Properties

Label 1680.j
Number of curves 8
Conductor 1680
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("1680.j1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1680.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1680.j1 1680p7 [0, -1, 0, -30732800, 65587209600] [4] 49152  
1680.j2 1680p5 [0, -1, 0, -1920800, 1025280000] [2, 4] 24576  
1680.j3 1680p8 [0, -1, 0, -1908800, 1038710400] [4] 49152  
1680.j4 1680p3 [0, -1, 0, -241120, -45479168] [2] 12288  
1680.j5 1680p4 [0, -1, 0, -120800, 15840000] [2, 4] 12288  
1680.j6 1680p2 [0, -1, 0, -17120, -499968] [2, 2] 6144  
1680.j7 1680p1 [0, -1, 0, 3360, -57600] [2] 3072 \(\Gamma_0(N)\)-optimal
1680.j8 1680p6 [0, -1, 0, 20320, 50499072] [4] 24576  

Rank

sage: E.rank()
 

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

Modular form 1680.2.a.j

sage: E.q_eigenform(10)
 
\( q - q^{3} + q^{5} + q^{7} + q^{9} + 4q^{11} - 2q^{13} - q^{15} + 2q^{17} - 4q^{19} + O(q^{20}) \)

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.