29.ae
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 69360.2.a.du

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} + q^{5} - 4 q^{7} + q^{9} + 2 q^{13} + q^{15} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 69360du

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
69360.cw8 69360du1 \([0, 1, 0, 6840, -666252]\) \(357911/2160\) \(-213553762467840\) \([2]\) \(221184\) \(1.4315\) \(\Gamma_0(N)\)-optimal
69360.cw6 69360du2 \([0, 1, 0, -85640, -8767500]\) \(702595369/72900\) \(7207439483289600\) \([2, 2]\) \(442368\) \(1.7780\)  
69360.cw7 69360du3 \([0, 1, 0, -62520, 19670100]\) \(-273359449/1536000\) \(-151860453310464000\) \([2]\) \(663552\) \(1.9808\)  
69360.cw5 69360du4 \([0, 1, 0, -316840, 59020340]\) \(35578826569/5314410\) \(525422338331811840\) \([2]\) \(884736\) \(2.1246\)  
69360.cw4 69360du5 \([0, 1, 0, -1334120, -593555532]\) \(2656166199049/33750\) \(3336777538560000\) \([2]\) \(884736\) \(2.1246\)  
69360.cw3 69360du6 \([0, 1, 0, -1542200, 735243348]\) \(4102915888729/9000000\) \(889807343616000000\) \([2, 2]\) \(1327104\) \(2.3273\)  
69360.cw1 69360du7 \([0, 1, 0, -24662200, 47132459348]\) \(16778985534208729/81000\) \(8008266092544000\) \([2]\) \(2654208\) \(2.6739\)  
69360.cw2 69360du8 \([0, 1, 0, -2097080, 158390100]\) \(10316097499609/5859375000\) \(579301656000000000000\) \([2]\) \(2654208\) \(2.6739\)