Properties

Label 2.57760.6t3.d.a
Dimension $2$
Group $D_{6}$
Conductor $57760$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(57760\)\(\medspace = 2^{5} \cdot 5 \cdot 19^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.6672435200.7
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.40.2t1.b.a
Projective image: $S_3$
Projective stem field: Galois closure of 3.1.14440.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 2x^{5} - 17x^{4} + 72x^{3} - 8x^{2} - 32x + 544 \) Copy content 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 $\mathbb{Q}_{ 17 }$: \( x^{2} + 16x + 3 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.