Properties

Label 2.88.10t6.b.a
Dimension 2
Group $D_5\times C_5$
Conductor $ 2^{3} \cdot 11 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$D_5\times C_5$
Conductor:$88= 2^{3} \cdot 11 $
Artin number field: Splitting field of 10.0.479756288.1 defined by $f= x^{10} - 2 x^{9} + x^{8} + 2 x^{7} - 3 x^{6} + 2 x^{4} + 2 x^{3} - x^{2} - 2 x + 1 $ over $\Q$
Size of Galois orbit: 4
Smallest containing permutation representation: $D_5\times C_5$
Parity: Odd
Determinant: 1.88.10t1.a.a
Projective image: $D_5$
Projective field: Galois closure of 5.1.937024.1

Galois action

Roots of defining polynomial

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

Generators of the action on the roots $r_1, \ldots, r_{ 10 }$

Cycle notation
$(2,9,8,5,3)$
$(1,9,6,2,7,3,4,5,10,8)$
$(1,10,4,7,6)(2,9,8,5,3)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 10 }$ Character value
$1$$1$$()$$2$
$5$$2$$(1,3)(2,10)(4,9)(5,6)(7,8)$$0$
$1$$5$$(1,6,7,4,10)(2,3,5,8,9)$$2 \zeta_{5}^{3}$
$1$$5$$(1,7,10,6,4)(2,5,9,3,8)$$2 \zeta_{5}$
$1$$5$$(1,4,6,10,7)(2,8,3,9,5)$$-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$
$1$$5$$(1,10,4,7,6)(2,9,8,5,3)$$2 \zeta_{5}^{2}$
$2$$5$$(2,9,8,5,3)$$\zeta_{5}^{2} + 1$
$2$$5$$(2,8,3,9,5)$$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5}$
$2$$5$$(2,5,9,3,8)$$\zeta_{5} + 1$
$2$$5$$(2,3,5,8,9)$$\zeta_{5}^{3} + 1$
$2$$5$$(1,10,4,7,6)(2,8,3,9,5)$$-\zeta_{5}^{3} - \zeta_{5} - 1$
$2$$5$$(1,4,6,10,7)(2,3,5,8,9)$$-\zeta_{5}^{2} - \zeta_{5} - 1$
$2$$5$$(1,7,10,6,4)(2,9,8,5,3)$$\zeta_{5}^{2} + \zeta_{5}$
$2$$5$$(1,6,7,4,10)(2,5,9,3,8)$$\zeta_{5}^{3} + \zeta_{5}$
$2$$5$$(1,4,6,10,7)(2,5,9,3,8)$$-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$
$2$$5$$(1,6,7,4,10)(2,9,8,5,3)$$\zeta_{5}^{3} + \zeta_{5}^{2}$
$5$$10$$(1,9,6,2,7,3,4,5,10,8)$$0$
$5$$10$$(1,2,4,8,6,3,10,9,7,5)$$0$
$5$$10$$(1,5,7,9,10,3,6,8,4,2)$$0$
$5$$10$$(1,8,10,5,4,3,7,2,6,9)$$0$
The blue line marks the conjugacy class containing complex conjugation.