Properties

Label 3.17280649.42t37.a.a
Dimension $3$
Group $\GL(3,2)$
Conductor $17280649$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $3$
Group: $\GL(3,2)$
Conductor: \(17280649\)\(\medspace = 4157^{2} \)
Artin stem field: Galois closure of 7.3.17280649.1
Galois orbit size: $2$
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 7.3.17280649.1

Defining polynomial

$f(x)$$=$ \( x^{7} - x^{6} - 4x^{5} + x^{4} + 8x^{3} + x^{2} - 4x - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.