Properties

Label 2.116.14t8.b.e
Dimension $2$
Group $C_7 \wr C_2$
Conductor $116$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_7 \wr C_2$
Conductor: \(116\)\(\medspace = 2^{2} \cdot 29 \)
Artin stem field: Galois closure of 14.0.9745585291264.1
Galois orbit size: $6$
Smallest permutation container: $C_7 \wr C_2$
Parity: odd
Determinant: 1.116.14t1.b.e
Projective image: $D_7$
Projective stem field: Galois closure of 7.1.38068692544.1

Defining polynomial

$f(x)$$=$ \( x^{14} - 4 x^{13} + 7 x^{12} - 4 x^{11} - 8 x^{10} + 24 x^{9} - 30 x^{8} + 16 x^{7} + 13 x^{6} - 38 x^{5} + 46 x^{4} - 36 x^{3} + 19 x^{2} - 6 x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{7} + 12x + 14 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( a^{6} + 5 a^{5} + 7 a^{4} + 16 a^{3} + 8 a^{2} + 16 a + 16 + \left(12 a^{6} + 16 a^{5} + 8 a^{4} + 16 a^{3} + 8 a^{2} + 15 a + 2\right)\cdot 17 + \left(12 a^{6} + 9 a^{5} + 6 a^{4} + 10 a^{3} + 4 a^{2} + 8 a + 13\right)\cdot 17^{2} + \left(14 a^{6} + 6 a^{5} + 4 a^{4} + 12 a^{3} + 7 a^{2} + 12 a + 6\right)\cdot 17^{3} + \left(14 a^{6} + 14 a^{5} + 8 a^{3} + 2 a^{2} + 11\right)\cdot 17^{4} + \left(3 a^{6} + a^{5} + 7 a^{4} + 10 a^{3} + 12 a^{2} + 6 a + 10\right)\cdot 17^{5} + \left(4 a^{6} + 15 a^{5} + 13 a^{4} + 11 a^{3} + 6 a^{2} + a + 1\right)\cdot 17^{6} + \left(7 a^{6} + 9 a^{5} + 9 a^{4} + 11 a^{3} + a^{2} + 2 a + 15\right)\cdot 17^{7} + \left(11 a^{6} + 9 a^{5} + 15 a^{4} + 13 a^{3} + 13 a^{2} + 4 a + 2\right)\cdot 17^{8} + \left(7 a^{6} + a^{5} + 9 a^{4} + 5 a^{3} + 8 a^{2} + 15 a + 4\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 6 a^{6} + 8 a^{5} + 15 a^{4} + 5 a^{3} + 15 a^{2} + 3 a + 8 + \left(15 a^{6} + 5 a^{4} + 10 a^{3} + 5 a^{2} + 6 a + 9\right)\cdot 17 + \left(5 a^{6} + a^{5} + 11 a^{4} + 5 a^{3} + 13 a^{2} + 16 a + 15\right)\cdot 17^{2} + \left(11 a^{6} + 12 a^{5} + 15 a^{4} + 13 a^{3} + 4 a^{2} + 7 a + 13\right)\cdot 17^{3} + \left(12 a^{6} + 13 a^{5} + 7 a^{4} + 4 a^{3} + 8 a^{2} + 11 a + 16\right)\cdot 17^{4} + \left(16 a^{6} + 12 a^{4} + 9 a^{3} + 14 a^{2} + 3 a + 11\right)\cdot 17^{5} + \left(3 a^{6} + 8 a^{5} + 13 a^{4} + 11 a^{3} + 9 a^{2} + 3 a + 2\right)\cdot 17^{6} + \left(9 