Properties

Label 3.88209.12t33.b
Dimension 3
Group $A_5$
Conductor $ 3^{6} \cdot 11^{2}$
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$88209= 3^{6} \cdot 11^{2} $
Artin number field: Splitting field of $f= x^{5} - x^{4} + 7 x^{3} + 3 x^{2} - 30 x + 21 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $A_5$
Parity: Even
Projective image: $A_5$
Projective field: Galois closure of 5.1.10673289.2

Galois action

Roots of defining polynomial

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