Properties

Label 3.2e4_743e2.42t37.1c1
Dimension 3
Group $\GL(3,2)$
Conductor $ 2^{4} \cdot 743^{2}$
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$3$
Group:$\GL(3,2)$
Conductor:$8832784= 2^{4} \cdot 743^{2} $
Artin number field: Splitting field of $f= x^{7} - x^{6} + x^{5} + 5 x^{4} - 6 x^{3} + 4 x^{2} + 8 x - 8 $ 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 $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{3} + 2 x + 27 $
Roots:
$r_{ 1 }$ $=$ $ 6 a^{2} + 5 a + 13 + \left(19 a^{2} + 17 a + 3\right)\cdot 29 + \left(4 a^{2} + 23\right)\cdot 29^{2} + \left(20 a^{2} + 25 a + 11\right)\cdot 29^{3} + \left(5 a^{2} + 5 a + 6\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 8 a^{2} + 13 a + 6 + \left(16 a^{2} + 28 a + 9\right)\cdot 29 + \left(25 a^{2} + 17 a + 12\right)\cdot 29^{2} + \left(11 a^{2} + 18 a + 10\right)\cdot 29^{3} + \left(4 a + 28\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 12 a^{2} + 9 a + 25 + \left(15 a^{2} + 13 a + 27\right)\cdot 29 + \left(5 a^{2} + 18 a + 14\right)\cdot 29^{2} + \left(6 a^{2} + 26 a + 10\right)\cdot 29^{3} + \left(4 a^{2} + 11 a + 25\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 15 a^{2} + 11 a + 25 + \left(22 a^{2} + 12 a + 7\right)\cdot 29 + \left(27 a^{2} + 10 a + 15\right)\cdot 29^{2} + \left(25 a^{2} + 14 a + 19\right)\cdot 29^{3} + \left(22 a^{2} + 18 a + 19\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 17 + 15\cdot 29 + 13\cdot 29^{2} + 9\cdot 29^{3} + 2\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 20 a^{2} + 22 a + 26 + \left(17 a^{2} + 21 a + 1\right)\cdot 29 + \left(25 a^{2} + 20 a + 3\right)\cdot 29^{2} + \left(11 a^{2} + a + 18\right)\cdot 29^{3} + \left(7 a^{2} + 23 a\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 26 a^{2} + 27 a + 5 + \left(24 a^{2} + 22 a + 21\right)\cdot 29 + \left(26 a^{2} + 18 a + 4\right)\cdot 29^{2} + \left(10 a^{2} + 7\right)\cdot 29^{3} + \left(17 a^{2} + 23 a + 4\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$

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

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

Character values on conjugacy classes

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