Properties

Label 2.13100.6t3.b.a
Dimension $2$
Group $D_{6}$
Conductor $13100$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(13100\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 131 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 6.0.4496182000.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.131.2t1.a.a
Projective image: $S_3$
Projective stem field: 3.1.524.1

Defining polynomial

$f(x)$$=$\(x^{6} - 2 x^{5} + 23 x^{4} - 74 x^{3} + 173 x^{2} - 572 x + 1331\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 8.

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.