Properties

Label 9.102...881.20t145.a.a
Dimension $9$
Group $S_6$
Conductor $1.022\times 10^{25}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $9$
Group: $S_6$
Conductor: \(102\!\cdots\!881\)\(\medspace = 14731^{6}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.0.14731.1
Galois orbit size: $1$
Smallest permutation container: 20T145
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_6$
Projective field: Galois closure of 6.0.14731.1

Defining polynomial

$f(x)$$=$$ x^{6} - x^{5} + x^{3} - x^{2} + 1 $.

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:
$r_{ 1 }$ $=$ $ 21 + 42\cdot 47 + 46\cdot 47^{2} + 10\cdot 47^{3} + 41\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 12 + 18\cdot 47 + 28\cdot 47^{2} + 9\cdot 47^{3} + 6\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 15 a + 10 + \left(29 a + 17\right)\cdot 47 + \left(46 a + 29\right)\cdot 47^{2} + \left(12 a + 40\right)\cdot 47^{3} + \left(38 a + 3\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 42 a + 11 + \left(7 a + 14\right)\cdot 47 + \left(8 a + 14\right)\cdot 47^{2} + \left(45 a + 12\right)\cdot 47^{3} + \left(26 a + 30\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 32 a + 40 + \left(17 a + 13\right)\cdot 47 + 46\cdot 47^{2} + \left(34 a + 19\right)\cdot 47^{3} + \left(8 a + 20\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 5 a + 1 + \left(39 a + 35\right)\cdot 47 + \left(38 a + 22\right)\cdot 47^{2} + a\cdot 47^{3} + \left(20 a + 39\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$

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$$1$$()$$9$
$15$$2$$(1,2)(3,4)(5,6)$$-3$
$15$$2$$(1,2)$$-3$
$45$$2$$(1,2)(3,4)$$1$
$40$$3$$(1,2,3)(4,5,6)$$0$
$40$$3$$(1,2,3)$$0$
$90$$4$$(1,2,3,4)(5,6)$$1$
$90$$4$$(1,2,3,4)$$1$
$144$$5$$(1,2,3,4,5)$$-1$
$120$$6$$(1,2,3,4,5,6)$$0$
$120$$6$$(1,2,3)(4,5)$$0$

The blue line marks the conjugacy class containing complex conjugation.