Properties

Label 5.5_9923.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 5 \cdot 9923 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$49615= 5 \cdot 9923 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + 3 x^{4} - x^{3} + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.5_9923.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{2} + 69 x + 7 $
Roots:
$r_{ 1 }$ $=$ $ 54 + 17\cdot 71 + 11\cdot 71^{2} + 26\cdot 71^{3} + 23\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 23 + 56\cdot 71 + 61\cdot 71^{2} + 21\cdot 71^{3} + 44\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 8 + 65\cdot 71 + 39\cdot 71^{2} + 37\cdot 71^{3} + 35\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 4 }$ $=$ $ a + 1 + \left(41 a + 45\right)\cdot 71 + \left(25 a + 3\right)\cdot 71^{2} + \left(3 a + 39\right)\cdot 71^{3} + \left(26 a + 47\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 55 + 44\cdot 71 + 11\cdot 71^{2} + 68\cdot 71^{3} + 36\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 70 a + 3 + \left(29 a + 55\right)\cdot 71 + \left(45 a + 13\right)\cdot 71^{2} + \left(67 a + 20\right)\cdot 71^{3} + \left(44 a + 25\right)\cdot 71^{4} +O\left(71^{ 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.