Properties

Label 2.77.10t6.b
Dimension 2
Group $D_5\times C_5$
Conductor $ 7 \cdot 11 $
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$D_5\times C_5$
Conductor:$77= 7 \cdot 11 $
Artin number field: Splitting field of $f= x^{10} - 3 x^{9} + 7 x^{8} - 12 x^{7} + 15 x^{6} - 15 x^{5} + 12 x^{4} - 7 x^{3} + 4 x^{2} - 2 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 5.1.717409.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $ x^{5} + 3 x + 18 $
Roots:
$r_{ 1 }$ $=$ $ 3 a^{4} + 9 a^{2} + 9 a + 14 + \left(15 a^{4} + 15 a^{3} + 3 a^{2} + 20 a + 13\right)\cdot 23 + \left(22 a^{4} + 15 a^{3} + 16 a^{2} + 9 a + 1\right)\cdot 23^{2} + \left(17 a^{4} + a^{3} + 17 a^{2} + 18 a + 15\right)\cdot 23^{3} + \left(6 a^{4} + a^{3} + 7 a^{2} + 8 a + 19\right)\cdot 23^{4} + \left(22 a^{4} + 12 a^{3} + 18 a^{2} + 11 a + 22\right)\cdot 23^{5} + \left(2 a^{4} + 15 a^{3} + 18 a^{2} + 12 a + 5\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$
$r_{ 2 }$ $=$ $ 4 a^{4} + 20 a^{3} + 16 a^{2} + a + 8 + \left(13 a^{4} + 8 a^{3} + 2 a^{2} + 16 a + 8\right)\cdot 23 + \left(21 a^{4} + 11 a^{3} + 14 a + 3\right)\cdot 23^{2} + \left(3 a^{4} + 3 a^{3} + 10 a^{2} + 14 a + 19\right)\cdot 23^{3} + \left(a^{3} + 4 a^{2} + 19 a + 10\right)\cdot 23^{4} + \left(12 a^{4} + 17 a^{3} + 12 a^{2} + 8 a + 13\right)\cdot 23^{5} + \left(4 a^{4} + 19 a^{3} + 5 a^{2} + 16\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$
$r_{ 3 }$ $=$ $ 6 a^{4} + 21 a^{3} + 16 a^{2} + 6 a + 22 + \left(18 a^{4} + 17 a^{3} + 18 a^{2} + 13 a + 15\right)\cdot 23 + \left(3 a^{4} + 10 a^{3} + 8 a^{2} + 17 a + 6\right)\cdot 23^{2} + \left(3 a^{4} + 10 a^{3} + 10 a^{2} + 12 a + 17\right)\cdot 23^{3} + \left(13 a^{4} + 4 a^{3} + 3 a + 9\right)\cdot 23^{4} + \left(12 a^{4} + 9 a^{3} + 9 a^{2} + 7 a + 19\right)\cdot 23^{5} + \left(12 a^{4} + 16 a^{3} + 20 a^{2} + 8 a + 12\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$
$r_{ 4 }$ $=$ $ 11 a^{4} + 22 a^{3} + 16 a^{2} + 14 a + 11 + \left(3 a^{4} + 17 a^{3} + 4 a^{2} + 13 a + 3\right)\cdot 23 + \left(7 a^{4} + 21 a^{3} + 16 a^{2} + 4 a + 10\right)\cdot 23^{2} + \left(5 a^{4} + 8 a^{3} + 9 a^{2} + 14 a + 22\right)\cdot 23^{3} + \left(10 a^{4} + a^{3} + 2 a^{2} + 11\right)\cdot 23^{4} + \left(22 a^{4} + 10 a^{3} + 16 a^{2} + 13 a + 15\right)\cdot 23^{5} + \left(10 a^{4} + 21 a^{2} + 13\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$
$r_{ 5 }$ $=$ $ 12 a^{4} + 7 a^{3} + 13 a + 8 + \left(5 a^{4} + 21 a^{3} + 13 a^{2} + 9 a + 4\right)\cdot 23 + \left(10 a^{4} + 16 a^{3} + 12 a^{2} + 11 a + 13\right)\cdot 23^{2} + \left(7 a^{4} + 19 a^{3} + 12 a^{2} + 17 a + 3\right)\cdot 23^{3} + \left(13 a^{4} + 8 a^{3} + 11 a^{2} + 8 a + 3\right)\cdot 23^{4} + \left(11 a^{4} + 7 a^{3} + 13 a^{2} + 20\right)\cdot 23^{5} + \left(7 a^{4} + 18 a^{3} + 7 a^{2} + 7 a + 16\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$
