Properties

Label 1680.g
Number of curves 8
Conductor 1680
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("1680.g1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1680.g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1680.g1 1680m7 [0, -1, 0, -5619720, 5129544432] [4] 27648  
1680.g2 1680m6 [0, -1, 0, -351240, 80233200] [2, 2] 13824  
1680.g3 1680m8 [0, -1, 0, -325640, 92398320] [2] 27648  
1680.g4 1680m4 [0, -1, 0, -69720, 6984432] [4] 9216  
1680.g5 1680m3 [0, -1, 0, -23560, 1065712] [2] 6912  
1680.g6 1680m2 [0, -1, 0, -9240, -176400] [2, 2] 4608  
1680.g7 1680m1 [0, -1, 0, -7960, -270608] [2] 2304 \(\Gamma_0(N)\)-optimal
1680.g8 1680m5 [0, -1, 0, 30760, -1328400] [2] 9216  

Rank

sage: E.rank()
 

The elliptic curves in class 1680.g have rank \(1\).

Modular form 1680.2.a.g

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.