a^{6} + 15 a^{5} + 4 a^{4} + a^{3} + 6 a^{2} + 2 a + 11\right)\cdot 17^{7} + \left(8 a^{5} + 6 a^{4} + 8 a^{2} + 9 a + 3\right)\cdot 17^{8} + \left(16 a^{6} + 16 a^{5} + 15 a^{4} + 11 a^{3} + 16 a + 16\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 6 a^{6} + 16 a^{5} + 12 a^{4} + 8 a^{3} + 2 a + 14 + \left(4 a^{6} + 8 a^{5} + 16 a^{4} + 9 a^{3} + 13 a^{2} + 16 a + 3\right)\cdot 17 + \left(16 a^{6} + 4 a^{5} + 13 a^{4} + 6 a^{3} + 15 a^{2} + 13 a + 13\right)\cdot 17^{2} + \left(3 a^{6} + 11 a^{5} + 2 a^{4} + 13 a^{3} + 7 a^{2} + 2 a + 14\right)\cdot 17^{3} + \left(8 a^{5} + 16 a^{4} + 8 a^{3} + 3 a^{2} + 12 a + 13\right)\cdot 17^{4} + \left(13 a^{5} + 9 a^{4} + 11 a^{2} + 4 a + 9\right)\cdot 17^{5} + \left(4 a^{6} + 12 a^{5} + 11 a^{4} + a^{3} + 16 a^{2} + 8 a + 6\right)\cdot 17^{6} + \left(8 a^{6} + a^{5} + 5 a^{4} + 12 a^{3} + 16 a^{2} + 4 a + 3\right)\cdot 17^{7} + \left(3 a^{6} + 13 a^{5} + 16 a^{4} + 12 a^{3} + 14 a^{2} + 5 a + 6\right)\cdot 17^{8} + \left(6 a^{6} + 15 a^{5} + 11 a^{4} + 7 a^{3} + 15 a^{2} + 11 a + 13\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 8 a^{6} + 3 a^{5} + 7 a^{4} + 3 a^{3} + 10 a^{2} + 7 a + 3 + \left(14 a^{6} + 8 a^{4} + 8 a^{3} + 5 a^{2} + 10 a + 13\right)\cdot 17 + \left(2 a^{6} + 8 a^{5} + 7 a^{4} + 6 a^{3} + 11 a^{2} + 16 a + 13\right)\cdot 17^{2} + \left(15 a^{6} + 6 a^{5} + 16 a^{4} + 12 a^{3} + 8 a^{2} + 2 a + 3\right)\cdot 17^{3} + \left(10 a^{6} + 8 a^{5} + a^{4} + 13 a^{3} + 13 a^{2} + 14 a + 14\right)\cdot 17^{4} + \left(9 a^{6} + 5 a^{5} + 3 a^{4} + 5 a^{3} + 4 a^{2} + 10 a + 6\right)\cdot 17^{5} + \left(7 a^{6} + 2 a^{5} + 14 a^{4} + 9 a^{3} + a^{2} + 16 a + 4\right)\cdot 17^{6} + \left(6 a^{6} + a^{5} + 10 a^{4} + 4 a^{3} + 3 a^{2} + 12 a + 9\right)\cdot 17^{7} + \left(15 a^{6} + 8 a^{5} + 5 a^{4} + 4 a^{3} + 11 a^{2} + 11 a + 9\right)\cdot 17^{8} + \left(11 a^{6} + 12 a^{5} + 12 a^{4} + 14 a^{3} + 15 a^{2} + 1\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 8 a^{6} + 5 a^{5} + 9 a^{4} + a^{3} + 15 a^{2} + 4 a + 3 + \left(4 a^{6} + 13 a^{5} + 8 a^{4} + 12 a^{3} + 9 a^{2} + a\right)\cdot 17 + \left(13 a^{5} + 7 a^{4} + 12 a^{3} + 6 a^{2} + 7 a + 9\right)\cdot 17^{2} + \left(15 a^{6} + 