Properties

Label 3.2e3_11_19.6t11.2c1
Dimension 3
Group $S_4\times C_2$
Conductor $ 2^{3} \cdot 11 \cdot 19 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$3$
Group:$S_4\times C_2$
Conductor:$1672= 2^{3} \cdot 11 \cdot 19 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + 2 x^{4} + x^{3} + x^{2} + 2 x + 2 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_4\times C_2$
Parity: Even
Determinant: 1.2e3_11_19.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 19 a + 5 + \left(36 a + 9\right)\cdot 37 + \left(16 a + 15\right)\cdot 37^{2} + \left(20 a + 4\right)\cdot 37^{3} + \left(2 a + 31\right)\cdot 37^{4} + \left(33 a + 1\right)\cdot 37^{5} + 15 a\cdot 37^{6} +O\left(37^{ 7 }\right)$
$r_{ 2 }$ $=$ $ 27 a + 5 + \left(12 a + 20\right)\cdot 37 + \left(31 a + 29\right)\cdot 37^{2} + \left(4 a + 31\right)\cdot 37^{3} + \left(12 a + 7\right)\cdot 37^{4} + 9 a\cdot 37^{5} + \left(3 a + 12\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$
$r_{ 3 }$ $=$ $ 18 a + 7 + 25\cdot 37 + \left(20 a + 9\right)\cdot 37^{2} + \left(16 a + 32\right)\cdot 37^{3} + \left(34 a + 20\right)\cdot 37^{4} + \left(3 a + 20\right)\cdot 37^{5} + \left(21 a + 30\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$
$r_{ 4 }$ $=$ $ 10 a + 2 + \left(24 a + 7\right)\cdot 37 + \left(5 a + 31\right)\cdot 37^{2} + \left(32 a + 19\right)\cdot 37^{3} + \left(24 a + 14\right)\cdot 37^{4} + \left(27 a + 25\right)\cdot 37^{5} + \left(33 a + 15\right)\cdot 37^{6} +O\left(37^{ 7 }\right)$
$r_{ 5 }$ $=$ $ 5 + 19\cdot 37 + 35\cdot 37^{4} + 20\cdot 37^{5} + 3\cdot 37^{6} +O\left(37^{ 7 }\right)$
$r_{ 6 }$ $=$ $ 14 + 30\cdot 37 + 24\cdot 37^{2} + 22\cdot 37^{3} + 37^{4} + 5\cdot 37^{5} + 12\cdot 37^{6} +O\left(37^{ 7 }\right)$

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

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

Character values on conjugacy classes

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