Properties

Label 182070cj
Number of curves 8
Conductor 182070
CM no
Rank 2
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("182070.bi1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 182070cj

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
182070.bi7 182070cj1 [1, -1, 0, 546156, -117201200] [2] 5242880 \(\Gamma_0(N)\)-optimal
182070.bi6 182070cj2 [1, -1, 0, -2783124, -1047402032] [2, 2] 10485760  
182070.bi5 182070cj3 [1, -1, 0, -19637604, 32752572160] [2, 2] 20971520  
182070.bi4 182070cj4 [1, -1, 0, -39197124, -94420180832] [2] 20971520  
182070.bi2 182070cj5 [1, -1, 0, -312250104, 2123820019660] [2, 2] 41943040  
182070.bi8 182070cj6 [1, -1, 0, 3303216, 104681219188] [2] 41943040  
182070.bi1 182070cj7 [1, -1, 0, -4996000854, 135920780694310] [2] 83886080  
182070.bi3 182070cj8 [1, -1, 0, -310299354, 2151664635010] [2] 83886080  

Rank

sage: E.rank()
 

The elliptic curves in class 182070cj have rank \(2\).

Modular form 182070.2.a.bi

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.