Properties

Label 1694f
Number of curves 6
Conductor 1694
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("1694.e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1694f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1694.e5 1694f1 [1, 0, 0, -63, -395] [2] 480 \(\Gamma_0(N)\)-optimal
1694.e4 1694f2 [1, 0, 0, -1273, -17577] [2] 960  
1694.e6 1694f3 [1, 0, 0, 542, 8196] [2] 1440  
1694.e3 1694f4 [1, 0, 0, -4298, 88540] [2] 2880  
1694.e2 1694f5 [1, 0, 0, -20633, 1142329] [2] 4320  
1694.e1 1694f6 [1, 0, 0, -330393, 73068601] [2] 8640  

Rank

sage: E.rank()
 

The elliptic curves in class 1694f have rank \(1\).

Modular form 1694.2.a.e

sage: E.q_eigenform(10)
 
\( q + q^{2} - 2q^{3} + q^{4} - 2q^{6} - q^{7} + q^{8} + q^{9} - 2q^{12} + 4q^{13} - q^{14} + q^{16} - 6q^{17} + q^{18} - 2q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.