Properties

Label 5.20349121.10t13.a.a
Dimension $5$
Group $S_5$
Conductor $20349121$
Root number $1$
Indicator $1$

Related objects

Learn more

Basic invariants

Dimension: $5$
Group: $S_5$
Conductor: \(20349121\)\(\medspace = 13^{2} \cdot 347^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 5.3.4511.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_5$
Projective stem field: 5.3.4511.1

Defining polynomial

$f(x)$$=$\(x^{5} - x^{3} - 2 x^{2} + 1\)  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 $\mathbb{Q}_{ 37 }$: \(x^{2} + 33 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 7 a + 16 + \left(5 a + 23\right)\cdot 37 + \left(20 a + 6\right)\cdot 37^{2} + \left(35 a + 36\right)\cdot 37^{3} + \left(22 a + 25\right)\cdot 37^{4} +O(37^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 21 + 6\cdot 37 + 6\cdot 37^{2} + 5\cdot 37^{3} + 27\cdot 37^{4} +O(37^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 18 a + 16 + \left(6 a + 36\right)\cdot 37 + \left(10 a + 27\right)\cdot 37^{2} + \left(8 a + 36\right)\cdot 37^{3} + \left(11 a + 24\right)\cdot 37^{4} +O(37^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 30 a + 7 + 31 a\cdot 37 + \left(16 a + 8\right)\cdot 37^{2} + \left(a + 10\right)\cdot 37^{3} + \left(14 a + 8\right)\cdot 37^{4} +O(37^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 19 a + 14 + \left(30 a + 7\right)\cdot 37 + \left(26 a + 25\right)\cdot 37^{2} + \left(28 a + 22\right)\cdot 37^{3} + \left(25 a + 24\right)\cdot 37^{4} +O(37^{5})\)  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$$1$$()$$5$
$10$$2$$(1,2)$$1$
$15$$2$$(1,2)(3,4)$$1$
$20$$3$$(1,2,3)$$-1$
$30$$4$$(1,2,3,4)$$-1$
$24$$5$$(1,2,3,4,5)$$0$
$20$$6$$(1,2,3)(4,5)$$1$

The blue line marks the conjugacy class containing complex conjugation.