Properties

Label 2.93.10t6.b
Dimension 2
Group $D_5\times C_5$
Conductor $ 3 \cdot 31 $
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$D_5\times C_5$
Conductor:$93= 3 \cdot 31 $
Artin number field: Splitting field of $f= x^{10} + 2 x^{8} - 3 x^{7} + 3 x^{6} - 7 x^{5} + 8 x^{4} - 7 x^{3} + 7 x^{2} - 4 x + 1 $ over $\Q$
Size of Galois orbit: 4
Smallest containing permutation representation: $D_5\times C_5$
Parity: Odd
Projective image: $D_5$
Projective field: Galois closure of 5.1.8311689.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{5} + 4 x + 11 $
Roots:
$r_{ 1 }$ $=$ $ 6 a^{4} + 8 a^{3} + 10 a^{2} + 8 a + 5 + \left(8 a^{4} + 10 a^{3} + 6 a^{2} + 7 a\right)\cdot 13 + \left(7 a^{4} + 8 a^{3} + 7 a^{2} + 12 a + 9\right)\cdot 13^{2} + \left(9 a^{4} + 11 a^{3} + 2 a^{2} + 8 a\right)\cdot 13^{3} + \left(9 a^{4} + 12 a^{3} + 11 a^{2} + 7 a + 3\right)\cdot 13^{4} + \left(4 a^{4} + 11 a^{3} + a^{2} + 12 a + 3\right)\cdot 13^{5} + \left(6 a^{4} + 5 a^{3} + a^{2} + 11 a + 1\right)\cdot 13^{6} + \left(4 a^{4} + 7 a + 3\right)\cdot 13^{7} + \left(7 a^{4} + 7 a^{3} + 3 a^{2} + 6 a + 11\right)\cdot 13^{8} +O\left(13^{ 9 }\right)$
$r_{ 2 }$ $=$ $ 2 a^{4} + 10 a^{3} + 3 a^{2} + 12 a + \left(2 a^{4} + 9 a^{3} + 3 a + 1\right)\cdot 13 + \left(4 a^{4} + 2 a^{3} + 11 a^{2} + 3\right)\cdot 13^{2} + \left(11 a^{4} + a^{3} + 8 a^{2} + 5 a + 6\right)\cdot 13^{3} + \left(8 a^{4} + 3 a^{3} + 5\right)\cdot 13^{4} + \left(12 a^{4} + 12 a^{3} + 12 a^{2} + 6 a + 10\right)\cdot 13^{5} + \left(12 a^{4} + 6 a^{3} + a^{2} + 3 a + 6\right)\cdot 13^{6} + \left(a^{4} + 5 a^{2} + 5 a + 5\right)\cdot 13^{7} + \left(4 a^{4} + 3 a^{3} + 3 a^{2} + 6 a + 11\right)\cdot 13^{8} +O\left(13^{ 9 }\right)$
$r_{ 3 }$ $=$ $ 8 a^{4} + 6 a^{3} + 3 a^{2} + 11 a + 1 + \left(11 a^{3} + 4 a^{2} + 9 a + 9\right)\cdot 13 + \left(7 a^{4} + 3 a^{3} + 2 a^{2} + 11 a + 9\right)\cdot 13^{2} + \left(9 a^{4} + 6 a^{3} + 4 a^{2} + 5\right)\cdot 13^{3} + \left(4 a^{4} + 2 a^{3} + 11 a^{2} + 3 a + 10\right)\cdot 13^{4} + \left(12 a^{4} + 2 a^{3} + 7 a^{2} + 6 a + 6\right)\cdot 13^{5} + \left(2 a^{4} + a^{2} + 6 a + 8\right)\cdot 13^{6} + \left(3 a^{4} + 6 a^{3} + 4 a^{2} + 10 a + 1\right)\cdot 13^{7} + \left(11 a^{3} + 2 a^{2} + 4\right)\cdot 13^{8} +O\left(13^{ 9 }\right)$
