Properties

Label 3.40000.12t33.e
Dimension 3
Group $A_5$
Conductor $ 2^{6} \cdot 5^{4}$
Frobenius-Schur indicator 1

Related objects

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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} - 20 x^{2} + 110 x + 116 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $A_5$
Parity: Even
Projective image: $A_5$
Projective field: Galois closure of 5.1.25000000.4

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 197 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 1 + 35\cdot 197 + 165\cdot 197^{2} + 58\cdot 197^{3} + 14\cdot 197^{4} +O\left(197^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 85 + 40\cdot 197 + 153\cdot 197^{2} + 25\cdot 197^{3} + 182\cdot 197^{4} +O\left(197^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 154 + 93\cdot 197 + 186\cdot 197^{2} + 155\cdot 197^{3} + 60\cdot 197^{4} +O\left(197^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 164 + 150\cdot 197 + 104\cdot 197^{2} + 75\cdot 197^{3} + 110\cdot 197^{4} +O\left(197^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 187 + 73\cdot 197 + 178\cdot 197^{2} + 77\cdot 197^{3} + 26\cdot 197^{4} +O\left(197^{ 5 }\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.