Properties

Label 2.1575.6t3.c
Dimension 2
Group $D_{6}$
Conductor $ 3^{2} \cdot 5^{2} \cdot 7 $
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$D_{6}$
Conductor:$1575= 3^{2} \cdot 5^{2} \cdot 7 $
Artin number field: Splitting field of $f= x^{6} - x^{5} - x^{4} + 4 x^{3} + x^{2} - 6 x + 9 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{6}$
Parity: Odd
Projective image: $S_3$
Projective field: Galois closure of 3.1.175.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{2} + 12 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ a + 3 + \left(3 a + 3\right)\cdot 13 + \left(6 a + 2\right)\cdot 13^{2} + \left(11 a + 9\right)\cdot 13^{3} + \left(a + 3\right)\cdot 13^{4} + \left(5 a + 8\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 4 a + 10 + \left(10 a + 12\right)\cdot 13 + \left(12 a + 12\right)\cdot 13^{2} + \left(11 a + 9\right)\cdot 13^{3} + \left(10 a + 3\right)\cdot 13^{4} + \left(7 a + 7\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 12 + 8\cdot 13 + 3\cdot 13^{2} + 10\cdot 13^{3} + 11\cdot 13^{4} + 11\cdot 13^{5} +O\left(13^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 12 a + 4 + \left(9 a + 5\right)\cdot 13 + \left(6 a + 5\right)\cdot 13^{2} + \left(a + 1\right)\cdot 13^{3} + \left(11 a + 7\right)\cdot 13^{4} + \left(7 a + 11\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 10 + 2\cdot 13 + 12\cdot 13^{2} + 11\cdot 13^{3} + 9\cdot 13^{4} + 8\cdot 13^{5} +O\left(13^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 9 a + 1 + \left(2 a + 6\right)\cdot 13 + 2\cdot 13^{2} + \left(a + 9\right)\cdot 13^{3} + \left(2 a + 2\right)\cdot 13^{4} + \left(5 a + 4\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character values
$c1$
$1$ $1$ $()$ $2$
$1$ $2$ $(1,6)(2,4)(3,5)$ $-2$
$3$ $2$ $(1,2)(3,5)(4,6)$ $0$
$3$ $2$ $(1,5)(3,6)$ $0$
$2$ $3$ $(1,4,5)(2,3,6)$ $-1$
$2$ $6$ $(1,3,4,6,5,2)$ $1$
The blue line marks the conjugacy class containing complex conjugation.