Properties

Label 2.2008.7t2.a
Dimension 2
Group $D_{7}$
Conductor $ 2^{3} \cdot 251 $
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$D_{7}$
Conductor:$2008= 2^{3} \cdot 251 $
Artin number field: Splitting field of $f= x^{7} - 3 x^{6} + 6 x^{5} - 14 x^{4} + 15 x^{3} - x^{2} + 82 x - 134 $ over $\Q$
Size of Galois orbit: 3
Smallest containing permutation representation: $D_{7}$
Parity: Odd
Projective image: $D_7$
Projective field: Galois closure of 7.1.8096384512.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 17 a + 15 + \left(41 a + 5\right)\cdot 47 + \left(16 a + 23\right)\cdot 47^{2} + \left(22 a + 6\right)\cdot 47^{3} + \left(41 a + 46\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 46 a + 22 + \left(12 a + 26\right)\cdot 47 + \left(18 a + 26\right)\cdot 47^{2} + \left(29 a + 44\right)\cdot 47^{3} + \left(38 a + 31\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 30 a + 2 + \left(5 a + 24\right)\cdot 47 + \left(30 a + 15\right)\cdot 47^{2} + \left(24 a + 34\right)\cdot 47^{3} + \left(5 a + 12\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$ $=$ $ a + 20 + \left(34 a + 6\right)\cdot 47 + \left(28 a + 3\right)\cdot 47^{2} + \left(17 a + 38\right)\cdot 47^{3} + \left(8 a + 32\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 2 a + 22 + \left(36 a + 26\right)\cdot 47 + \left(23 a + 15\right)\cdot 47^{2} + \left(4 a + 10\right)\cdot 47^{3} + \left(6 a + 31\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 37 + 2\cdot 47 + 30\cdot 47^{2} + 11\cdot 47^{3} + 41\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 45 a + 26 + \left(10 a + 2\right)\cdot 47 + \left(23 a + 27\right)\cdot 47^{2} + \left(42 a + 42\right)\cdot 47^{3} + \left(40 a + 38\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

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

Character values on conjugacy classes

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