Properties

Label 3.2e2_83.6t11.1c1
Dimension 3
Group $S_4\times C_2$
Conductor $ 2^{2} \cdot 83 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$S_4\times C_2$
Conductor:$332= 2^{2} \cdot 83 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + 3 x^{4} - 9 x^{3} + 7 x^{2} - 4 x - 2 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_4\times C_2$
Parity: Even
Determinant: 1.2e2_83.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
$r_{ 1 }$ $=$ $ 20 a + 19 + \left(15 a + 13\right)\cdot 41 + \left(35 a + 24\right)\cdot 41^{2} + \left(7 a + 16\right)\cdot 41^{3} + \left(25 a + 8\right)\cdot 41^{4} + \left(4 a + 9\right)\cdot 41^{5} + \left(7 a + 17\right)\cdot 41^{6} + \left(26 a + 28\right)\cdot 41^{7} + \left(4 a + 32\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$
$r_{ 2 }$ $=$ $ 11 a + 39 + \left(33 a + 13\right)\cdot 41 + \left(11 a + 40\right)\cdot 41^{2} + \left(24 a + 30\right)\cdot 41^{3} + \left(5 a + 24\right)\cdot 41^{4} + \left(8 a + 32\right)\cdot 41^{5} + 19\cdot 41^{6} + \left(25 a + 22\right)\cdot 41^{7} + \left(14 a + 37\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$
$r_{ 3 }$ $=$ $ 21 + 17\cdot 41 + 31\cdot 41^{2} + 34\cdot 41^{3} + 41^{4} + 30\cdot 41^{5} + 33\cdot 41^{6} + 7\cdot 41^{7} + 5\cdot 41^{8} +O\left(41^{ 9 }\right)$
$r_{ 4 }$ $=$ $ 30 a + 31 + \left(7 a + 20\right)\cdot 41 + \left(29 a + 1\right)\cdot 41^{2} + \left(16 a + 10\right)\cdot 41^{3} + \left(35 a + 17\right)\cdot 41^{4} + \left(32 a + 10\right)\cdot 41^{5} + \left(40 a + 12\right)\cdot 41^{6} + \left(15 a + 15\right)\cdot 41^{7} + \left(26 a + 15\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$
$r_{ 5 }$ $=$ $ 17 + 17\cdot 41 + 33\cdot 41^{2} + 25\cdot 41^{3} + 35\cdot 41^{4} + 41^{5} + 6\cdot 41^{6} + 31\cdot 41^{7} + 11\cdot 41^{8} +O\left(41^{ 9 }\right)$
$r_{ 6 }$ $=$ $ 21 a + 38 + \left(25 a + 39\right)\cdot 41 + \left(5 a + 32\right)\cdot 41^{2} + \left(33 a + 4\right)\cdot 41^{3} + \left(15 a + 35\right)\cdot 41^{4} + \left(36 a + 38\right)\cdot 41^{5} + \left(33 a + 33\right)\cdot 41^{6} + \left(14 a + 17\right)\cdot 41^{7} + \left(36 a + 20\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$3$
$1$$2$$(1,6)(2,4)(3,5)$$-3$
$3$$2$$(1,6)$$1$
$3$$2$$(1,6)(3,5)$$-1$
$6$$2$$(2,3)(4,5)$$-1$
$6$$2$$(1,6)(2,3)(4,5)$$1$
$8$$3$$(1,3,2)(4,6,5)$$0$
$6$$4$$(1,5,6,3)$$-1$
$6$$4$$(1,6)(2,5,4,3)$$1$
$8$$6$$(1,5,4,6,3,2)$$0$
The blue line marks the conjugacy class containing complex conjugation.