Properties

Label 4.18927429625.10t12.b.a
Dimension $4$
Group $S_5$
Conductor $18927429625$
Root number $1$
Indicator $1$

Related objects

Learn more

Basic invariants

Dimension: $4$
Group: $S_5$
Conductor: \(18927429625\)\(\medspace = 5^{3} \cdot 13^{3} \cdot 41^{3}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 5.1.2665.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Determinant: 1.2665.2t1.a.a
Projective image: $S_5$
Projective stem field: 5.1.2665.1

Defining polynomial

$f(x)$$=$\(x^{5} - x^{4} + x^{3} - x^{2} + 2 x - 1\)  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 $\mathbb{Q}_{ 23 }$: \(x^{2} + 21 x + 5\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 2 + 21\cdot 23 + 4\cdot 23^{2} + 23^{3} + 5\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 10 + 12\cdot 23 + 15\cdot 23^{2} + 5\cdot 23^{3} + 4\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 9 + 9\cdot 23 + 2\cdot 23^{2} + 11\cdot 23^{3} + 2\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 13 a + \left(3 a + 16\right)\cdot 23 + \left(17 a + 7\right)\cdot 23^{2} + \left(17 a + 16\right)\cdot 23^{3} + \left(9 a + 4\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 10 a + 3 + \left(19 a + 10\right)\cdot 23 + \left(5 a + 15\right)\cdot 23^{2} + \left(5 a + 11\right)\cdot 23^{3} + \left(13 a + 6\right)\cdot 23^{4} +O(23^{5})\)  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.