Properties

Label 2.131.5t2.1
Dimension 2
Group $D_{5}$
Conductor $ 131 $
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$2$
Group:$D_{5}$
Conductor:$131 $
Artin number field: Splitting field of $f= x^{5} - x^{4} + 2 x^{3} - x^{2} + x + 2 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $D_{5}$
Parity: Odd

Galois action

Roots of defining polynomial

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

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

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

Character values on conjugacy classes

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