Properties

Label 5.3e8_7e2_23e2.6t15.1c1
Dimension 5
Group $A_6$
Conductor $ 3^{8} \cdot 7^{2} \cdot 23^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$5$
Group:$A_6$
Conductor:$170067681= 3^{8} \cdot 7^{2} \cdot 23^{2} $
Artin number field: Splitting field of $f= x^{6} - 3 x^{5} - 9 x^{4} + 14 x^{3} + 27 x^{2} - 3 x - 10 $ 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 $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 27 + 19\cdot 31 + 20\cdot 31^{2} + 2\cdot 31^{3} + 10\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 20 a + 29 + \left(23 a + 28\right)\cdot 31 + 11 a\cdot 31^{2} + \left(23 a + 12\right)\cdot 31^{3} + 30\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 14 + 5\cdot 31 + 29\cdot 31^{2} + 20\cdot 31^{3} + 24\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 12 a + 13 + \left(14 a + 29\right)\cdot 31 + \left(17 a + 25\right)\cdot 31^{2} + \left(13 a + 15\right)\cdot 31^{3} + \left(29 a + 2\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 19 a + 6 + \left(16 a + 15\right)\cdot 31 + \left(13 a + 15\right)\cdot 31^{2} + \left(17 a + 25\right)\cdot 31^{3} + \left(a + 16\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 11 a + 7 + \left(7 a + 25\right)\cdot 31 + 19 a\cdot 31^{2} + \left(7 a + 16\right)\cdot 31^{3} + \left(30 a + 8\right)\cdot 31^{4} +O\left(31^{ 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)$$-1$
$40$$3$$(1,2,3)$$2$
$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.