Properties

Label 4.3017.5t5.b.a
Dimension $4$
Group $S_5$
Conductor $3017$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $S_5$
Conductor: \(3017\)\(\medspace = 7 \cdot 431 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.1.3017.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Determinant: 1.3017.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.1.3017.1

Defining polynomial

$f(x)$$=$ \( x^{5} - x^{2} + 1 \) 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:
$r_{ 1 }$ $=$ \( 4 a + 22 + \left(9 a + 13\right)\cdot 23 + \left(17 a + 13\right)\cdot 23^{2} + \left(16 a + 5\right)\cdot 23^{3} + \left(2 a + 8\right)\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 15 a + 11 + \left(12 a + 22\right)\cdot 23 + \left(8 a + 14\right)\cdot 23^{2} + \left(3 a + 16\right)\cdot 23^{3} + \left(18 a + 16\right)\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 11 + 17\cdot 23 + 4\cdot 23^{2} + 10\cdot 23^{3} + 20\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 8 a + 18 + \left(10 a + 9\right)\cdot 23 + \left(14 a + 19\right)\cdot 23^{2} + \left(19 a + 14\right)\cdot 23^{3} + \left(4 a + 3\right)\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 19 a + 7 + \left(13 a + 5\right)\cdot 23 + \left(5 a + 16\right)\cdot 23^{2} + \left(6 a + 21\right)\cdot 23^{3} + \left(20 a + 19\right)\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

Cycle notation
$(1,2)$
$(1,2,3,4,5)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character value
$1$$1$$()$$4$
$10$$2$$(1,2)$$2$
$15$$2$$(1,2)(3,4)$$0$
$20$$3$$(1,2,3)$$1$
$30$$4$$(1,2,3,4)$$0$
$24$$5$$(1,2,3,4,5)$$-1$
$20$$6$$(1,2,3)(4,5)$$-1$

The blue line marks the conjugacy class containing complex conjugation.