Properties

Label 7.881...216.8t37.a.a
Dimension $7$
Group $\GL(3,2)$
Conductor $8.814\times 10^{13}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $7$
Group: $\GL(3,2)$
Conductor: \(88136346505216\)\(\medspace = 2^{12} \cdot 383^{4} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.3.9388096.1
Galois orbit size: $1$
Smallest permutation container: $\PSL(2,7)$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $\GL(3,2)$
Projective stem field: Galois closure of 7.3.9388096.1

Defining polynomial

$f(x)$$=$ \( x^{7} - x^{6} + 2x^{5} + 2x^{4} - 5x^{3} + 7x^{2} - 5x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{3} + x + 14 \) Copy content Toggle raw display

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

Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 7 }$ Character value
$1$$1$$()$$7$
$21$$2$$(1,7)(2,6)$$-1$
$56$$3$$(1,4,7)(2,6,3)$$1$
$42$$4$$(1,6,4,3)(2,5)$$-1$
$24$$7$$(1,2,5,6,4,3,7)$$0$
$24$$7$$(1,6,7,5,3,2,4)$$0$

The blue line marks the conjugacy class containing complex conjugation.