Properties

Label 10.2e26_5e12.30t176.1c1
Dimension 10
Group $S_6$
Conductor $ 2^{26} \cdot 5^{12}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$10$
Group:$S_6$
Conductor:$16384000000000000= 2^{26} \cdot 5^{12} $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + 5 x^{4} - 10 x^{2} + 8 x - 6 $ 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 $\Q_{ 193 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$: $ x^{2} + 192 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 188 + 45\cdot 193 + 5\cdot 193^{2} + 143\cdot 193^{3} + 165\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 40 + 28\cdot 193 + 38\cdot 193^{2} + 152\cdot 193^{3} + 166\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 138 a + 69 + \left(177 a + 124\right)\cdot 193 + \left(10 a + 170\right)\cdot 193^{2} + \left(6 a + 11\right)\cdot 193^{3} + \left(5 a + 142\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 154 + 160\cdot 193 + 14\cdot 193^{2} + 52\cdot 193^{3} + 88\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 55 a + 14 + \left(15 a + 164\right)\cdot 193 + \left(182 a + 3\right)\cdot 193^{2} + \left(186 a + 7\right)\cdot 193^{3} + \left(187 a + 141\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 116 + 55\cdot 193 + 153\cdot 193^{2} + 19\cdot 193^{3} + 68\cdot 193^{4} +O\left(193^{ 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.