Properties

Label 2.175.6t3.a
Dimension $2$
Group $D_{6}$
Conductor $175$
Indicator $1$

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{6}$
Conductor:\(175\)\(\medspace = 5^{2} \cdot 7 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.2.153125.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
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 $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{2} + 18 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 15 a + 1 + \left(5 a + 1\right)\cdot 19 + \left(17 a + 13\right)\cdot 19^{2} + 13\cdot 19^{3} + \left(6 a + 3\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 6 a + 2 + \left(13 a + 12\right)\cdot 19 + \left(16 a + 9\right)\cdot 19^{2} + \left(9 a + 5\right)\cdot 19^{3} + \left(12 a + 8\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 7 + 9\cdot 19 + 10\cdot 19^{2} + 3\cdot 19^{3} + 14\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 13 a + 8 + 5 a\cdot 19 + \left(2 a + 13\right)\cdot 19^{2} + \left(9 a + 17\right)\cdot 19^{3} + \left(6 a + 10\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 4 a + 16 + \left(13 a + 10\right)\cdot 19 + \left(a + 5\right)\cdot 19^{2} + \left(18 a + 16\right)\cdot 19^{3} + \left(12 a + 8\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 5 + 4\cdot 19 + 5\cdot 19^{2} + 11\cdot 19^{4} +O\left(19^{ 5 }\right)$

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

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

Character values on conjugacy classes

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