Properties

Label 3.1670792.42t37.b.a
Dimension $3$
Group $\GL(3,2)$
Conductor $1670792$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $3$
Group: $\GL(3,2)$
Conductor: \(1670792\)\(\medspace = 2^{3} \cdot 457^{2} \)
Artin stem field: Galois closure of 7.3.13366336.2
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 7.3.13366336.2

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{3} + x + 28 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 22 a^{2} + 2 a + 2 + \left(23 a^{2} + 25 a + 21\right)\cdot 31 + \left(24 a + 1\right)\cdot 31^{2} + \left(4 a^{2} + 21 a + 1\right)\cdot 31^{3} + \left(19 a^{2} + 27 a + 14\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 8 a^{2} + 17 a + 3 + \left(22 a^{2} + 8 a + 20\right)\cdot 31 + \left(30 a^{2} + 18 a + 21\right)\cdot 31^{2} + \left(24 a^{2} + 4 a + 4\right)\cdot 31^{3} + \left(5 a^{2} + 14 a + 5\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( a^{2} + 12 a + 19 + \left(16 a^{2} + 28 a + 5\right)\cdot 31 + \left(30 a^{2} + 18 a + 11\right)\cdot 31^{2} + \left(a^{2} + 4 a + 20\right)\cdot 31^{3} + \left(6 a^{2} + 20 a + 15\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 24 + 8\cdot 31 + 10\cdot 31^{2} + 4\cdot 31^{3} + 8\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 2 a^{2} + 4 a + 17 + \left(18 a^{2} + 16 a + 24\right)\cdot 31 + \left(15 a^{2} + 11 a + 5\right)\cdot 31^{2} + \left(30 a^{2} + 10\right)\cdot 31^{3} + \left(5 a^{2} + 6 a + 10\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 15 a^{2} + 23 a + 5 + \left(29 a^{2} + 20 a + 1\right)\cdot 31 + \left(a^{2} + a + 7\right)\cdot 31^{2} + \left(27 a^{2} + 3 a + 18\right)\cdot 31^{3} + \left(25 a^{2} + 13 a + 23\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 14 a^{2} + 4 a + 25 + \left(14 a^{2} + 25 a + 11\right)\cdot 31 + \left(13 a^{2} + 17 a + 4\right)\cdot 31^{2} + \left(4 a^{2} + 27 a + 3\right)\cdot 31^{3} + \left(30 a^{2} + 11 a + 16\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.