Properties

Label 195195.bu
Number of curves $6$
Conductor $195195$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("bu1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 195195.bu

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
195195.bu1 195195bt6 [1, 0, 1, -515314804, -4502577363919] [2] 17694720  
195195.bu2 195195bt4 [1, 0, 1, -32207179, -70354769119] [2, 2] 8847360  
195195.bu3 195195bt5 [1, 0, 1, -32047474, -71086984603] [2] 17694720  
195195.bu4 195195bt3 [1, 0, 1, -4300209, 1809641017] [2] 8847360  
195195.bu5 195195bt2 [1, 0, 1, -2022934, -1087963693] [2, 2] 4423680  
195195.bu6 195195bt1 [1, 0, 1, 5911, -50818129] [2] 2211840 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 195195.bu have rank \(1\).

Complex multiplication

The elliptic curves in class 195195.bu do not have complex multiplication.

Modular form 195195.2.a.bu

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

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.