↑

Properties

Label 29370bp
Number of curves $4$
Conductor $29370$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve([1, 0, 0, -11842820, 15197648400]) E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 29370bp have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(5\)\(1 - T\)
\(11\)\(1 + T\)
\(89\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 3 T + 7 T^{2}\) 1.7.ad
\(13\) \( 1 - 3 T + 13 T^{2}\) 1.13.ad
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + T + 23 T^{2}\) 1.23.b
\(29\) \( 1 - 5 T + 29 T^{2}\) 1.29.af
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 29370bp do not have complex multiplication.

Modular form 29370.2.a.bp

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - 2 q^{7} + q^{8} + q^{9} + q^{10} + q^{11} + q^{12} - 6 q^{13} - 2 q^{14} + q^{15} + q^{16} - 2 q^{17} + q^{18} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 29370bp

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
29370.bm4 29370bp1 \([1, 0, 0, -11842820, 15197648400]\) \(183691516586815867994210881/6511602493440000000000\) \(6511602493440000000000\) \([10]\) \(2560000\) \(2.9563\) \(\Gamma_0(N)\)-optimal
29370.bm3 29370bp2 \([1, 0, 0, -187842820, 990906448400]\) \(733005968209216932163418210881/972182437259299200000\) \(972182437259299200000\) \([10]\) \(5120000\) \(3.3029\)  
29370.bm2 29370bp3 \([1, 0, 0, -1048084820, -13060028592000]\) \(127324800640445734294052812418881/521410035333380777456400\) \(521410035333380777456400\) \([2]\) \(12800000\) \(3.7610\)  
29370.bm1 29370bp4 \([1, 0, 0, -1064189920, -12637949352220]\) \(133284956652244710243152075681281/8135392425834393812901934620\) \(8135392425834393812901934620\) \([2]\) \(25600000\) \(4.1076\)