$r_{ 4 }$ $=$ $ 6 a^{4} + a^{3} + 8 a^{2} + a + 5 + \left(a^{4} + 9 a^{3} + 9 a^{2} + 3 a + 1\right)\cdot 13 + \left(6 a^{4} + 12 a^{3} + 11 a^{2} + 8 a + 4\right)\cdot 13^{2} + \left(12 a^{4} + 9 a^{3} + 11 a^{2} + 12 a + 7\right)\cdot 13^{3} + \left(6 a^{4} + 4 a^{3} + 6 a + 4\right)\cdot 13^{4} + \left(2 a^{4} + 2 a^{3} + 5 a^{2} + 5 a + 1\right)\cdot 13^{5} + \left(12 a^{4} + 11 a^{3} + 3 a^{2} + 5 a + 12\right)\cdot 13^{6} + \left(4 a^{4} + 9 a^{3} + 11 a^{2} + 1\right)\cdot 13^{7} + \left(10 a^{4} + 8 a^{3} + 7 a^{2}\right)\cdot 13^{8} +O\left(13^{ 9 }\right)$
$r_{ 5 }$ $=$ $ 11 a^{4} + 2 a^{3} + 10 a^{2} + 11 a + \left(6 a^{4} + 6 a^{3} + 10 a^{2} + 7 a + 7\right)\cdot 13 + \left(a^{4} + 12 a^{3} + 5 a^{2} + 3 a + 12\right)\cdot 13^{2} + \left(10 a^{4} + 5 a^{3} + 4 a^{2} + 10 a + 12\right)\cdot 13^{3} + \left(10 a^{2} + 8 a + 1\right)\cdot 13^{4} + \left(a^{4} + 2 a^{3} + 7 a^{2} + 11 a + 10\right)\cdot 13^{5} + \left(7 a^{4} + 8 a^{3} + 2 a^{2} + 9 a + 2\right)\cdot 13^{6} + \left(4 a^{3} + 2 a^{2} + 3 a + 5\right)\cdot 13^{7} + \left(6 a^{4} + 7 a^{3} + 10 a^{2} + 11 a\right)\cdot 13^{8} +O\left(13^{ 9 }\right)$
$r_{ 6 }$ $=$ $ 7 a^{4} + 8 a^{3} + 5 a^{2} + 8 a + 8 + \left(11 a^{4} + 8 a^{3} + 12 a^{2} + 10 a + 11\right)\cdot 13 + \left(3 a^{4} + 6 a^{3} + a^{2} + 3 a + 1\right)\cdot 13^{2} + \left(4 a^{4} + 10 a^{3} + 8 a^{2} + 10 a + 2\right)\cdot 13^{3} + \left(9 a^{4} + 5 a^{3} + 11 a^{2} + 4 a + 11\right)\cdot 13^{4} + \left(10 a^{4} + 5 a^{3} + 6 a^{2} + 9 a + 9\right)\cdot 13^{5} + \left(a^{4} + 8 a^{3} + 5 a^{2} + 12 a + 6\right)\cdot 13^{6} + \left(a^{4} + 9 a^{3} + 2 a^{2} + 12 a + 4\right)\cdot 13^{7} + \left(a^{4} + 12 a^{3} + 11 a^{2} + 6 a + 5\right)\cdot 13^{8} +O\left(13^{ 9 }\right)$
$r_{ 7 }$ $=$ $ 6 a^{4} + 7 a^{3} + 4 a^{2} + 4 a + 10 + \left(2 a^{4} + 10 a^{3} + 2 a^{2} + 3 a\right)\cdot 13 + \left(6 a^{4} + 8 a^{3} + 2 a^{2} + 11 a + 4\right)\cdot 13^{2} + \left(5 a^{4} + 3 a^{3} + 10 a^{2} + 2 a + 3\right)\cdot 13^{3} + \left(8 a^{4} + 11 a^{3} + 3 a^{2} + 11 a + 8\right)\cdot 13^{4} + \left(10 a^{4} + 5 a^{3} + 2 a^{2} + 11 a + 9\right)\cdot 13^{5} + \left(6 a^{4} + 4 a^{3} + 4 a^{2} + 8 a + 9\right)\cdot 13^{6} + \left(2 a^{4} + 10 a^{3} + a^{2} + 7 a + 3\right)\cdot 13^{7} + \left(8 a^{4} + 8 a^{3} + 10 a^{2} + 7 a + 7\right)\cdot 13^{8} +O\left(13^{ 9 }\right)$
