Properties

Label 5.2e6_7e2_23e2.6t15.2c1
Dimension 5
Group $A_6$
Conductor $ 2^{6} \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:$1658944= 2^{6} \cdot 7^{2} \cdot 23^{2} $
Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 5 x^{4} - 8 x^{2} + 4 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 $\Q_{ 83 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$: $ x^{2} + 82 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 22 a + 65 + \left(15 a + 64\right)\cdot 83 + \left(58 a + 53\right)\cdot 83^{2} + \left(23 a + 19\right)\cdot 83^{3} + \left(18 a + 62\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 4 a + 33 + \left(55 a + 25\right)\cdot 83 + \left(61 a + 17\right)\cdot 83^{2} + \left(66 a + 26\right)\cdot 83^{3} + \left(69 a + 56\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 79 a + 37 + \left(27 a + 76\right)\cdot 83 + \left(21 a + 23\right)\cdot 83^{2} + \left(16 a + 31\right)\cdot 83^{3} + \left(13 a + 59\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 61 a + 4 + \left(67 a + 58\right)\cdot 83 + \left(24 a + 13\right)\cdot 83^{2} + \left(59 a + 68\right)\cdot 83^{3} + \left(64 a + 56\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 6 + 21\cdot 83 + 3\cdot 83^{2} + 50\cdot 83^{3} + 79\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 24 + 3\cdot 83 + 54\cdot 83^{2} + 53\cdot 83^{3} + 17\cdot 83^{4} +O\left(83^{ 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.