Properties

Label 20160.bz
Number of curves $8$
Conductor $20160$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("20160.bz1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 20160.bz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
20160.bz1 20160bn7 [0, 0, 0, -3715788, -1734477712] [2] 884736  
20160.bz2 20160bn4 [0, 0, 0, -3318348, -2326649488] [2] 294912  
20160.bz3 20160bn6 [0, 0, 0, -1555788, 727058288] [2, 2] 442368  
20160.bz4 20160bn3 [0, 0, 0, -1544268, 738638192] [2] 221184  
20160.bz5 20160bn2 [0, 0, 0, -207948, -36150928] [2, 2] 147456  
20160.bz6 20160bn5 [0, 0, 0, -46668, -90792592] [2] 294912  
20160.bz7 20160bn1 [0, 0, 0, -23628, 491888] [2] 73728 \(\Gamma_0(N)\)-optimal
20160.bz8 20160bn8 [0, 0, 0, 419892, 2447480432] [2] 884736  

Rank

sage: E.rank()
 

The elliptic curves in class 20160.bz have rank \(1\).

Modular form 20160.2.a.bz

sage: E.q_eigenform(10)
 
\( q - q^{5} + q^{7} - 2q^{13} + 6q^{17} + 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.