Properties

Label 5.6848689.10t13.a.a
Dimension $5$
Group $\PGL(2,5)$
Conductor $6848689$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $\PGL(2,5)$
Conductor: \(6848689\)\(\medspace = 2617^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 6.2.17923019113.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_5$
Projective stem field: 6.2.17923019113.1

Defining polynomial

$f(x)$$=$\(x^{6} - x^{5} - 18 x^{4} + 3 x^{3} + 209 x^{2} + 15 x - 1064\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \(x^{2} + 24 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 3 + 23\cdot 29 + 27\cdot 29^{3} + 2\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 12 + 29 + 21\cdot 29^{2} + 4\cdot 29^{3} + 10\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 9 a + 15 + \left(23 a + 15\right)\cdot 29 + \left(3 a + 1\right)\cdot 29^{2} + \left(11 a + 7\right)\cdot 29^{3} + \left(4 a + 20\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 20 a + 2 + \left(5 a + 7\right)\cdot 29 + \left(25 a + 26\right)\cdot 29^{2} + 17 a\cdot 29^{3} + \left(24 a + 2\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 17 a + \left(24 a + 25\right)\cdot 29 + \left(25 a + 9\right)\cdot 29^{2} + \left(28 a + 22\right)\cdot 29^{3} + \left(8 a + 17\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 12 a + 27 + \left(4 a + 14\right)\cdot 29 + \left(3 a + 27\right)\cdot 29^{2} + 24\cdot 29^{3} + \left(20 a + 4\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,6,2,4,5,3)$
$(1,5)(2,4)(3,6)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.