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Properties

Label 29040co
Number of curves $6$
Conductor $29040$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("co1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 29040co have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1 + T\)
\(11\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 3 T + 7 T^{2}\) 1.7.ad
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 - T + 19 T^{2}\) 1.19.ab
\(23\) \( 1 - 2 T + 23 T^{2}\) 1.23.ac
\(29\) \( 1 - 8 T + 29 T^{2}\) 1.29.ai
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 29040co do not have complex multiplication.

Modular form 29040.2.a.co

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} + q^{5} + q^{9} + 2 q^{13} - q^{15} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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 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 29040co

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
29040.bm5 29040co1 \([0, -1, 0, 9640, 1409520]\) \(13651919/126720\) \(-919520091832320\) \([2]\) \(92160\) \(1.5525\) \(\Gamma_0(N)\)-optimal
29040.bm4 29040co2 \([0, -1, 0, -145240, 19747312]\) \(46694890801/3920400\) \(28447652841062400\) \([2, 2]\) \(184320\) \(1.8990\)  
29040.bm3 29040co3 \([0, -1, 0, -493720, -110723600]\) \(1834216913521/329422500\) \(2390393051228160000\) \([2, 2]\) \(368640\) \(2.2456\)  
29040.bm2 29040co4 \([0, -1, 0, -2274840, 1321358832]\) \(179415687049201/1443420\) \(10473908546027520\) \([2]\) \(368640\) \(2.2456\)  
29040.bm6 29040co5 \([0, -1, 0, 958280, -640413200]\) \(13411719834479/32153832150\) \(-233318297753543270400\) \([2]\) \(737280\) \(2.5922\)  
29040.bm1 29040co6 \([0, -1, 0, -7521400, -7936748048]\) \(6484907238722641/283593750\) \(2057845257600000000\) \([2]\) \(737280\) \(2.5922\)