Properties

Label 223440hg
Number of curves $4$
Conductor $223440$
CM no
Rank $2$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("hg1")
 
E.isogeny_class()
 

Elliptic curves in class 223440hg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
223440.j4 223440hg1 \([0, -1, 0, 964, 19776]\) \(3286064/7695\) \(-231759118080\) \([2]\) \(294912\) \(0.86475\) \(\Gamma_0(N)\)-optimal
223440.j3 223440hg2 \([0, -1, 0, -7856, 224400]\) \(445138564/81225\) \(9785384985600\) \([2, 2]\) \(589824\) \(1.2113\)  
223440.j1 223440hg3 \([0, -1, 0, -119576, 15954576]\) \(784767874322/35625\) \(8583671040000\) \([2]\) \(1179648\) \(1.5579\)  
223440.j2 223440hg4 \([0, -1, 0, -37256, -2550960]\) \(23735908082/1954815\) \(471003197306880\) \([2]\) \(1179648\) \(1.5579\)  

Rank

sage: E.rank()
 

The elliptic curves in class 223440hg have rank \(2\).

Complex multiplication

The elliptic curves in class 223440hg do not have complex multiplication.

Modular form 223440.2.a.hg

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{9} - 4 q^{11} - 2 q^{13} + q^{15} - 6 q^{17} - q^{19} + O(q^{20})\) 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.