Properties

Label 10.3e14_277e4.30t176.1c1
Dimension 10
Group $S_6$
Conductor $ 3^{14} \cdot 277^{4}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$10$
Group:$S_6$
Conductor:$28158962038780329= 3^{14} \cdot 277^{4} $
Artin number field: Splitting field of $f= x^{6} - 3 x^{4} + 3 x^{2} - 3 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 30T176
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 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $ x^{2} + 21 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 2 + 7\cdot 23 + 20\cdot 23^{2} + 11\cdot 23^{3} + 19\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 3 a + 14 + \left(5 a + 17\right)\cdot 23 + \left(8 a + 3\right)\cdot 23^{2} + \left(4 a + 12\right)\cdot 23^{3} + \left(8 a + 21\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 17 + 3\cdot 23 + 15\cdot 23^{2} + 19\cdot 23^{3} +O\left(23^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 7 a + 1 + \left(6 a + 5\right)\cdot 23 + \left(5 a + 5\right)\cdot 23^{2} + \left(2 a + 18\right)\cdot 23^{3} + \left(2 a + 18\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 20 a + 20 + \left(17 a + 1\right)\cdot 23 + \left(14 a + 15\right)\cdot 23^{2} + \left(18 a + 12\right)\cdot 23^{3} + \left(14 a + 10\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 16 a + 15 + \left(16 a + 10\right)\cdot 23 + \left(17 a + 9\right)\cdot 23^{2} + \left(20 a + 17\right)\cdot 23^{3} + \left(20 a + 20\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$10$
$15$$2$$(1,2)(3,4)(5,6)$$-2$
$15$$2$$(1,2)$$2$
$45$$2$$(1,2)(3,4)$$-2$
$40$$3$$(1,2,3)(4,5,6)$$1$
$40$$3$$(1,2,3)$$1$
$90$$4$$(1,2,3,4)(5,6)$$0$
$90$$4$$(1,2,3,4)$$0$
$144$$5$$(1,2,3,4,5)$$0$
$120$$6$$(1,2,3,4,5,6)$$1$
$120$$6$$(1,2,3)(4,5)$$-1$
The blue line marks the conjugacy class containing complex conjugation.