Properties

Label 5.35099.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 35099 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$35099 $
Artin number field: Splitting field of $f= x^{6} + x^{4} + x^{2} - x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.35099.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 28 + 57\cdot 59 + 10\cdot 59^{2} + 51\cdot 59^{3} + 47\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 12 a + 46 + \left(53 a + 47\right)\cdot 59 + \left(26 a + 22\right)\cdot 59^{2} + \left(23 a + 51\right)\cdot 59^{3} + \left(4 a + 30\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 35 + 14\cdot 59 + 50\cdot 59^{2} + 14\cdot 59^{3} + 42\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 47 a + 58 + \left(5 a + 29\right)\cdot 59 + \left(32 a + 55\right)\cdot 59^{2} + \left(35 a + 47\right)\cdot 59^{3} + \left(54 a + 11\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 43 + 34\cdot 59 + 25\cdot 59^{2} + 42\cdot 59^{3} + 6\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 26 + 51\cdot 59 + 11\cdot 59^{2} + 28\cdot 59^{3} + 37\cdot 59^{4} +O\left(59^{ 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$$1$$()$$5$
$15$$2$$(1,2)(3,4)(5,6)$$-1$
$15$$2$$(1,2)$$3$
$45$$2$$(1,2)(3,4)$$1$
$40$$3$$(1,2,3)(4,5,6)$$-1$
$40$$3$$(1,2,3)$$2$
$90$$4$$(1,2,3,4)(5,6)$$-1$
$90$$4$$(1,2,3,4)$$1$
$144$$5$$(1,2,3,4,5)$$0$
$120$$6$$(1,2,3,4,5,6)$$-1$
$120$$6$$(1,2,3)(4,5)$$0$
The blue line marks the conjugacy class containing complex conjugation.