Properties

Label 10.3e16_5e12.30t176.2c1
Dimension 10
Group $S_6$
Conductor $ 3^{16} \cdot 5^{12}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$10$
Group:$S_6$
Conductor:$10509453369140625= 3^{16} \cdot 5^{12} $
Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 15 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 $\Q_{ 97 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $ x^{2} + 96 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 84 a + 32 + \left(61 a + 94\right)\cdot 97 + \left(29 a + 80\right)\cdot 97^{2} + \left(91 a + 22\right)\cdot 97^{3} + \left(81 a + 3\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 26 a + 39 + \left(59 a + 92\right)\cdot 97 + \left(79 a + 25\right)\cdot 97^{2} + \left(78 a + 44\right)\cdot 97^{3} + \left(20 a + 18\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 75 a + 32 + \left(85 a + 93\right)\cdot 97 + \left(7 a + 34\right)\cdot 97^{2} + \left(16 a + 92\right)\cdot 97^{3} + \left(21 a + 57\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 22 a + 10 + \left(11 a + 7\right)\cdot 97 + \left(89 a + 54\right)\cdot 97^{2} + \left(80 a + 3\right)\cdot 97^{3} + \left(75 a + 63\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 71 a + 65 + \left(37 a + 28\right)\cdot 97 + \left(17 a + 46\right)\cdot 97^{2} + \left(18 a + 43\right)\cdot 97^{3} + \left(76 a + 57\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 13 a + 19 + \left(35 a + 72\right)\cdot 97 + \left(67 a + 48\right)\cdot 97^{2} + \left(5 a + 84\right)\cdot 97^{3} + \left(15 a + 90\right)\cdot 97^{4} +O\left(97^{ 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.