12 a^{5} + 11 a^{4} + 2 a^{3} + 5 a^{2} + a + 9\right)\cdot 17^{3} + \left(15 a^{5} + 6 a^{4} + 15 a^{3} + 6 a^{2} + 11 a + 3\right)\cdot 17^{4} + \left(8 a^{6} + 7 a^{5} + 16 a^{4} + 7 a^{3} + 9 a^{2} + 3 a\right)\cdot 17^{5} + \left(4 a^{6} + 10 a^{5} + 15 a^{4} + 5 a^{3} + a^{2} + 11 a + 9\right)\cdot 17^{6} + \left(13 a^{5} + 10 a^{4} + 9 a^{3} + 2 a^{2} + 2 a + 1\right)\cdot 17^{7} + \left(11 a^{6} + 10 a^{5} + 9 a^{4} + 12 a^{3} + 15 a^{2} + 4 a + 1\right)\cdot 17^{8} + \left(3 a^{6} + 3 a^{5} + a^{4} + 3 a^{3} + 13 a^{2} + 5 a + 4\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 8 a^{6} + 13 a^{5} + 5 a^{4} + 16 a^{3} + a^{2} + 5 a + 3 + \left(a^{6} + 7 a^{5} + 9 a^{3} + 5 a^{2} + 15 a + 8\right)\cdot 17 + \left(a^{6} + 3 a^{5} + 14 a^{4} + 2 a^{3} + 14 a^{2} + 3 a\right)\cdot 17^{2} + \left(10 a^{6} + 11 a^{5} + 13 a^{4} + 10 a^{3} + 4 a^{2} + 9 a + 12\right)\cdot 17^{3} + \left(15 a^{6} + 2 a^{5} + a^{4} + 5 a^{3} + 5 a^{2} + 11 a + 6\right)\cdot 17^{4} + \left(16 a^{6} + 4 a^{5} + 5 a^{4} + 10 a^{3} + 12 a^{2} + a + 6\right)\cdot 17^{5} + \left(3 a^{6} + 12 a^{4} + 9 a^{3} + 15 a^{2} + 2 a + 11\right)\cdot 17^{6} + \left(5 a^{5} + 11 a^{4} + 7 a^{3} + 12 a^{2} + 14 a + 3\right)\cdot 17^{7} + \left(3 a^{6} + 5 a^{5} + 6 a^{4} + 4 a^{3} + 3 a^{2} + 16 a + 6\right)\cdot 17^{8} + \left(5 a^{6} + 13 a^{5} + a^{4} + 8 a^{3} + 13 a^{2} + 5 a\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 10 a^{6} + 5 a^{4} + a^{3} + 4 a^{2} + 12 a + 9 + \left(3 a^{6} + 4 a^{5} + 12 a^{3} + 14 a^{2} + 14 a + 10\right)\cdot 17 + \left(3 a^{6} + 15 a^{5} + 12 a^{4} + 13 a^{3} + 12 a^{2} + 8 a + 12\right)\cdot 17^{2} + \left(12 a^{6} + 6 a^{5} + 12 a^{4} + 7 a^{3} + 12 a^{2} + 8 a + 4\right)\cdot 17^{3} + \left(2 a^{6} + 12 a^{5} + 15 a^{4} + 11 a^{2} + 12 a + 10\right)\cdot 17^{4} + \left(4 a^{6} + 14 a^{5} + 14 a^{4} + 15 a^{2} + 16 a + 13\right)\cdot 17^{5} + \left(7 a^{6} + a^{5} + 4 a^{4} + 11 a^{3} + 14 a^{2} + 10\right)\cdot 17^{6} + \left(4 a^{6} + 3 a^{5} + 6 a^{4} + 8 a^{3} + 7 a^{2} + 14 a\right)\cdot 17^{7} + \left(7 a^{6} + 8 a^{5} + 3 a^{4} + 7 a^{3} + 8 a^{2} + 15 a + 11\right)\cdot 17^{8} + \left(13 a^{6} + a^{5} + 14 a^{4} + 4 a^{3} + 