Properties

Label 72128.bg
Number of curves $4$
Conductor $72128$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bg1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 72128.bg have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(7\)\(1\)
\(23\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + 3 T^{2}\) 1.3.a
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 72128.bg do not have complex multiplication.

Modular form 72128.2.a.bg

Copy content sage:E.q_eigenform(10)
 
\(q + 2 q^{5} - 3 q^{9} + 4 q^{11} + 6 q^{13} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 72128.bg

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
72128.bg1 72128bw4 \([0, 0, 0, -387884, 92961232]\) \(209267191953/55223\) \(1703131408498688\) \([2]\) \(491520\) \(1.9070\)  
72128.bg2 72128bw2 \([0, 0, 0, -27244, 1070160]\) \(72511713/25921\) \(799429028478976\) \([2, 2]\) \(245760\) \(1.5605\)  
72128.bg3 72128bw1 \([0, 0, 0, -11564, -466480]\) \(5545233/161\) \(4965397692416\) \([2]\) \(122880\) \(1.2139\) \(\Gamma_0(N)\)-optimal
72128.bg4 72128bw3 \([0, 0, 0, 82516, 7524048]\) \(2014698447/1958887\) \(-60413993723625472\) \([2]\) \(491520\) \(1.9070\)