Properties

Label 1680k
Number of curves $8$
Conductor $1680$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("1680.b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1680k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1680.b7 1680k1 [0, -1, 0, -656, 2496] [2] 1152 \(\Gamma_0(N)\)-optimal
1680.b5 1680k2 [0, -1, 0, -5776, -165440] [2, 2] 2304  
1680.b4 1680k3 [0, -1, 0, -42896, 3433920] [2] 3456  
1680.b2 1680k4 [0, -1, 0, -92176, -10740800] [2] 4608  
1680.b6 1680k5 [0, -1, 0, -1296, -419904] [2] 4608  
1680.b3 1680k6 [0, -1, 0, -43216, 3380416] [2, 2] 6912  
1680.b1 1680k7 [0, -1, 0, -103216, -7995584] [2] 13824  
1680.b8 1680k8 [0, -1, 0, 11664, 11327040] [2] 13824  

Rank

sage: E.rank()
 

The elliptic curves in class 1680k have rank \(0\).

Modular form 1680.2.a.b

sage: E.q_eigenform(10)
 
\( q - q^{3} - q^{5} - q^{7} + q^{9} + 2q^{13} + q^{15} - 6q^{17} + 4q^{19} + O(q^{20}) \)

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 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.