5 a^{2} + 2\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 10 a^{6} + 5 a^{5} + 12 a^{4} + 5 a^{3} + 16 a^{2} + 3 + \left(5 a^{6} + 7 a^{5} + 10 a^{3} + 16 a^{2} + 10 a + 6\right)\cdot 17 + \left(7 a^{6} + 9 a^{5} + 6 a^{4} + 16 a^{3} + 4 a^{2} + 15 a + 8\right)\cdot 17^{2} + \left(a^{6} + 14 a^{5} + 13 a^{4} + 2 a^{3} + 14 a^{2} + 11 a + 11\right)\cdot 17^{3} + \left(4 a^{6} + 9 a^{5} + 9 a^{4} + 9 a^{2} + 6 a + 8\right)\cdot 17^{4} + \left(2 a^{6} + a^{5} + 8 a^{4} + 3 a^{3} + 12 a^{2} + 12 a + 8\right)\cdot 17^{5} + \left(15 a^{6} + 16 a^{5} + 7 a^{4} + 11 a^{2} + a + 10\right)\cdot 17^{6} + \left(12 a^{6} + 5 a^{5} + 9 a^{4} + 3 a^{3} + 3 a^{2} + a + 2\right)\cdot 17^{7} + \left(9 a^{6} + 15 a^{5} + 12 a^{4} + 4 a^{3} + 2 a^{2} + 3 a + 6\right)\cdot 17^{8} + \left(14 a^{5} + 16 a^{4} + 14 a^{2} + 8 a + 5\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 10 a^{6} + 9 a^{5} + 6 a^{4} + 6 a^{3} + 13 a^{2} + 5 a + 9 + \left(10 a^{6} + 8 a^{4} + 16 a^{3} + 11 a^{2} + 11 a + 14\right)\cdot 17 + \left(14 a^{6} + 13 a^{5} + 6 a^{4} + 14 a^{3} + 2 a^{2} + 8 a + 15\right)\cdot 17^{2} + \left(13 a^{6} + 12 a^{5} + 6 a^{4} + 8 a^{3} + 4 a^{2} + 13 a + 4\right)\cdot 17^{3} + \left(5 a^{6} + 5 a^{5} + 8 a^{4} + 15 a^{3} + 8 a^{2} + 5 a + 3\right)\cdot 17^{4} + \left(8 a^{6} + 3 a^{5} + 11 a^{4} + 15 a^{3} + 2 a^{2} + 7 a + 3\right)\cdot 17^{5} + \left(2 a^{6} + 8 a^{5} + 12 a^{4} + 2 a^{3} + 11 a^{2} + 10 a + 3\right)\cdot 17^{6} + \left(7 a^{6} + 16 a^{5} + 12 a^{4} + 14 a^{3} + 6 a^{2} + 9\right)\cdot 17^{7} + \left(16 a^{6} + 12 a^{5} + 10 a^{4} + 12 a^{3} + a^{2} + 10 a + 10\right)\cdot 17^{8} + \left(2 a^{6} + 2 a^{5} + 16 a^{4} + 6 a^{3} + 12 a^{2} + 11 a + 11\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 10 }$ $=$ \( 10 a^{6} + 12 a^{5} + 6 a^{4} + 4 a^{3} + 16 a^{2} + 16 a + 3 + \left(14 a^{6} + 4 a^{5} + 14 a^{4} + 12 a^{3} + 5 a^{2} + 3 a + 16\right)\cdot 17 + \left(13 a^{6} + 16 a^{5} + 8 a^{4} + 16 a^{3} + 7 a^{2} + 2 a + 9\right)\cdot 17^{2} + \left(6 a^{6} + 11 a^{5} + 8 a^{4} + 4 a^{3} + 14 a^{2} + 8\right)\cdot 17^{3} + \left(9 a^{6} + 3 a^{5} + 2 a^{4} + 16 a^{3} + 8 a^{2} + a + 7\right)\cdot 17^{4} + \left(5 a^{6} + 