Properties

Label 2.18775.6t3.a.a
Dimension $2$
Group $D_{6}$
Conductor $18775$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(18775\)\(\medspace = 5^{2} \cdot 751 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 6.0.52945593875.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.751.2t1.a.a
Projective image: $S_3$
Projective stem field: 3.1.751.1

Defining polynomial

$f(x)$$=$\(x^{6} - 2 x^{5} + 47 x^{4} - 51 x^{3} + 534 x^{2} - 115 x + 945\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 9.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \(x^{2} + 16 x + 3\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 11 a + 5 + \left(6 a + 10\right)\cdot 17 + \left(5 a + 11\right)\cdot 17^{2} + \left(5 a + 3\right)\cdot 17^{3} + \left(a + 4\right)\cdot 17^{4} + \left(16 a + 16\right)\cdot 17^{5} + \left(11 a + 14\right)\cdot 17^{6} + \left(9 a + 10\right)\cdot 17^{7} + \left(8 a + 6\right)\cdot 17^{8} +O(17^{9})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 3 a + 11 + \left(13 a + 3\right)\cdot 17 + 6\cdot 17^{2} + \left(3 a + 13\right)\cdot 17^{3} + \left(6 a + 10\right)\cdot 17^{4} + \left(4 a + 7\right)\cdot 17^{5} + \left(14 a + 8\right)\cdot 17^{6} + \left(16 a + 11\right)\cdot 17^{7} + \left(10 a + 16\right)\cdot 17^{8} +O(17^{9})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 10 + 16\cdot 17 + 16\cdot 17^{2} + 4\cdot 17^{3} + 9\cdot 17^{4} + 3\cdot 17^{5} + 7\cdot 17^{6} + 8\cdot 17^{7} + 6\cdot 17^{8} +O(17^{9})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 14 a + 14 + \left(3 a + 13\right)\cdot 17 + \left(16 a + 10\right)\cdot 17^{2} + \left(13 a + 15\right)\cdot 17^{3} + \left(10 a + 13\right)\cdot 17^{4} + \left(12 a + 5\right)\cdot 17^{5} + \left(2 a + 1\right)\cdot 17^{6} + 14\cdot 17^{7} + \left(6 a + 10\right)\cdot 17^{8} +O(17^{9})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 14 + 12\cdot 17^{2} + 9\cdot 17^{3} + 12\cdot 17^{4} + 3\cdot 17^{5} + 8\cdot 17^{6} + 14\cdot 17^{7} + 4\cdot 17^{8} +O(17^{9})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 6 a + 16 + \left(10 a + 5\right)\cdot 17 + \left(11 a + 10\right)\cdot 17^{2} + \left(11 a + 3\right)\cdot 17^{3} + 15 a\cdot 17^{4} + 14\cdot 17^{5} + \left(5 a + 10\right)\cdot 17^{6} + \left(7 a + 8\right)\cdot 17^{7} + \left(8 a + 5\right)\cdot 17^{8} +O(17^{9})\)  Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,4)(2,6)(3,5)$$-2$
$3$$2$$(2,3)(5,6)$$0$
$3$$2$$(1,2)(3,5)(4,6)$$0$
$2$$3$$(1,5,6)(2,4,3)$$-1$
$2$$6$$(1,2,5,4,6,3)$$1$

The blue line marks the conjugacy class containing complex conjugation.