↑

Properties

Label 139425.f
Number of curves $6$
Conductor $139425$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve("f1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 139425.f have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(3\)\(1 + T\)
\(5\)\(1\)
\(11\)\(1 - T\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + T + 2 T^{2}\) 1.2.b
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 139425.f do not have complex multiplication.

Modular form 139425.2.a.f

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} + q^{6} + 3 q^{8} + q^{9} + q^{11} + q^{12} - q^{16} + 6 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 139425.f

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
139425.f1 139425n3 \([1, 1, 1, -29000488, 60099209906]\) \(35765103905346817/1287\) \(97064112234375\) \([2]\) \(5505024\) \(2.6283\)  
139425.f2 139425n6 \([1, 1, 1, -12713113, -16900020844]\) \(3013001140430737/108679952667\) \(8196521463322650046875\) \([2]\) \(11010048\) \(2.9749\)  
139425.f3 139425n4 \([1, 1, 1, -2002738, 729256406]\) \(11779205551777/3763454409\) \(283835556444545015625\) \([2, 2]\) \(5505024\) \(2.6283\)  
139425.f4 139425n2 \([1, 1, 1, -1812613, 938393906]\) \(8732907467857/1656369\) \(124921512445640625\) \([2, 2]\) \(2752512\) \(2.2817\)  
139425.f5 139425n1 \([1, 1, 1, -101488, 17808656]\) \(-1532808577/938223\) \(-70759737818859375\) \([2]\) \(1376256\) \(1.9351\) \(\Gamma_0(N)\)-optimal
139425.f6 139425n5 \([1, 1, 1, 5665637, 4977536156]\) \(266679605718863/296110251723\) \(-22332306687638154796875\) \([2]\) \(11010048\) \(2.9749\)