Properties

Label 3.3e2_79e2.12t33.1c1
Dimension 3
Group $A_5$
Conductor $ 3^{2} \cdot 79^{2}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$56169= 3^{2} \cdot 79^{2} $
Artin number field: Splitting field of $f= x^{5} - 79 x^{2} + 474 x - 711 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $A_5$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $ x^{2} + 21 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 11 a + 7 + \left(22 a + 15\right)\cdot 23 + \left(12 a + 7\right)\cdot 23^{2} + \left(14 a + 3\right)\cdot 23^{3} + \left(15 a + 13\right)\cdot 23^{4} + \left(15 a + 18\right)\cdot 23^{5} + \left(12 a + 20\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$
$r_{ 2 }$ $=$ $ 12 a + 6 + 3\cdot 23 + \left(10 a + 11\right)\cdot 23^{2} + \left(8 a + 19\right)\cdot 23^{3} + \left(7 a + 6\right)\cdot 23^{4} + \left(7 a + 11\right)\cdot 23^{5} + \left(10 a + 7\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$
$r_{ 3 }$ $=$ $ 8 a + 9 + 16\cdot 23 + \left(17 a + 16\right)\cdot 23^{2} + \left(20 a + 3\right)\cdot 23^{3} + \left(19 a + 16\right)\cdot 23^{4} + 22 a\cdot 23^{5} + 12\cdot 23^{6} +O\left(23^{ 7 }\right)$
$r_{ 4 }$ $=$ $ 22 + 23 + 6\cdot 23^{2} + 14\cdot 23^{3} + 20\cdot 23^{4} + 11\cdot 23^{5} + 14\cdot 23^{6} +O\left(23^{ 7 }\right)$
$r_{ 5 }$ $=$ $ 15 a + 2 + \left(22 a + 9\right)\cdot 23 + \left(5 a + 4\right)\cdot 23^{2} + \left(2 a + 5\right)\cdot 23^{3} + \left(3 a + 12\right)\cdot 23^{4} + 3\cdot 23^{5} + \left(22 a + 14\right)\cdot 23^{6} +O\left(23^{ 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 value
$1$$1$$()$$3$
$15$$2$$(1,2)(3,4)$$-1$
$20$$3$$(1,2,3)$$0$
$12$$5$$(1,2,3,4,5)$$-\zeta_{5}^{3} - \zeta_{5}^{2}$
$12$$5$$(1,3,4,5,2)$$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
The blue line marks the conjugacy class containing complex conjugation.