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Properties

Label 164730.ci
Number of curves $4$
Conductor $164730$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ci1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 164730.ci have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 164730.ci do not have complex multiplication.

Modular form 164730.2.a.ci

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + 2 q^{7} + q^{8} + q^{9} - q^{10} - 2 q^{11} - q^{12} + 4 q^{13} + 2 q^{14} + q^{15} + q^{16} + q^{18} - q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 164730.ci

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
164730.ci1 164730bd4 \([1, 1, 1, -15265975611, -726003584520867]\) \(16300610738133468173382620881/2228489100\) \(53790309416997900\) \([2]\) \(122880000\) \(4.0254\)  
164730.ci2 164730bd3 \([1, 1, 1, -954123391, -11344106285611]\) \(-3979640234041473454886161/1471455901872240\) \(-35517368361898422184560\) \([2]\) \(61440000\) \(3.6789\)  
164730.ci3 164730bd2 \([1, 1, 1, -25416111, -42501389067]\) \(75224183150104868881/11219310000000000\) \(270806869257390000000000\) \([2]\) \(24576000\) \(3.2207\)  
164730.ci4 164730bd1 \([1, 1, 1, 2697809, -3625460491]\) \(89962967236397039/287450726400000\) \(-6938361742580121600000\) \([2]\) \(12288000\) \(2.8741\) \(\Gamma_0(N)\)-optimal