Properties

Label 6720bw
Number of curves 8
Conductor 6720
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("6720.bi1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 6720bw

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
6720.bi7 6720bw1 [0, 1, 0, -31841, -2196705] [2] 18432 \(\Gamma_0(N)\)-optimal
6720.bi6 6720bw2 [0, 1, 0, -36961, -1448161] [2, 2] 36864  
6720.bi5 6720bw3 [0, 1, 0, -94241, 8431455] [2] 55296  
6720.bi4 6720bw4 [0, 1, 0, -278881, 55596575] [2] 73728  
6720.bi8 6720bw5 [0, 1, 0, 123039, -10504161] [2] 73728  
6720.bi2 6720bw6 [0, 1, 0, -1404961, 640460639] [2, 2] 110592  
6720.bi1 6720bw7 [0, 1, 0, -22478881, 41013876575] [2] 221184  
6720.bi3 6720bw8 [0, 1, 0, -1302561, 737883999] [2] 221184  

Rank

sage: E.rank()
 

The elliptic curves in class 6720bw have rank \(1\).

Modular form 6720.2.a.bi

sage: E.q_eigenform(10)
 
\( q + q^{3} - q^{5} - q^{7} + q^{9} - 2q^{13} - q^{15} - 6q^{17} + 8q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.