Properties

Label 397488gk
Number of curves $4$
Conductor $397488$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("gk1")
 
E.isogeny_class()
 

Elliptic curves in class 397488gk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
397488.gk3 397488gk1 \([0, 1, 0, -354384, -81216108]\) \(4649101309/6804\) \(7203483237629952\) \([2]\) \(2764800\) \(1.9444\) \(\Gamma_0(N)\)-optimal
397488.gk4 397488gk2 \([0, 1, 0, -252464, -128792364]\) \(-1680914269/5786802\) \(-6126562493604274176\) \([2]\) \(5529600\) \(2.2910\)  
397488.gk1 397488gk3 \([0, 1, 0, -10597344, 13270038900]\) \(124318741396429/51631104\) \(54662520900107108352\) \([2]\) \(13824000\) \(2.7491\)  
397488.gk2 397488gk4 \([0, 1, 0, -8966624, 17494908276]\) \(-75306487574989/81352871712\) \(-86129342697009390944256\) \([2]\) \(27648000\) \(3.0957\)  

Rank

sage: E.rank()
 

The elliptic curves in class 397488gk have rank \(1\).

Complex multiplication

The elliptic curves in class 397488gk do not have complex multiplication.

Modular form 397488.2.a.gk

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2 q^{5} + q^{9} - 2 q^{15} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.