Properties

Label 2.2563.6t3.b.a
Dimension 2
Group $D_{6}$
Conductor $ 11 \cdot 233 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$D_{6}$
Conductor:$2563= 11 \cdot 233 $
Artin number field: Splitting field of 6.0.72258659.1 defined by $f= x^{6} - 3 x^{5} + 18 x^{4} - 31 x^{3} + 43 x^{2} - 28 x + 20 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{6}$
Parity: Odd
Determinant: 1.2563.2t1.a.a
Projective image: $S_3$
Projective field: Galois closure of 3.1.2563.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 40 a + 15 + \left(44 a + 5\right)\cdot 47 + \left(11 a + 31\right)\cdot 47^{2} + \left(7 a + 24\right)\cdot 47^{3} + \left(20 a + 40\right)\cdot 47^{4} + \left(18 a + 17\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 40 a + \left(44 a + 39\right)\cdot 47 + \left(11 a + 36\right)\cdot 47^{2} + \left(7 a + 19\right)\cdot 47^{3} + \left(20 a + 20\right)\cdot 47^{4} + \left(18 a + 12\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 7 a + 33 + \left(2 a + 41\right)\cdot 47 + \left(35 a + 15\right)\cdot 47^{2} + \left(39 a + 22\right)\cdot 47^{3} + \left(26 a + 6\right)\cdot 47^{4} + \left(28 a + 29\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 42 + 34\cdot 47 + 34\cdot 47^{2} + 37\cdot 47^{3} + 42\cdot 47^{4} + 44\cdot 47^{5} +O\left(47^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 6 + 12\cdot 47 + 12\cdot 47^{2} + 9\cdot 47^{3} + 4\cdot 47^{4} + 2\cdot 47^{5} +O\left(47^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 7 a + 1 + \left(2 a + 8\right)\cdot 47 + \left(35 a + 10\right)\cdot 47^{2} + \left(39 a + 27\right)\cdot 47^{3} + \left(26 a + 26\right)\cdot 47^{4} + \left(28 a + 34\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$

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

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

Character values on conjugacy classes

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