Properties

Label 2.2e2_13.6t5.1c2
Dimension 2
Group $S_3\times C_3$
Conductor $ 2^{2} \cdot 13 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$S_3\times C_3$
Conductor:$52= 2^{2} \cdot 13 $
Artin number field: Splitting field of $f= x^{6} - x^{4} - 2 x^{3} + 2 x + 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $S_3\times C_3$
Parity: Odd
Determinant: 1.2e2_13.6t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 9 a + 19 + \left(14 a + 8\right)\cdot 31 + \left(17 a + 2\right)\cdot 31^{2} + \left(25 a + 4\right)\cdot 31^{3} + \left(23 a + 25\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 6 a + 21 + \left(23 a + 13\right)\cdot 31 + \left(18 a + 21\right)\cdot 31^{2} + \left(26 a + 15\right)\cdot 31^{3} + \left(25 a + 21\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 25 a + 2 + \left(7 a + 23\right)\cdot 31 + \left(12 a + 4\right)\cdot 31^{2} + \left(4 a + 19\right)\cdot 31^{3} + \left(5 a + 15\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 11 a + 27 + \left(9 a + 5\right)\cdot 31 + \left(14 a + 11\right)\cdot 31^{2} + \left(20 a + 10\right)\cdot 31^{3} + \left(28 a + 4\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 22 a + 6 + \left(16 a + 28\right)\cdot 31 + \left(13 a + 22\right)\cdot 31^{2} + \left(5 a + 6\right)\cdot 31^{3} + \left(7 a + 16\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 20 a + 18 + \left(21 a + 13\right)\cdot 31 + \left(16 a + 30\right)\cdot 31^{2} + \left(10 a + 5\right)\cdot 31^{3} + \left(2 a + 10\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$

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

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

Character values on conjugacy classes

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