11 a^{5} + 10 a^{2} + 9 a + 13\right)\cdot 17^{5} + \left(15 a^{6} + a^{5} + 7 a^{4} + 12 a^{3} + 9 a^{2} + 12 a + 2\right)\cdot 17^{6} + \left(7 a^{6} + 9 a^{4} + 9 a^{3} + 11 a^{2} + 5 a + 7\right)\cdot 17^{7} + \left(16 a^{6} + 5 a^{5} + 12 a^{4} + 15 a^{3} + 5 a^{2} + 16 a + 14\right)\cdot 17^{8} + \left(4 a^{6} + a^{5} + 6 a^{4} + 13 a^{3} + 10 a^{2} + 6 a + 6\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 11 }$ $=$ \( 13 a^{6} + 6 a^{5} + 16 a^{4} + a^{3} + 8 a^{2} + 3 a + 12 + \left(16 a^{6} + 3 a^{5} + 7 a^{4} + 13 a^{3} + 16 a^{2} + 7 a + 16\right)\cdot 17 + \left(8 a^{5} + 8 a^{4} + 11 a^{3} + 3 a + 15\right)\cdot 17^{2} + \left(4 a^{6} + 10 a^{5} + 12 a^{4} + 12 a^{3} + 6 a^{2} + 10 a + 6\right)\cdot 17^{3} + \left(6 a^{6} + 5 a^{5} + 10 a^{4} + 3 a^{3} + 10 a^{2} + 15 a + 11\right)\cdot 17^{4} + \left(7 a^{5} + 14 a^{4} + 2 a^{3} + 13 a^{2} + 7 a + 8\right)\cdot 17^{5} + \left(a^{6} + 4 a^{5} + 11 a^{4} + 11 a^{3} + 15 a^{2} + 7 a + 13\right)\cdot 17^{6} + \left(a^{6} + 2 a^{5} + 3 a^{4} + 11 a^{3} + 14 a^{2} + 12 a + 4\right)\cdot 17^{7} + \left(16 a^{6} + 7 a^{5} + 11 a^{4} + 6 a^{2} + 11 a + 10\right)\cdot 17^{8} + \left(16 a^{6} + 2 a^{5} + 12 a^{4} + 11 a^{3} + 12 a^{2} + 6 a + 8\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 12 }$ $=$ \( 15 a^{6} + 2 a^{5} + 15 a^{4} + 10 a^{3} + 9 a^{2} + 8 a + 1 + \left(7 a^{6} + 9 a^{5} + 15 a^{3} + 12 a^{2} + 5 a + 3\right)\cdot 17 + \left(2 a^{6} + 15 a^{4} + 14 a^{3} + 5 a^{2} + 4 a + 14\right)\cdot 17^{2} + \left(13 a^{6} + 2 a^{5} + 16 a^{4} + 11 a^{3} + 16 a^{2} + a\right)\cdot 17^{3} + \left(14 a^{6} + 16 a^{5} + 13 a^{4} + 8 a^{3} + 7 a + 2\right)\cdot 17^{4} + \left(16 a^{6} + 12 a^{5} + 5 a^{4} + 8 a^{3} + 7 a^{2} + 5 a + 1\right)\cdot 17^{5} + \left(15 a^{6} + 12 a^{5} + 16 a^{4} + 3 a^{3} + 4 a^{2} + 8 a + 12\right)\cdot 17^{6} + \left(a^{6} + a^{5} + 2 a^{4} + 12 a^{3} + 13 a^{2} + 13 a + 1\right)\cdot 17^{7} + \left(13 a^{6} + 4 a^{5} + 11 a^{4} + 15 a^{3} + 8 a + 9\right)\cdot 17^{8} + \left(2 a^{6} + 11 a^{5} + 16 a^{4} + 3 a^{3} + 8 a^{2} + 10 a + 8\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 13 }$ $=$ \( 15 a^{6} + 16 a^{5} + 2 a^{4} + 3 a^{3} + 15 