Properties

Label 10.3e16_11e8.30t176.1c1
Dimension 10
Group $S_6$
Conductor $ 3^{16} \cdot 11^{8}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$10$
Group:$S_6$
Conductor:$9227446944279201= 3^{16} \cdot 11^{8} $
Artin number field: Splitting field of $f= x^{6} + 3 x^{4} - 2 x^{3} + 6 x^{2} + 9 x + 5 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 30T176
Parity: Even
Determinant: 1.1.1t1.1c1

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:
$r_{ 1 }$ $=$ $ 8 + 31\cdot 47 + 30\cdot 47^{2} + 28\cdot 47^{3} + 34\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 41 a + 8 + \left(41 a + 26\right)\cdot 47 + \left(15 a + 42\right)\cdot 47^{2} + \left(3 a + 41\right)\cdot 47^{3} + \left(10 a + 44\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 6 a + 43 + \left(5 a + 21\right)\cdot 47 + \left(31 a + 32\right)\cdot 47^{2} + \left(43 a + 32\right)\cdot 47^{3} + \left(36 a + 14\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 42 + 31\cdot 47 + 30\cdot 47^{2} + 38\cdot 47^{3} + 45\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 27 a + 40 + \left(2 a + 25\right)\cdot 47 + \left(12 a + 38\right)\cdot 47^{2} + \left(4 a + 24\right)\cdot 47^{3} + \left(13 a + 36\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 20 a + \left(44 a + 4\right)\cdot 47 + \left(34 a + 13\right)\cdot 47^{2} + \left(42 a + 21\right)\cdot 47^{3} + \left(33 a + 11\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$$1$$()$$10$
$15$$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.