$r_{ 8 }$ $=$ $ 4 a^{4} + 7 a^{3} + 10 a^{2} + 7 a + 1 + \left(2 a^{4} + 3 a^{3} + 6 a^{2} + 3 a + 8\right)\cdot 13 + \left(2 a^{4} + 7 a^{3} + a^{2} + 12 a + 1\right)\cdot 13^{2} + \left(6 a^{4} + 7 a^{3} + 8 a^{2} + 2 a + 8\right)\cdot 13^{3} + \left(3 a^{4} + 10\right)\cdot 13^{4} + \left(6 a^{4} + 10 a^{3} + 2 a^{2} + 6 a\right)\cdot 13^{5} + \left(3 a^{3} + 8 a^{2} + 11 a + 5\right)\cdot 13^{6} + \left(a^{4} + 10 a^{2} + 4 a + 9\right)\cdot 13^{7} + \left(7 a^{4} + 7 a^{3} + 11 a^{2} + 8 a + 3\right)\cdot 13^{8} +O\left(13^{ 9 }\right)$
$r_{ 9 }$ $=$ $ 11 a^{4} + 2 a^{3} + 10 a^{2} + 9 a + \left(2 a^{4} + 10 a^{3} + 6 a^{2} + 2\right)\cdot 13 + \left(12 a^{4} + 3 a^{3} + a^{2} + 8 a + 5\right)\cdot 13^{2} + \left(12 a^{4} + 11 a^{3} + 8 a^{2} + 12 a + 6\right)\cdot 13^{3} + \left(3 a^{4} + 7 a^{3} + 12 a^{2} + 4\right)\cdot 13^{4} + \left(10 a^{4} + 2 a^{3} + 6 a^{2} + 3\right)\cdot 13^{5} + \left(9 a^{4} + a^{3} + 5 a^{2} + 9 a + 6\right)\cdot 13^{6} + \left(7 a^{4} + a^{3} + 9 a^{2} + 9 a + 7\right)\cdot 13^{7} + \left(3 a^{4} + 3 a^{3} + 8 a^{2} + 4 a + 5\right)\cdot 13^{8} +O\left(13^{ 9 }\right)$
$r_{ 10 }$ $=$ $ 4 a^{4} + a^{3} + 2 a^{2} + 7 a + 9 + \left(11 a^{3} + 5 a^{2} + a + 10\right)\cdot 13 + \left(a^{4} + 10 a^{3} + 6 a^{2} + 6 a\right)\cdot 13^{2} + \left(9 a^{4} + 9 a^{3} + 11 a^{2} + 11 a + 12\right)\cdot 13^{3} + \left(8 a^{4} + 2 a^{3} + a^{2} + 7 a + 4\right)\cdot 13^{4} + \left(6 a^{4} + 10 a^{3} + 12 a^{2} + 8 a + 9\right)\cdot 13^{5} + \left(4 a^{4} + a^{3} + 4 a^{2} + 11 a + 5\right)\cdot 13^{6} + \left(11 a^{4} + 9 a^{3} + 5 a^{2} + a + 9\right)\cdot 13^{7} + \left(3 a^{4} + 8 a^{3} + 9 a^{2} + 12 a + 2\right)\cdot 13^{8} +O\left(13^{ 9 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 10 }$ Character values
$c1$ $c2$ $c3$ $c4$
$1$ $1$ $()$ $2$ $2$ $2$ $2$
$5$ $2$ $(1,6)(2,5)(3,8)(4,7)(9,10)$ $0$ $0$ $0$ $0$
$1$ $5$ $(1,2,10,3,4)(5,9,8,7,6)$ $2 \zeta_{5}^{3}$ $-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$ $2 \zeta_{5}$ $2 \zeta_{5}^{2}$
