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Properties

Label 28224fv
Number of curves $6$
Conductor $28224$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("fv1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 28224fv have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 3 T + 5 T^{2}\) 1.5.ad
\(11\) \( 1 + T + 11 T^{2}\) 1.11.b
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 + 4 T + 17 T^{2}\) 1.17.e
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 + 7 T + 29 T^{2}\) 1.29.h
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 28224fv do not have complex multiplication.

Modular form 28224.2.a.fv

Copy content sage:E.q_eigenform(10)
 
\(q + 2 q^{5} - 4 q^{11} - 2 q^{13} + 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 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 28224fv

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
28224.et5 28224fv1 \([0, 0, 0, 1176, -19208]\) \(2048/3\) \(-263473523712\) \([2]\) \(24576\) \(0.87691\) \(\Gamma_0(N)\)-optimal
28224.et4 28224fv2 \([0, 0, 0, -7644, -192080]\) \(35152/9\) \(12646729138176\) \([2, 2]\) \(49152\) \(1.2235\)  
28224.et3 28224fv3 \([0, 0, 0, -42924, 3265360]\) \(1556068/81\) \(455282248974336\) \([2, 2]\) \(98304\) \(1.5701\)  
28224.et2 28224fv4 \([0, 0, 0, -113484, -14713328]\) \(28756228/3\) \(16862305517568\) \([2]\) \(98304\) \(1.5701\)  
28224.et6 28224fv5 \([0, 0, 0, 27636, 12946192]\) \(207646/6561\) \(-73755724333842432\) \([2]\) \(196608\) \(1.9166\)  
28224.et1 28224fv6 \([0, 0, 0, -677964, 214860688]\) \(3065617154/9\) \(101173833105408\) \([2]\) \(196608\) \(1.9166\)