Properties

Label 10.2e6_3e12_5e12.30t176.2c1
Dimension 10
Group $S_6$
Conductor $ 2^{6} \cdot 3^{12} \cdot 5^{12}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$10$
Group:$S_6$
Conductor:$8303765625000000= 2^{6} \cdot 3^{12} \cdot 5^{12} $
Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 5 x^{3} - 5 $ 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 $\Q_{ 157 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 157 }$: $ x^{2} + 152 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 119 + 129\cdot 157 + 27\cdot 157^{2} + 55\cdot 157^{3} + 65\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 137 + 77\cdot 157 + 64\cdot 157^{2} + 115\cdot 157^{3} + 25\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 139 + 29\cdot 157 + 32\cdot 157^{2} + 97\cdot 157^{3} + 6\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 156 a + 20 + \left(84 a + 66\right)\cdot 157 + \left(15 a + 146\right)\cdot 157^{2} + \left(56 a + 142\right)\cdot 157^{3} + \left(55 a + 78\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 5 }$ $=$ $ a + 15 + \left(72 a + 21\right)\cdot 157 + \left(141 a + 139\right)\cdot 157^{2} + \left(100 a + 93\right)\cdot 157^{3} + \left(101 a + 142\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 44 + 146\cdot 157 + 60\cdot 157^{2} + 123\cdot 157^{3} + 151\cdot 157^{4} +O\left(157^{ 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.