$1$ $5$ $(1,10,4,2,3)(5,8,6,9,7)$ $2 \zeta_{5}$ $2 \zeta_{5}^{3}$ $2 \zeta_{5}^{2}$ $-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$
$1$ $5$ $(1,3,2,4,10)(5,7,9,6,8)$ $-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$ $2 \zeta_{5}^{2}$ $2 \zeta_{5}^{3}$ $2 \zeta_{5}$
$1$ $5$ $(1,4,3,10,2)(5,6,7,8,9)$ $2 \zeta_{5}^{2}$ $2 \zeta_{5}$ $-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$ $2 \zeta_{5}^{3}$
$2$ $5$ $(1,2,10,3,4)$ $\zeta_{5}^{3} + 1$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5}$ $\zeta_{5} + 1$ $\zeta_{5}^{2} + 1$
$2$ $5$ $(1,10,4,2,3)$ $\zeta_{5} + 1$ $\zeta_{5}^{3} + 1$ $\zeta_{5}^{2} + 1$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5}$
$2$ $5$ $(1,3,2,4,10)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5}$ $\zeta_{5}^{2} + 1$ $\zeta_{5}^{3} + 1$ $\zeta_{5} + 1$
$2$ $5$ $(1,4,3,10,2)$ $\zeta_{5}^{2} + 1$ $\zeta_{5} + 1$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5}$ $\zeta_{5}^{3} + 1$
$2$ $5$ $(1,10,4,2,3)(5,9,8,7,6)$ $\zeta_{5}^{3} + \zeta_{5}$ $-\zeta_{5}^{2} - \zeta_{5} - 1$ $\zeta_{5}^{2} + \zeta_{5}$ $-\zeta_{5}^{3} - \zeta_{5} - 1$
$2$ $5$ $(1,4,3,10,2)(5,8,6,9,7)$ $\zeta_{5}^{2} + \zeta_{5}$ $\zeta_{5}^{3} + \zeta_{5}$ $-\zeta_{5}^{3} - \zeta_{5} - 1$ $-\zeta_{5}^{2} - \zeta_{5} - 1$
$2$ $5$ $(1,2,10,3,4)(5,7,9,6,8)$ $-\zeta_{5}^{2} - \zeta_{5} - 1$ $-\zeta_{5}^{3} - \zeta_{5} - 1$ $\zeta_{5}^{3} + \zeta_{5}$ $\zeta_{5}^{2} + \zeta_{5}$
$2$ $5$ $(1,3,2,4,10)(5,6,7,8,9)$ $-\zeta_{5}^{3} - \zeta_{5} - 1$ $\zeta_{5}^{2} + \zeta_{5}$ $-\zeta_{5}^{2} - \zeta_{5} - 1$ $\zeta_{5}^{3} + \zeta_{5}$
$2$ $5$ $(1,3,2,4,10)(5,8,6,9,7)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ $\zeta_{5}^{3} + \zeta_{5}^{2}$ $\zeta_{5}^{3} + \zeta_{5}^{2}$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$
$2$ $5$ $(1,2,10,3,4)(5,6,7,8,9)$ $\zeta_{5}^{3} + \zeta_{5}^{2}$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ $\zeta_{5}^{3} + \zeta_{5}^{2}$
$5$ $10$ $(1,8,2,7,10,6,3,5,4,9)$ $0$ $0$ $0$ $0$
$5$ $10$ $(1,7,3,9,2,6,4,8,10,5)$ $0$ $0$ $0$ $0$
$5$ $10$ $(1,5,10,8,4,6,2,9,3,7)$ $0$ $0$ $0$ $0$
$5$ $10$ $(1,9,4,5,3,6,10,7,2,8)$ $0$ $0$ $0$ $0$
The blue line marks the conjugacy class containing complex conjugation.