Properties

Label 7.29079798784.8t36.a
Dimension $7$
Group $C_2^3:(C_7: C_3)$
Conductor $29079798784$
Indicator $1$

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Basic invariants

Dimension:$7$
Group:$C_2^3:(C_7: C_3)$
Conductor:\(29079798784\)\(\medspace = 2^{10} \cdot 73^{4} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 8.0.29079798784.1
Galois orbit size: $1$
Smallest permutation container: $C_2^3:(C_7: C_3)$
Parity: even
Projective image: $F_8:C_3$
Projective field: Galois closure of 8.0.29079798784.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{3} + 2x + 27 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 6 a^{2} + 3 a + 15 + \left(17 a^{2} + 25 a + 27\right)\cdot 29 + \left(15 a^{2} + 27 a + 9\right)\cdot 29^{2} + \left(7 a^{2} + 18 a + 8\right)\cdot 29^{3} + \left(4 a^{2} + 15 a + 1\right)\cdot 29^{4} + \left(12 a^{2} + 2 a + 4\right)\cdot 29^{5} + \left(20 a^{2} + 5 a + 2\right)\cdot 29^{6} + \left(8 a^{2} + 21 a + 3\right)\cdot 29^{7} + \left(20 a^{2} + 27 a + 23\right)\cdot 29^{8} + \left(9 a + 22\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 24 a^{2} + 5 a + 15 + \left(14 a^{2} + 11 a + 10\right)\cdot 29 + \left(19 a^{2} + 22 a + 4\right)\cdot 29^{2} + \left(22 a^{2} + 6 a + 18\right)\cdot 29^{3} + \left(9 a^{2} + 9 a + 26\right)\cdot 29^{4} + \left(12 a^{2} + 9 a + 15\right)\cdot 29^{5} + \left(3 a^{2} + 24 a + 18\right)\cdot 29^{6} + \left(2 a^{2} + 15 a + 8\right)\cdot 29^{7} + \left(21 a^{2} + 20 a + 6\right)\cdot 29^{8} + \left(3 a^{2} + 18 a + 14\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 8 a^{2} + 9 a + 13 + \left(9 a^{2} + 21 a + 22\right)\cdot 29 + \left(14 a^{2} + 7 a + 16\right)\cdot 29^{2} + \left(27 a^{2} + 27 a + 24\right)\cdot 29^{3} + \left(13 a^{2} + 4 a + 12\right)\cdot 29^{4} + \left(22 a^{2} + 15 a\right)\cdot 29^{5} + \left(24 a + 15\right)\cdot 29^{6} + \left(28 a^{2} + 12 a + 4\right)\cdot 29^{7} + \left(5 a^{2} + 22 a + 15\right)\cdot 29^{8} + \left(12 a^{2} + 21 a + 25\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 18 + 13\cdot 29 + 28\cdot 29^{2} + 11\cdot 29^{3} + 23\cdot 29^{4} + 28\cdot 29^{5} + 3\cdot 29^{6} + 14\cdot 29^{7} + 21\cdot 29^{8} + 12\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 26 a^{2} + 15 a + 8 + \left(4 a^{2} + 25 a + 26\right)\cdot 29 + \left(24 a^{2} + 27 a\right)\cdot 29^{2} + \left(7 a^{2} + 23 a + 8\right)\cdot 29^{3} + \left(5 a^{2} + 14 a + 1\right)\cdot 29^{4} + \left(23 a^{2} + 4 a + 11\right)\cdot 29^{5} + \left(24 a^{2} + 9 a + 8\right)\cdot 29^{6} + \left(27 a^{2} + 4\right)\cdot 29^{7} + \left(a^{2} + 15 a\right)\cdot 29^{8} + \left(13 a^{2} + 17 a + 17\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 14 + 11\cdot 29^{2} + 7\cdot 29^{4} + 9\cdot 29^{5} + 23\cdot 29^{7} + 26\cdot 29^{8} + 9\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 28 a^{2} + 15 a + 25 + \left(22 a^{2} + 13 a + 15\right)\cdot 29 + \left(23 a^{2} + 13 a + 1\right)\cdot 29^{2} + \left(2 a^{2} + a + 2\right)\cdot 29^{3} + \left(5 a^{2} + 3 a + 12\right)\cdot 29^{4} + \left(17 a^{2} + 19 a + 20\right)\cdot 29^{5} + \left(18 a^{2} + 20 a + 28\right)\cdot 29^{6} + \left(18 a^{2} + a + 25\right)\cdot 29^{7} + \left(16 a^{2} + 3 a + 27\right)\cdot 29^{8} + \left(6 a^{2} + 17 a + 20\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 24 a^{2} + 11 a + 10 + \left(17 a^{2} + 19 a + 28\right)\cdot 29 + \left(18 a^{2} + 16 a + 13\right)\cdot 29^{2} + \left(18 a^{2} + 8 a + 13\right)\cdot 29^{3} + \left(19 a^{2} + 10 a + 2\right)\cdot 29^{4} + \left(28 a^{2} + 7 a + 26\right)\cdot 29^{5} + \left(18 a^{2} + 3 a + 9\right)\cdot 29^{6} + \left(a^{2} + 6 a + 3\right)\cdot 29^{7} + \left(21 a^{2} + 27 a + 24\right)\cdot 29^{8} + \left(21 a^{2} + a + 21\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation
$(1,5,3,6,8,4)(2,7)$
$(1,3,2,5,4,8,6)$
$(1,7)(2,6)(3,5)(4,8)$
$(1,8)(2,3)(4,7)(5,6)$
$(1,6)(2,7)(3,4)(5,8)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character values
$c1$
$1$ $1$ $()$ $7$
$7$ $2$ $(1,6)(2,7)(3,4)(5,8)$ $-1$
$28$ $3$ $(1,3,8)(4,5,6)$ $1$
$28$ $3$ $(1,8,3)(4,6,5)$ $1$
$28$ $6$ $(1,5,3,6,8,4)(2,7)$ $-1$
$28$ $6$ $(1,4,8,6,3,5)(2,7)$ $-1$
$24$ $7$ $(1,3,2,5,4,8,6)$ $0$
$24$ $7$ $(1,5,6,2,8,3,4)$ $0$
The blue line marks the conjugacy class containing complex conjugation.