Properties

Label 5.5e8_17e2.6t15.1c1
Dimension 5
Group $A_6$
Conductor $ 5^{8} \cdot 17^{2}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$A_6$
Conductor:$112890625= 5^{8} \cdot 17^{2} $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + 5 x^{2} - 11 x - 13 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $A_6$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 16 a + 29 + \left(29 a + 4\right)\cdot 47 + \left(12 a + 2\right)\cdot 47^{2} + \left(21 a + 23\right)\cdot 47^{3} + \left(29 a + 37\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 31 + 33\cdot 47 + 4\cdot 47^{2} + 34\cdot 47^{3} + 24\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 31 a + 14 + 17 a\cdot 47 + \left(34 a + 45\right)\cdot 47^{2} + \left(25 a + 5\right)\cdot 47^{3} + \left(17 a + 28\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 9 a + 41 + \left(10 a + 26\right)\cdot 47 + \left(42 a + 1\right)\cdot 47^{2} + \left(13 a + 20\right)\cdot 47^{3} + \left(23 a + 45\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 38 a + 12 + \left(36 a + 38\right)\cdot 47 + \left(4 a + 28\right)\cdot 47^{2} + \left(33 a + 5\right)\cdot 47^{3} + \left(23 a + 31\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 16 + 37\cdot 47 + 11\cdot 47^{2} + 5\cdot 47^{3} + 21\cdot 47^{4} +O\left(47^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,2,3)$
$(1,2)(3,4,5,6)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$5$
$45$$2$$(1,2)(3,4)$$1$
$40$$3$$(1,2,3)(4,5,6)$$2$
$40$$3$$(1,2,3)$$-1$
$90$$4$$(1,2,3,4)(5,6)$$-1$
$72$$5$$(1,2,3,4,5)$$0$
$72$$5$$(1,3,4,5,2)$$0$
The blue line marks the conjugacy class containing complex conjugation.