Properties

Label 20160ce
Number of curves 8
Conductor 20160
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("20160.cz1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 20160ce

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
20160.cz7 20160ce1 [0, 0, 0, 120948, -12199696] [2] 196608 \(\Gamma_0(N)\)-optimal
20160.cz6 20160ce2 [0, 0, 0, -616332, -109225744] [2, 2] 393216  
20160.cz4 20160ce3 [0, 0, 0, -8680332, -9840860944] [2] 786432  
20160.cz5 20160ce4 [0, 0, 0, -4348812, 3412742384] [2, 2] 786432  
20160.cz2 20160ce5 [0, 0, 0, -69148812, 221322182384] [2, 2] 1572864  
20160.cz8 20160ce6 [0, 0, 0, 731508, 10909262576] [2] 1572864  
20160.cz1 20160ce7 [0, 0, 0, -1106380812, 14164624511984] [4] 3145728  
20160.cz3 20160ce8 [0, 0, 0, -68716812, 224224012784] [2] 3145728  

Rank

sage: E.rank()
 

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

Modular form 20160.2.a.cz

sage: E.q_eigenform(10)
 
\( q + q^{5} - q^{7} - 4q^{11} + 2q^{13} - 2q^{17} - 4q^{19} + O(q^{20}) \)

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 Cremona numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 2 & 2 \\ 8 & 4 & 8 & 2 & 4 & 1 & 8 & 8 \\ 16 & 8 & 16 & 4 & 2 & 8 & 1 & 4 \\ 16 & 8 & 16 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.