Properties

Label 5.45872.6t16.a.a
Dimension $5$
Group $S_6$
Conductor $45872$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $S_6$
Conductor: \(45872\)\(\medspace = 2^{4} \cdot 47 \cdot 61 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.45872.1
Galois orbit size: $1$
Smallest permutation container: $S_6$
Parity: odd
Determinant: 1.2867.2t1.a.a
Projective image: $S_6$
Projective stem field: Galois closure of 6.0.45872.1

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{5} - x^{3} + x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{2} + 21x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 4 + 3\cdot 23 + 16\cdot 23^{2} + 18\cdot 23^{3} + 6\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 8 + 10\cdot 23 + 21\cdot 23^{2} + 6\cdot 23^{3} + 18\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( a + 10 + \left(4 a + 2\right)\cdot 23 + \left(21 a + 15\right)\cdot 23^{2} + \left(20 a + 4\right)\cdot 23^{3} + \left(4 a + 21\right)\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 22 a + 12 + \left(18 a + 9\right)\cdot 23 + \left(a + 7\right)\cdot 23^{2} + \left(2 a + 2\right)\cdot 23^{3} + \left(18 a + 10\right)\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 10 a + 8 + 3\cdot 23 + \left(14 a + 2\right)\cdot 23^{2} + \left(22 a + 14\right)\cdot 23^{3} + \left(13 a + 3\right)\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 13 a + 5 + \left(22 a + 17\right)\cdot 23 + \left(8 a + 6\right)\cdot 23^{2} + 22\cdot 23^{3} + \left(9 a + 8\right)\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display

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.