Properties

Label 7.115...904.8t36.a.a
Dimension $7$
Group $C_2^3:(C_7: C_3)$
Conductor $1.157\times 10^{12}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $7$
Group: $C_2^3:(C_7: C_3)$
Conductor: \(1157018619904\)\(\medspace = 2^{12} \cdot 7^{10} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.0.1157018619904.1
Galois orbit size: $1$
Smallest permutation container: $C_2^3:(C_7: C_3)$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $F_8:C_3$
Projective stem field: Galois closure of 8.0.1157018619904.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 4x^{7} + 14x^{6} - 14x^{5} + 14x^{4} + 28x^{3} + 14x^{2} - 18x + 23 \) 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 $\mathbb{Q}_{ 17 }$: \( x^{3} + x + 14 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.