Properties

Label 3.5e4_13e2.12t33.1
Dimension 3
Group $A_5$
Conductor $ 5^{4} \cdot 13^{2}$
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$105625= 5^{4} \cdot 13^{2} $
Artin number field: Splitting field of $f= x^{5} + 5 x^{3} - 10 x^{2} - 45 $ 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 $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 16 + 34\cdot 37^{2} + 11\cdot 37^{3} + 15\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 23 a + 29 + \left(4 a + 30\right)\cdot 37 + \left(6 a + 24\right)\cdot 37^{2} + \left(7 a + 3\right)\cdot 37^{3} + \left(32 a + 22\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 14 a + 10 + \left(32 a + 26\right)\cdot 37 + \left(30 a + 7\right)\cdot 37^{2} + \left(29 a + 26\right)\cdot 37^{3} + \left(4 a + 32\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 14 a + \left(9 a + 15\right)\cdot 37 + \left(35 a + 30\right)\cdot 37^{2} + \left(12 a + 7\right)\cdot 37^{3} + \left(3 a + 20\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 23 a + 19 + \left(27 a + 1\right)\cdot 37 + \left(a + 14\right)\cdot 37^{2} + \left(24 a + 24\right)\cdot 37^{3} + \left(33 a + 20\right)\cdot 37^{4} +O\left(37^{ 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.