Properties

Label 10.2e20_3e20.30t176.2c1
Dimension 10
Group $S_6$
Conductor $ 2^{20} \cdot 3^{20}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$10$
Group:$S_6$
Conductor:$3656158440062976= 2^{20} \cdot 3^{20} $
Artin number field: Splitting field of $f= x^{6} + 3 x^{4} - 4 x^{3} - 6 x - 2 $ 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 $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 4 a + 10 + \left(38 a + 30\right)\cdot 67 + \left(38 a + 23\right)\cdot 67^{2} + \left(9 a + 51\right)\cdot 67^{3} + \left(19 a + 23\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 49 a + 60 + \left(2 a + 65\right)\cdot 67 + \left(17 a + 22\right)\cdot 67^{2} + \left(53 a + 27\right)\cdot 67^{3} + \left(55 a + 58\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 13 a + 66 + \left(15 a + 25\right)\cdot 67 + \left(61 a + 49\right)\cdot 67^{2} + \left(52 a + 49\right)\cdot 67^{3} + \left(31 a + 63\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 63 a + 26 + \left(28 a + 44\right)\cdot 67 + \left(28 a + 5\right)\cdot 67^{2} + \left(57 a + 51\right)\cdot 67^{3} + \left(47 a + 23\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 18 a + 55 + \left(64 a + 27\right)\cdot 67 + \left(49 a + 21\right)\cdot 67^{2} + \left(13 a + 22\right)\cdot 67^{3} + \left(11 a + 27\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 54 a + 51 + \left(51 a + 6\right)\cdot 67 + \left(5 a + 11\right)\cdot 67^{2} + \left(14 a + 66\right)\cdot 67^{3} + \left(35 a + 3\right)\cdot 67^{4} +O\left(67^{ 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.