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Properties

Label 189618.bt
Number of curves $2$
Conductor $189618$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bt1")
 
E.isogeny_class()
 

Elliptic curves in class 189618.bt

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
189618.bt1 189618l1 \([1, 0, 0, -6668321, 5675394633]\) \(6793805286030262681/1048227429629952\) \(5059593591384718983168\) \([2]\) \(25288704\) \(2.8881\) \(\Gamma_0(N)\)-optimal
189618.bt2 189618l2 \([1, 0, 0, 11610719, 31320887753]\) \(35862531227445945959/108547797844556928\) \(-523939487566287981082752\) \([2]\) \(50577408\) \(3.2346\)  

Rank

sage: E.rank()
 

The elliptic curves in class 189618.bt have rank \(1\).

Complex multiplication

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

Modular form 189618.2.a.bt

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.