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Properties

Label 28314.bg
Number of curves $4$
Conductor $28314$
CM no
Rank $1$
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 28314.bg have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(11\)\(1\)
\(13\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(7\) \( 1 + 4 T + 7 T^{2}\) 1.7.e
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 28314.bg do not have complex multiplication.

Modular form 28314.2.a.bg

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - 2 q^{5} - 4 q^{7} + q^{8} - 2 q^{10} + q^{13} - 4 q^{14} + q^{16} + 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 28314.bg

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
28314.bg1 28314ch4 \([1, -1, 1, -14447786, -20996206239]\) \(258252149810350513/1938176193096\) \(2503092471661842942024\) \([2]\) \(2211840\) \(2.9361\)  
28314.bg2 28314ch2 \([1, -1, 1, -1510466, 169249281]\) \(295102348042033/161237583936\) \(208233175052292945984\) \([2, 2]\) \(1105920\) \(2.5895\)  
28314.bg3 28314ch1 \([1, -1, 1, -1161986, 481766145]\) \(134351465835313/205590528\) \(265513581641797632\) \([4]\) \(552960\) \(2.2430\) \(\Gamma_0(N)\)-optimal
28314.bg4 28314ch3 \([1, -1, 1, 5851174, 1329443745]\) \(17154149157653327/10519679024712\) \(-13585828504576707449928\) \([2]\) \(2211840\) \(2.9361\)