Properties

Label 4.181057.5t5.a.a
Dimension $4$
Group $S_5$
Conductor $181057$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $4$
Group: $S_5$
Conductor: \(181057\)\(\medspace = 331 \cdot 547 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.5.181057.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Determinant: 1.181057.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.5.181057.1

Defining polynomial

$f(x)$$=$ \( x^{5} - 2x^{4} - 4x^{3} + 7x^{2} + 2x - 3 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \( x^{2} + 33x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 35 a + 6 + \left(35 a + 9\right)\cdot 37 + \left(23 a + 15\right)\cdot 37^{2} + \left(20 a + 11\right)\cdot 37^{3} + a\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 22 a + 15 + \left(5 a + 1\right)\cdot 37 + \left(27 a + 22\right)\cdot 37^{2} + \left(17 a + 24\right)\cdot 37^{3} + \left(35 a + 31\right)\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 2 a + 35 + \left(a + 6\right)\cdot 37 + \left(13 a + 1\right)\cdot 37^{2} + \left(16 a + 33\right)\cdot 37^{3} + \left(35 a + 22\right)\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 15 a + 29 + \left(31 a + 1\right)\cdot 37 + \left(9 a + 14\right)\cdot 37^{2} + \left(19 a + 31\right)\cdot 37^{3} + \left(a + 7\right)\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 28 + 17\cdot 37 + 21\cdot 37^{2} + 10\cdot 37^{3} + 11\cdot 37^{4} +O(37^{5})\) 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.