Properties

Label 1680.f
Number of curves 4
Conductor 1680
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("1680.f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1680.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1680.f1 1680n3 [0, -1, 0, -1800, 30000] [4] 1024  
1680.f2 1680n2 [0, -1, 0, -120, 432] [2, 2] 512  
1680.f3 1680n1 [0, -1, 0, -40, -80] [2] 256 \(\Gamma_0(N)\)-optimal
1680.f4 1680n4 [0, -1, 0, 280, 2352] [2] 1024  

Rank

sage: E.rank()
 

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

Modular form 1680.2.a.f

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

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.