Properties

Label 3.2e6_5e4.12t33.2
Dimension 3
Group $A_5$
Conductor $ 2^{6} \cdot 5^{4}$
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$40000= 2^{6} \cdot 5^{4} $
Artin number field: Splitting field of $f= x^{5} + 10 x^{3} - 10 x^{2} + 35 x - 18 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $A_5$
Parity: Even

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character values
$c1$ $c2$
$1$ $1$ $()$ $3$ $3$
$15$ $2$ $(1,2)(3,4)$ $-1$ $-1$
$20$ $3$ $(1,2,3)$ $0$ $0$
$12$ $5$ $(1,2,3,4,5)$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
$12$ $5$ $(1,3,4,5,2)$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$
The blue line marks the conjugacy class containing complex conjugation.