Properties

Label 182070.bf
Number of curves $6$
Conductor $182070$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bf1")
 
E.isogeny_class()
 

Elliptic curves in class 182070.bf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
182070.bf1 182070cw5 \([1, -1, 0, -43696854, 111190080078]\) \(524388516989299201/3150\) \(55428306573150\) \([2]\) \(10485760\) \(2.7027\)  
182070.bf2 182070cw3 \([1, -1, 0, -2731104, 1737789228]\) \(128031684631201/9922500\) \(174599165705422500\) \([2, 2]\) \(5242880\) \(2.3561\)  
182070.bf3 182070cw6 \([1, -1, 0, -2549034, 1979323290]\) \(-104094944089921/35880468750\) \(-631363054559786718750\) \([2]\) \(10485760\) \(2.7027\)  
182070.bf4 182070cw4 \([1, -1, 0, -962424, -343239660]\) \(5602762882081/345888060\) \(6086345850689556060\) \([2]\) \(5242880\) \(2.3561\)  
182070.bf5 182070cw2 \([1, -1, 0, -182124, 23345280]\) \(37966934881/8643600\) \(152095273236723600\) \([2, 2]\) \(2621440\) \(2.0095\)  
182070.bf6 182070cw1 \([1, -1, 0, 25956, 2245968]\) \(109902239/188160\) \(-3310917512636160\) \([2]\) \(1310720\) \(1.6629\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 182070.bf have rank \(1\).

Complex multiplication

The elliptic curves in class 182070.bf do not have complex multiplication.

Modular form 182070.2.a.bf

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{5} - q^{7} - q^{8} - q^{10} + 4 q^{11} - 2 q^{13} + q^{14} + q^{16} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.