Properties

Label 7.126548911552.24t283.a.b
Dimension $7$
Group $C_2^3:(C_7: C_3)$
Conductor $126548911552$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $7$
Group: $C_2^3:(C_7: C_3)$
Conductor: \(126548911552\)\(\medspace = 2^{6} \cdot 7^{11} \)
Artin stem field: Galois closure of 8.0.18078415936.1
Galois orbit size: $2$
Smallest permutation container: 24T283
Parity: even
Determinant: 1.7.3t1.a.b
Projective image: $F_8:C_3$
Projective stem field: Galois closure of 8.0.18078415936.1

Defining polynomial

$f(x)$$=$ \( x^{8} - x^{7} + 7x^{6} + 7x^{5} + 7x^{4} + 7x^{3} + 7x^{2} + 5x + 2 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{3} + 4x + 17 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 9 a^{2} + 5 a + 12 + \left(9 a^{2} + 5 a + 13\right)\cdot 19 + \left(16 a^{2} + 4 a\right)\cdot 19^{2} + \left(7 a^{2} + 17 a + 17\right)\cdot 19^{3} + \left(5 a^{2} + 13 a + 17\right)\cdot 19^{4} + \left(9 a^{2} + 18 a + 9\right)\cdot 19^{5} + \left(14 a^{2} + 5 a + 2\right)\cdot 19^{6} + \left(9 a^{2} + 7 a + 14\right)\cdot 19^{7} + \left(14 a^{2} + 4 a + 8\right)\cdot 19^{8} + \left(18 a^{2} + 10 a + 9\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 16 + 17\cdot 19 + 15\cdot 19^{2} + 4\cdot 19^{3} + 18\cdot 19^{4} + 9\cdot 19^{5} + 6\cdot 19^{6} + 10\cdot 19^{7} + 8\cdot 19^{8} + 18\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 15 a^{2} + 12 a + 9 + \left(17 a^{2} + 3 a + 10\right)\cdot 19 + \left(3 a^{2} + 17 a + 11\right)\cdot 19^{2} + \left(2 a^{2} + 16 a + 14\right)\cdot 19^{3} + \left(4 a^{2} + 14 a + 1\right)\cdot 19^{4} + 11\cdot 19^{5} + \left(2 a^{2} + 3 a + 13\right)\cdot 19^{6} + \left(16 a^{2} + 8 a + 5\right)\cdot 19^{7} + \left(3 a^{2} + 18\right)\cdot 19^{8} + \left(2 a^{2} + 11 a + 15\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 17 + 5\cdot 19 + 2\cdot 19^{2} + 4\cdot 19^{3} + 11\cdot 19^{4} + 2\cdot 19^{5} + 8\cdot 19^{6} + 19^{7} + 11\cdot 19^{8} + 3\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 13 a^{2} + 11 a + 17 + \left(8 a^{2} + 13 a + 7\right)\cdot 19 + \left(10 a^{2} + 3 a + 14\right)\cdot 19^{2} + \left(15 a + 8\right)\cdot 19^{3} + \left(12 a^{2} + 10 a + 12\right)\cdot 19^{4} + \left(5 a^{2} + 7 a + 6\right)\cdot 19^{5} + \left(10 a^{2} + 14 a + 14\right)\cdot 19^{6} + \left(12 a^{2} + 5 a + 9\right)\cdot 19^{7} + \left(10 a^{2} + 17 a + 7\right)\cdot 19^{8} + \left(12 a^{2} + a + 3\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 4 a^{2} + 8 a + 12 + \left(2 a^{2} + 12 a + 9\right)\cdot 19 + \left(10 a^{2} + 9 a + 13\right)\cdot 19^{2} + \left(a^{2} + 4 a + 17\right)\cdot 19^{3} + \left(5 a^{2} + 6 a + 12\right)\cdot 19^{4} + \left(7 a^{2} + 4\right)\cdot 19^{5} + \left(17 a^{2} + 14\right)\cdot 19^{6} + \left(12 a^{2} + 11 a + 10\right)\cdot 19^{7} + \left(17 a^{2} + 4 a + 13\right)\cdot 19^{8} + \left(3 a^{2} + 5\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 14 a^{2} + 2 a + \left(10 a^{2} + 10 a + 17\right)\cdot 19 + \left(17 a^{2} + 16 a + 9\right)\cdot 19^{2} + \left(8 a^{2} + 3 a + 13\right)\cdot 19^{3} + \left(9 a^{2} + 9 a + 9\right)\cdot 19^{4} + \left(9 a^{2} + 18 a + 10\right)\cdot 19^{5} + \left(2 a^{2} + 9 a + 8\right)\cdot 19^{6} + \left(12 a^{2} + 3 a + 1\right)\cdot 19^{7} + \left(14 a + 16\right)\cdot 19^{8} + \left(17 a^{2} + 16 a + 4\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 2 a^{2} + 13 + \left(8 a^{2} + 12 a + 12\right)\cdot 19 + \left(17 a^{2} + 5 a + 7\right)\cdot 19^{2} + \left(16 a^{2} + 18 a + 14\right)\cdot 19^{3} + \left(a^{2} + a + 10\right)\cdot 19^{4} + \left(6 a^{2} + 11 a + 1\right)\cdot 19^{5} + \left(10 a^{2} + 4 a + 8\right)\cdot 19^{6} + \left(12 a^{2} + 2 a + 3\right)\cdot 19^{7} + \left(9 a^{2} + 16 a + 11\right)\cdot 19^{8} + \left(2 a^{2} + 16 a + 14\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$7$
$7$$2$$(1,5)(2,3)(4,8)(6,7)$$-1$
$28$$3$$(1,8,7)(3,6,4)$$-\zeta_{3} - 1$
$28$$3$$(1,7,8)(3,4,6)$$\zeta_{3}$
$28$$6$$(1,6,4,7,5,3)(2,8)$$\zeta_{3} + 1$
$28$$6$$(1,3,5,7,4,6)(2,8)$$-\zeta_{3}$
$24$$7$$(1,6,2,5,3,4,8)$$0$
$24$$7$$(1,5,8,2,4,6,3)$$0$

The blue line marks the conjugacy class containing complex conjugation.