Properties

Label 162624gg
Number of curves $4$
Conductor $162624$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("gg1")
 
E.isogeny_class()
 

Elliptic curves in class 162624gg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
162624.hx4 162624gg1 \([0, 1, 0, 26943, 1046175]\) \(4657463/3696\) \(-1716437504753664\) \([2]\) \(737280\) \(1.6112\) \(\Gamma_0(N)\)-optimal
162624.hx3 162624gg2 \([0, 1, 0, -127937, 8945055]\) \(498677257/213444\) \(99124265899524096\) \([2, 2]\) \(1474560\) \(1.9578\)  
162624.hx1 162624gg3 \([0, 1, 0, -1754177, 893294367]\) \(1285429208617/614922\) \(285572289853390848\) \([2]\) \(2949120\) \(2.3043\)  
162624.hx2 162624gg4 \([0, 1, 0, -979777, -367397857]\) \(223980311017/4278582\) \(1986990966440460288\) \([2]\) \(2949120\) \(2.3043\)  

Rank

sage: E.rank()
 

The elliptic curves in class 162624gg have rank \(1\).

Complex multiplication

The elliptic curves in class 162624gg do not have complex multiplication.

Modular form 162624.2.a.gg

sage: E.q_eigenform(10)
 
\(q + q^{3} + 2 q^{5} - q^{7} + q^{9} + 2 q^{13} + 2 q^{15} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.