Properties

Label 6.317159.7t7.a.a
Dimension $6$
Group $S_7$
Conductor $317159$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $6$
Group: $S_7$
Conductor: \(317159\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.317159.1
Galois orbit size: $1$
Smallest permutation container: $S_7$
Parity: odd
Determinant: 1.317159.2t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.317159.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: \( x^{2} + 45x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 34 a + 14 + \left(12 a + 39\right)\cdot 47 + \left(23 a + 3\right)\cdot 47^{2} + \left(22 a + 2\right)\cdot 47^{3} + \left(43 a + 46\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 20 a + 9 + \left(17 a + 26\right)\cdot 47 + \left(26 a + 19\right)\cdot 47^{2} + \left(27 a + 2\right)\cdot 47^{3} + \left(5 a + 37\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 11 a + 23 + \left(30 a + 21\right)\cdot 47 + \left(9 a + 4\right)\cdot 47^{2} + \left(7 a + 43\right)\cdot 47^{3} + \left(34 a + 21\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 36 a + 45 + \left(16 a + 23\right)\cdot 47 + \left(37 a + 40\right)\cdot 47^{2} + 39 a\cdot 47^{3} + \left(12 a + 36\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 16 + 5\cdot 47 + 27\cdot 47^{2} + 37\cdot 47^{3} + 9\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 27 a + 2 + \left(29 a + 41\right)\cdot 47 + \left(20 a + 7\right)\cdot 47^{2} + \left(19 a + 31\right)\cdot 47^{3} + \left(41 a + 20\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 13 a + 35 + \left(34 a + 30\right)\cdot 47 + \left(23 a + 37\right)\cdot 47^{2} + \left(24 a + 23\right)\cdot 47^{3} + \left(3 a + 16\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display

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.