Properties

Label 3.69696.12t33.a
Dimension 3
Group $A_5$
Conductor $ 2^{6} \cdot 3^{2} \cdot 11^{2}$
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$69696= 2^{6} \cdot 3^{2} \cdot 11^{2} $
Artin number field: Splitting field of $f= x^{5} - x^{4} - 4 x^{3} - 8 x^{2} + 25 x - 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $A_5$
Parity: Even
Projective image: $A_5$
Projective field: Galois closure of 5.1.8433216.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 449 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 107 + 396\cdot 449 + 197\cdot 449^{2} + 434\cdot 449^{3} + 397\cdot 449^{4} +O\left(449^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 133 + 215\cdot 449 + 174\cdot 449^{2} + 346\cdot 449^{3} + 332\cdot 449^{4} +O\left(449^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 140 + 24\cdot 449 + 378\cdot 449^{2} + 324\cdot 449^{3} + 233\cdot 449^{4} +O\left(449^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 190 + 17\cdot 449 + 398\cdot 449^{2} + 425\cdot 449^{3} + 262\cdot 449^{4} +O\left(449^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 329 + 244\cdot 449 + 198\cdot 449^{2} + 264\cdot 449^{3} + 119\cdot 449^{4} +O\left(449^{ 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.