Properties

Label 76050ek
Number of curves $4$
Conductor $76050$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("ek1")
 
E.isogeny_class()
 

Elliptic curves in class 76050ek

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
76050.fj3 76050ek1 \([1, -1, 1, -4595, -143373]\) \(-121945/32\) \(-2814995008800\) \([]\) \(129600\) \(1.1057\) \(\Gamma_0(N)\)-optimal
76050.fj4 76050ek2 \([1, -1, 1, 33430, 1058217]\) \(46969655/32768\) \(-2882554889011200\) \([]\) \(388800\) \(1.6550\)  
76050.fj2 76050ek3 \([1, -1, 1, -19805, 12663447]\) \(-25/2\) \(-68725464082031250\) \([]\) \(648000\) \(1.9104\)  
76050.fj1 76050ek4 \([1, -1, 1, -4772930, 4014794697]\) \(-349938025/8\) \(-274901856328125000\) \([]\) \(1944000\) \(2.4597\)  

Rank

sage: E.rank()
 

The elliptic curves in class 76050ek have rank \(1\).

Complex multiplication

The elliptic curves in class 76050ek do not have complex multiplication.

Modular form 76050.2.a.ek

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + 2 q^{7} + q^{8} - 3 q^{11} + 2 q^{14} + q^{16} - 3 q^{17} - 5 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

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.

\(\left(\begin{array}{rrrr} 1 & 3 & 5 & 15 \\ 3 & 1 & 15 & 5 \\ 5 & 15 & 1 & 3 \\ 15 & 5 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.