a^{2} + 14 a + 1 + \left(10 a^{6} + 9 a^{5} + 14 a^{3} + 10 a^{2} + 5 a + 12\right)\cdot 17 + \left(14 a^{6} + 12 a^{5} + 11 a^{4} + 4 a^{3} + 16 a^{2} + 2 a + 10\right)\cdot 17^{2} + \left(16 a^{6} + 8 a^{5} + 6 a^{4} + 10 a^{3} + 16 a^{2} + a + 2\right)\cdot 17^{3} + \left(8 a^{6} + 8 a^{5} + 16 a^{4} + 13 a^{3} + 3 a^{2} + 12 a + 8\right)\cdot 17^{4} + \left(5 a^{6} + 14 a^{5} + 13 a^{3} + 6 a^{2} + 4 a + 3\right)\cdot 17^{5} + \left(2 a^{6} + 10 a^{5} + 2 a^{4} + 12 a^{3} + 3 a^{2} + 7 a\right)\cdot 17^{6} + \left(8 a^{6} + 15 a^{5} + 2 a^{4} + 7 a^{3} + 16 a^{2} + 2 a + 12\right)\cdot 17^{7} + \left(15 a^{6} + 14 a^{5} + a^{4} + 6 a^{3} + 15 a^{2} + 13 a + 6\right)\cdot 17^{8} + \left(12 a^{6} + 5 a^{5} + 8 a^{4} + 5 a^{3} + 5 a^{2} + 16 a + 8\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display
$r_{ 14 }$ $=$ \( 16 a^{6} + 2 a^{5} + 2 a^{4} + 6 a^{3} + 6 a^{2} + 7 a + 4 + \left(13 a^{6} + 16 a^{5} + 4 a^{4} + 9 a^{3} + 16 a^{2} + 12 a + 2\right)\cdot 17 + \left(5 a^{6} + 2 a^{5} + 7 a^{4} + 14 a^{3} + a^{2} + 6 a\right)\cdot 17^{2} + \left(14 a^{6} + 8 a^{5} + 11 a^{4} + 11 a^{3} + 12 a^{2} + a + 1\right)\cdot 17^{3} + \left(11 a^{6} + 10 a^{5} + 6 a^{4} + 3 a^{3} + 8 a^{2} + 14 a + 1\right)\cdot 17^{4} + \left(3 a^{6} + 2 a^{5} + 8 a^{4} + 13 a^{3} + 3 a^{2} + 7 a + 4\right)\cdot 17^{5} + \left(14 a^{6} + 14 a^{5} + 9 a^{4} + 16 a^{3} + 13 a^{2} + 10 a + 13\right)\cdot 17^{6} + \left(9 a^{6} + 9 a^{5} + a^{4} + 4 a^{3} + a^{2} + 13 a + 2\right)\cdot 17^{7} + \left(13 a^{6} + 12 a^{5} + 13 a^{4} + 8 a^{3} + 11 a^{2} + 5 a + 4\right)\cdot 17^{8} + \left(13 a^{6} + 15 a^{5} + 8 a^{4} + 5 a^{3} + 16 a^{2} + 2 a + 10\right)\cdot 17^{9} +O(17^{10})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 14 }$

Cycle notation
$(1,7,9,3,4,5,6)(2,10,12,13,11,14,8)$
$(2,14,13,10,8,11,12)$
$(1,13,5,10,3,8,7,11,6,12,4,2,9,14)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 14 }$ Character value
$1$$1$$()$$2$
$7$$2$$(1,11)(2,3)(4,10)(5,12)(6,13)(7,14)(8,9)$$0$
$1$$7$$(1,5,3,7,6,4,9)(2,14,13,10,8,11,12)$$-2 \zeta_{7}^{5} - 2 \zeta_{7}^{4} - 2 \zeta_{7}^{3} - 2 \zeta_{7}^{2} - 2 \zeta_{7} - 2$
$1$$7$$(1,3,6,9,5,7,4)(2,13,8,12,14,10,11)$$2 \zeta_{7}^{5}$
