Properties

Label 5.7e2_149e2.6t15.2c1
Dimension 5
Group $A_6$
Conductor $ 7^{2} \cdot 149^{2}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$A_6$
Conductor:$1087849= 7^{2} \cdot 149^{2} $
Artin number field: Splitting field of $f= x^{6} - x^{5} + 2 x^{4} - 3 x^{2} + 2 x - 4 $ 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 $\Q_{ 113 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$: $ x^{2} + 101 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 95 + 112\cdot 113 + 10\cdot 113^{2} + 54\cdot 113^{3} + 94\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 97 a + 4 + \left(47 a + 63\right)\cdot 113 + \left(70 a + 76\right)\cdot 113^{2} + \left(107 a + 37\right)\cdot 113^{3} + \left(79 a + 10\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 16 a + 38 + \left(65 a + 88\right)\cdot 113 + \left(42 a + 82\right)\cdot 113^{2} + \left(5 a + 15\right)\cdot 113^{3} + \left(33 a + 71\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 29 + 19\cdot 113 + 77\cdot 113^{3} + 31\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 39 a + 79 + \left(31 a + 28\right)\cdot 113 + \left(77 a + 88\right)\cdot 113^{2} + \left(108 a + 28\right)\cdot 113^{3} + \left(41 a + 94\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 74 a + 95 + \left(81 a + 26\right)\cdot 113 + \left(35 a + 80\right)\cdot 113^{2} + \left(4 a + 12\right)\cdot 113^{3} + \left(71 a + 37\right)\cdot 113^{4} +O\left(113^{ 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.