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Properties

Label 177450.bk
Number of curves $4$
Conductor $177450$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bk1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 177450.bk have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(5\)\(1\)
\(7\)\(1 + T\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 8 T + 19 T^{2}\) 1.19.i
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 177450.bk do not have complex multiplication.

Modular form 177450.2.a.bk

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

Isogeny matrix

Copy content 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

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 177450.bk

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
177450.bk1 177450kh4 \([1, 1, 0, -1398263680375, -395595766231272875]\) \(4008766897254067912673785886329/1423480510711669921875000000\) \(107357320944182574748992919921875000000\) \([2]\) \(6936330240\) \(5.9985\)  
177450.bk2 177450kh2 \([1, 1, 0, -591050728375, 170392088831903125]\) \(302773487204995438715379645049/8911747415025000000000000\) \(672114103571396956640625000000000000\) \([2, 2]\) \(3468165120\) \(5.6519\)  
177450.bk3 177450kh1 \([1, 1, 0, -586810856375, 173019227999647125]\) \(296304326013275547793071733369/268420373544960000000\) \(20243966793908981760000000000000\) \([2]\) \(1734082560\) \(5.3053\) \(\Gamma_0(N)\)-optimal
177450.bk4 177450kh3 \([1, 1, 0, 148324271625, 568243121956903125]\) \(4784981304203817469820354951/1852343836482910078035000000\) \(-139701717203597479857031879921875000000\) \([2]\) \(6936330240\) \(5.9985\)