Properties

Label 6.317159.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 317159 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$317159 $
Artin number field: Splitting field of $f= x^{7} - 3 x^{6} + 4 x^{5} - 4 x^{4} + 4 x^{3} - 4 x^{2} + 2 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.317159.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 34 a + 14 + \left(12 a + 39\right)\cdot 47 + \left(23 a + 3\right)\cdot 47^{2} + \left(22 a + 2\right)\cdot 47^{3} + \left(43 a + 46\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 20 a + 9 + \left(17 a + 26\right)\cdot 47 + \left(26 a + 19\right)\cdot 47^{2} + \left(27 a + 2\right)\cdot 47^{3} + \left(5 a + 37\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 11 a + 23 + \left(30 a + 21\right)\cdot 47 + \left(9 a + 4\right)\cdot 47^{2} + \left(7 a + 43\right)\cdot 47^{3} + \left(34 a + 21\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 36 a + 45 + \left(16 a + 23\right)\cdot 47 + \left(37 a + 40\right)\cdot 47^{2} + 39 a\cdot 47^{3} + \left(12 a + 36\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 16 + 5\cdot 47 + 27\cdot 47^{2} + 37\cdot 47^{3} + 9\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 27 a + 2 + \left(29 a + 41\right)\cdot 47 + \left(20 a + 7\right)\cdot 47^{2} + \left(19 a + 31\right)\cdot 47^{3} + \left(41 a + 20\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 13 a + 35 + \left(34 a + 30\right)\cdot 47 + \left(23 a + 37\right)\cdot 47^{2} + \left(24 a + 23\right)\cdot 47^{3} + \left(3 a + 16\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 7 }$ Character value
$1$$1$$()$$6$
$21$$2$$(1,2)$$4$
$105$$2$$(1,2)(3,4)(5,6)$$0$
$105$$2$$(1,2)(3,4)$$2$
$70$$3$$(1,2,3)$$3$
$280$$3$$(1,2,3)(4,5,6)$$0$
$210$$4$$(1,2,3,4)$$2$
$630$$4$$(1,2,3,4)(5,6)$$0$
$504$$5$$(1,2,3,4,5)$$1$
$210$$6$$(1,2,3)(4,5)(6,7)$$-1$
$420$$6$$(1,2,3)(4,5)$$1$
$840$$6$$(1,2,3,4,5,6)$$0$
$720$$7$$(1,2,3,4,5,6,7)$$-1$
$504$$10$$(1,2,3,4,5)(6,7)$$-1$
$420$$12$$(1,2,3,4)(5,6,7)$$-1$
The blue line marks the conjugacy class containing complex conjugation.