Properties

Label 6.17923019113.20t30.a
Dimension $6$
Group $\PGL(2,5)$
Conductor $17923019113$
Indicator $1$

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Basic invariants

Dimension:$6$
Group:$\PGL(2,5)$
Conductor:\(17923019113\)\(\medspace = 2617^{3}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.2.17923019113.1
Galois orbit size: $1$
Smallest permutation container: 20T30
Parity: even
Projective image: $S_5$
Projective field: 6.2.17923019113.1

Galois action

Roots of defining polynomial

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 values
$c1$
$1$ $1$ $()$ $6$
$10$ $2$ $(1,5)(2,4)(3,6)$ $0$
$15$ $2$ $(2,3)(4,6)$ $-2$
$20$ $3$ $(1,2,5)(3,6,4)$ $0$
$30$ $4$ $(1,4,6,2)$ $0$
$24$ $5$ $(1,3,2,5,6)$ $1$
$20$ $6$ $(1,6,2,4,5,3)$ $0$
The blue line marks the conjugacy class containing complex conjugation.