↑

Properties

Label 248430.fk
Number of curves $4$
Conductor $248430$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve("fk1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 248430.fk have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 + T\)
\(5\)\(1 + T\)
\(7\)\(1\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 6 T + 11 T^{2}\) 1.11.g
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 - 8 T + 19 T^{2}\) 1.19.ai
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 248430.fk do not have complex multiplication.

Modular form 248430.2.a.fk

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 248430.fk

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
248430.fk1 248430fk3 \([1, 1, 1, -426252226, 3387077537423]\) \(15082569606665230489/7751016000\) \(4401563658477823656000\) \([2]\) \(83607552\) \(3.4851\)  
248430.fk2 248430fk4 \([1, 1, 1, -423933546, 3425751264879]\) \(-14837772556740428569/342100087875000\) \(-194268121024736623102875000\) \([2]\) \(167215104\) \(3.8316\)  
248430.fk3 248430fk1 \([1, 1, 1, -6281311, 2718181409]\) \(48264326765929/22299191460\) \(12663025175509254769860\) \([2]\) \(27869184\) \(2.9358\) \(\Gamma_0(N)\)-optimal
248430.fk4 248430fk2 \([1, 1, 1, 22122519, 20521702053]\) \(2108526614950391/1540302022350\) \(-874690157349134165116350\) \([2]\) \(55738368\) \(3.2823\)