Properties

Label 10.5e4_191e6.30t176.1c1
Dimension 10
Group $S_6$
Conductor $ 5^{4} \cdot 191^{6}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$10$
Group:$S_6$
Conductor:$30344516420400625= 5^{4} \cdot 191^{6} $
Artin number field: Splitting field of $f= x^{6} - x^{4} - 3 x^{3} + x + 3 $ 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 $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $ x^{2} + 70 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 40 + 15\cdot 73 + 43\cdot 73^{2} + 33\cdot 73^{3} + 30\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 47 + 27\cdot 73 + 48\cdot 73^{2} + 54\cdot 73^{3} + 59\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 61 a + 56 + \left(69 a + 68\right)\cdot 73 + \left(13 a + 32\right)\cdot 73^{2} + \left(32 a + 68\right)\cdot 73^{3} + \left(58 a + 5\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 58 + 32\cdot 73 + 12\cdot 73^{3} + 31\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 71 + 2\cdot 73 + 16\cdot 73^{2} + 45\cdot 73^{3} + 15\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 12 a + 20 + \left(3 a + 71\right)\cdot 73 + \left(59 a + 4\right)\cdot 73^{2} + \left(40 a + 5\right)\cdot 73^{3} + \left(14 a + 3\right)\cdot 73^{4} +O\left(73^{ 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.