Properties

Label 10.770...689.30t164.a.a
Dimension $10$
Group $S_6$
Conductor $7.702\times 10^{25}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $10$
Group: $S_6$
Conductor: \(770\!\cdots\!689\)\(\medspace = 20627^{6}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 6.0.20627.1
Galois orbit size: $1$
Smallest permutation container: 30T164
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_6$
Projective stem field: 6.0.20627.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 277 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 277 }$: \(x^{2} + 274 x + 5\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 62 a + 226 + \left(43 a + 116\right)\cdot 277 + \left(152 a + 9\right)\cdot 277^{2} + \left(231 a + 84\right)\cdot 277^{3} + \left(175 a + 85\right)\cdot 277^{4} +O(277^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 230 + 135\cdot 277 + 119\cdot 277^{2} + 146\cdot 277^{3} + 55\cdot 277^{4} +O(277^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 118 + 184\cdot 277 + 194\cdot 277^{2} + 65\cdot 277^{3} + 153\cdot 277^{4} +O(277^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 41 + 146\cdot 277 + 30\cdot 277^{2} + 9\cdot 277^{3} + 24\cdot 277^{4} +O(277^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 215 a + 135 + \left(233 a + 184\right)\cdot 277 + \left(124 a + 145\right)\cdot 277^{2} + \left(45 a + 72\right)\cdot 277^{3} + \left(101 a + 104\right)\cdot 277^{4} +O(277^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 82 + 63\cdot 277 + 54\cdot 277^{2} + 176\cdot 277^{3} + 131\cdot 277^{4} +O(277^{5})\)  Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$10$
$15$$2$$(1,2)(3,4)(5,6)$$2$
$15$$2$$(1,2)$$-2$
$45$$2$$(1,2)(3,4)$$-2$
$40$$3$$(1,2,3)(4,5,6)$$1$
$40$$3$$(1,2,3)$$1$
$90$$4$$(1,2,3,4)(5,6)$$0$
$90$$4$$(1,2,3,4)$$0$
$144$$5$$(1,2,3,4,5)$$0$
$120$$6$$(1,2,3,4,5,6)$$-1$
$120$$6$$(1,2,3)(4,5)$$1$

The blue line marks the conjugacy class containing complex conjugation.