Properties

Label 3.2e3_1733e2.42t37.1c1
Dimension 3
Group $\GL(3,2)$
Conductor $ 2^{3} \cdot 1733^{2}$
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$3$
Group:$\GL(3,2)$
Conductor:$24026312= 2^{3} \cdot 1733^{2} $
Artin number field: Splitting field of $f= x^{7} - 4 x^{5} - 2 x^{4} - 4 x^{3} - 8 x^{2} - 4 x - 4 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $\PSL(2,7)$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{3} + x + 40 $
Roots:
$r_{ 1 }$ $=$ $ 30 a + 40 + \left(13 a^{2} + 3 a + 4\right)\cdot 43 + \left(30 a^{2} + 22 a + 21\right)\cdot 43^{2} + \left(21 a^{2} + 21 a\right)\cdot 43^{3} + \left(30 a^{2} + 18\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 38 a^{2} + 5 a + 15 + \left(16 a^{2} + 34 a + 3\right)\cdot 43 + \left(26 a^{2} + 23 a + 2\right)\cdot 43^{2} + \left(14 a^{2} + 11 a + 32\right)\cdot 43^{3} + \left(13 a^{2} + 15 a + 5\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 40 + 34\cdot 43 + 18\cdot 43^{3} + 16\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 8 a^{2} + 21 a + 31 + \left(24 a^{2} + 20 a + 26\right)\cdot 43 + \left(31 a + 15\right)\cdot 43^{2} + \left(33 a^{2} + 13 a + 22\right)\cdot 43^{3} + \left(24 a^{2} + 4 a + 28\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 17 a^{2} + 15 a + 1 + \left(12 a^{2} + 14 a + 29\right)\cdot 43 + \left(26 a^{2} + 3 a + 30\right)\cdot 43^{2} + \left(27 a^{2} + 14 a + 40\right)\cdot 43^{3} + \left(31 a^{2} + 40 a + 17\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 35 a^{2} + 35 a + 6 + \left(5 a^{2} + 18 a\right)\cdot 43 + \left(12 a^{2} + 32 a + 9\right)\cdot 43^{2} + \left(31 a^{2} + 7 a + 21\right)\cdot 43^{3} + \left(30 a^{2} + 38 a + 32\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 31 a^{2} + 23 a + 39 + \left(13 a^{2} + 37 a + 29\right)\cdot 43 + \left(33 a^{2} + 15 a + 6\right)\cdot 43^{2} + \left(17 a + 37\right)\cdot 43^{3} + \left(41 a^{2} + 30 a + 9\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 7 }$ Character value
$1$$1$$()$$3$
$21$$2$$(1,7)(2,4)$$-1$
$56$$3$$(1,2,5)(3,7,6)$$0$
$42$$4$$(2,3,5,7)(4,6)$$1$
$24$$7$$(1,7,4,6,2,3,5)$$\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$
$24$$7$$(1,6,5,4,3,7,2)$$-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$
The blue line marks the conjugacy class containing complex conjugation.