$1$$7$$(1,7,9,3,4,5,6)(2,10,12,13,11,14,8)$$2 \zeta_{7}^{4}$
$1$$7$$(1,6,5,4,3,9,7)(2,8,14,11,13,12,10)$$2 \zeta_{7}^{3}$
$1$$7$$(1,4,7,5,9,6,3)(2,11,10,14,12,8,13)$$2 \zeta_{7}^{2}$
$1$$7$$(1,9,4,6,7,3,5)(2,12,11,8,10,13,14)$$2 \zeta_{7}$
$2$$7$$(1,3,6,9,5,7,4)(2,14,13,10,8,11,12)$$-\zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$
$2$$7$$(1,6,5,4,3,9,7)(2,13,8,12,14,10,11)$$\zeta_{7}^{5} + \zeta_{7}^{3}$
$2$$7$$(1,9,4,6,7,3,5)(2,10,12,13,11,14,8)$$\zeta_{7}^{4} + \zeta_{7}$
$2$$7$$(1,5,3,7,6,4,9)(2,8,14,11,13,12,10)$$-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$
$2$$7$$(1,7,9,3,4,5,6)(2,11,10,14,12,8,13)$$\zeta_{7}^{4} + \zeta_{7}^{2}$
$2$$7$$(1,4,7,5,9,6,3)(2,12,11,8,10,13,14)$$\zeta_{7}^{2} + \zeta_{7}$
$2$$7$$(2,14,13,10,8,11,12)$$-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7}$
$2$$7$$(2,13,8,12,14,10,11)$$\zeta_{7}^{5} + 1$
$2$$7$$(2,10,12,13,11,14,8)$$\zeta_{7}^{4} + 1$
$2$$7$$(2,8,14,11,13,12,10)$$\zeta_{7}^{3} + 1$
$2$$7$$(2,11,10,14,12,8,13)$$\zeta_{7}^{2} + 1$
$2$$7$$(2,12,11,8,10,13,14)$$\zeta_{7} + 1$
$2$$7$$(1,7,9,3,4,5,6)(2,8,14,11,13,12,10)$$\zeta_{7}^{4} + \zeta_{7}^{3}$
$2$$7$$(1,9,4,6,7,3,5)(2,14,13,10,8,11,12)$$-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$
$2$$7$$(1,3,6,9,5,7,4)(2,11,10,14,12,8,13)$$\zeta_{7}^{5} + \zeta_{7}^{2}$
$2$$7$$(1,6,5,4,3,9,7)(2,11,10,14,12,8,13)$$\zeta_{7}^{3} + \zeta_{7}^{2}$
$2$$7$$(1,5,3,7,6,4,9)(2,10,12,13,11,14,8)$$-\zeta_{7}^{5} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$
$2$$7$$(1,4,7,5,9,6,3)(2,14,13,10,8,11,12)$$-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7} - 1$
$2$$7$$(1,3,6,9,5,7,4)(2,12,11,8,10,13,14)$$\zeta_{7}^{5} + \zeta_{7}$
$2$$7$$(1,9,4,6,7,3,5)(2,8,14,11,13,12,10)$$\zeta_{7}^{3} + \zeta_{7}$
$2$$7$$(1,7,9,3,4,5,6)(2,13,8,12,14,10,11)$$\zeta_{7}^{5} + \zeta_{7}^{4}$
$7$$14$$(1,13,5,10,3,8,7,11,6,12,4,2,9,14)$$0$
$7$$14$$(1,10,7,12,9,13,3,11,4,14,5,8,6,2)$$0$
$7$$14$$(1,8,4,13,7,2,5,11,9,10,6,14,3,12)$$0$
$7$$14$$(1,12,3,14,6,10,9,11,5,2,7,13,4,8)$$0$
$7$$14$$(1,2,6,8,5,14,4,11,3,13,9,12,7,10)$$0$
$7$$14$$(1,14,9,2,4,12,6,11,7,8,3,10,5,13)$$0$

The blue line marks the conjugacy class containing complex conjugation.