Properties

Label 189618.a
Number of curves $4$
Conductor $189618$
CM no
Rank $2$
Graph

Related objects

Downloads

Learn more

Show commands for: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 189618.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
189618.a1 189618bi4 \([1, 1, 0, -327356, -66950970]\) \(803760366578833/65593817586\) \(316608829068463074\) \([2]\) \(3538944\) \(2.1007\)  
189618.a2 189618bi2 \([1, 1, 0, -68786, 5707200]\) \(7457162887153/1370924676\) \(6617191564438884\) \([2, 2]\) \(1769472\) \(1.7542\)  
189618.a3 189618bi1 \([1, 1, 0, -65406, 6410916]\) \(6411014266033/296208\) \(1429739440272\) \([2]\) \(884736\) \(1.4076\) \(\Gamma_0(N)\)-optimal
189618.a4 189618bi3 \([1, 1, 0, 135704, 33395146]\) \(57258048889007/132611470002\) \(-640090236908883618\) \([2]\) \(3538944\) \(2.1007\)  

Rank

sage: E.rank()
 

The elliptic curves in class 189618.a have rank \(2\).

Complex multiplication

The elliptic curves in class 189618.a do not have complex multiplication.

Modular form 189618.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - 2q^{5} + q^{6} - 4q^{7} - q^{8} + q^{9} + 2q^{10} - q^{11} - q^{12} + 4q^{14} + 2q^{15} + q^{16} + q^{17} - q^{18} - 4q^{19} + O(q^{20})\)  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 LMFDB 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 LMFDB labels.