Properties

Label 10.2e16_5e16.30t176.2c1
Dimension 10
Group $S_6$
Conductor $ 2^{16} \cdot 5^{16}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$10$
Group:$S_6$
Conductor:$10000000000000000= 2^{16} \cdot 5^{16} $
Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 5 x^{4} - 5 x^{3} + 10 x^{2} - 4 x - 8 $ 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:
$r_{ 1 }$ $=$ $ 45 a + 25 + \left(3 a + 23\right)\cdot 47 + \left(29 a + 13\right)\cdot 47^{2} + \left(7 a + 1\right)\cdot 47^{3} + \left(24 a + 25\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 8 + 8\cdot 47 + 45\cdot 47^{2} + 9\cdot 47^{3} +O\left(47^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 2 a + 21 + \left(43 a + 33\right)\cdot 47 + \left(17 a + 20\right)\cdot 47^{2} + \left(39 a + 34\right)\cdot 47^{3} + \left(22 a + 18\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 15 + 38\cdot 47 + 20\cdot 47^{2} + 41\cdot 47^{3} + 12\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 7 a + 7 + \left(18 a + 4\right)\cdot 47 + \left(39 a + 37\right)\cdot 47^{2} + \left(29 a + 16\right)\cdot 47^{3} + \left(44 a + 12\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 40 a + 21 + \left(28 a + 33\right)\cdot 47 + \left(7 a + 3\right)\cdot 47^{2} + \left(17 a + 37\right)\cdot 47^{3} + \left(2 a + 24\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.