$r_{ 6 }$ $=$ $ 12 a^{4} + 15 a^{3} + 3 a^{2} + 9 a + 18 + \left(6 a^{4} + 9 a^{3} + 21 a^{2} + 9 a + 10\right)\cdot 23 + \left(3 a^{4} + 6 a^{3} + 20 a + 5\right)\cdot 23^{2} + \left(a^{4} + 7 a^{3} + 5 a^{2} + 7 a + 3\right)\cdot 23^{3} + \left(4 a^{4} + 17 a^{3} + 11 a^{2} + 20\right)\cdot 23^{4} + \left(14 a^{4} + 2 a^{3} + 8 a^{2} + 15 a + 4\right)\cdot 23^{5} + \left(17 a^{4} + 2 a^{3} + 13 a^{2} + 20 a + 11\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$
$r_{ 7 }$ $=$ $ 13 a^{4} + 14 a^{3} + 18 a^{2} + 16 a + 2 + \left(4 a^{4} + 14 a^{3} + 21 a^{2} + 16 a + 6\right)\cdot 23 + \left(10 a^{4} + 18 a^{3} + 19 a^{2} + 11 a + 8\right)\cdot 23^{2} + \left(9 a^{4} + 15 a^{3} + 10 a^{2} + 19 a + 9\right)\cdot 23^{3} + \left(18 a^{4} + 21 a^{3} + 4 a^{2} + 21 a + 22\right)\cdot 23^{4} + \left(7 a^{4} + 6 a^{3} + a + 7\right)\cdot 23^{5} + \left(7 a^{3} + 8 a^{2} + 16 a + 20\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$
$r_{ 8 }$ $=$ $ 15 a^{4} + 12 a^{3} + 16 a^{2} + 16 a + 6 + \left(8 a^{4} + 8 a^{3} + 6 a^{2} + 5 a + 7\right)\cdot 23 + \left(4 a^{4} + 20 a^{3} + 2 a^{2} + 7 a + 8\right)\cdot 23^{2} + \left(16 a^{4} + 4 a^{3} + a^{2} + 6 a + 15\right)\cdot 23^{3} + \left(7 a^{4} + 21 a^{3} + 21 a^{2} + 22 a + 12\right)\cdot 23^{4} + \left(13 a^{4} + 11 a^{3} + 20 a + 10\right)\cdot 23^{5} + \left(18 a^{4} + 7 a^{3} + 4 a^{2} + a + 20\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$
$r_{ 9 }$ $=$ $ 17 a^{4} + 3 a^{3} + 17 a^{2} + 19 a + 20 + \left(14 a^{4} + 19 a^{3} + 17 a^{2} + 9 a + 21\right)\cdot 23 + \left(19 a^{4} + 6 a^{3} + 16 a^{2} + 19 a + 21\right)\cdot 23^{2} + \left(16 a^{4} + 13 a^{3} + 7 a^{2} + 11 a + 16\right)\cdot 23^{3} + \left(17 a^{4} + 20 a^{2} + 4 a + 13\right)\cdot 23^{4} + \left(15 a^{4} + 19 a^{3} + 21 a^{2} + 11 a + 2\right)\cdot 23^{5} + \left(11 a^{4} + 15 a^{3} + 8 a^{2} + 16 a + 13\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$
$r_{ 10 }$ $=$ $ 22 a^{4} + a^{3} + 4 a^{2} + 12 a + 9 + \left(a^{4} + 5 a^{3} + 5 a^{2}\right)\cdot 23 + \left(12 a^{4} + 9 a^{3} + 21 a^{2} + 21 a + 13\right)\cdot 23^{2} + \left(10 a^{4} + 6 a^{3} + 6 a^{2} + 14 a + 15\right)\cdot 23^{3} + \left(14 a^{3} + 8 a^{2} + a + 13\right)\cdot 23^{4} + \left(6 a^{4} + 18 a^{3} + 14 a^{2} + 2 a + 20\right)\cdot 23^{5} + \left(5 a^{4} + 11 a^{3} + 6 a^{2} + 8 a + 6\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$

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

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

Character values on conjugacy classes

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