Properties

Label 5.5611284433.6t14.b.a
Dimension $5$
Group $\PGL(2,5)$
Conductor $5611284433$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $\PGL(2,5)$
Conductor: \(5611284433\)\(\medspace = 1777^{3}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 6.2.5611284433.1
Galois orbit size: $1$
Smallest permutation container: $\PGL(2,5)$
Parity: even
Determinant: 1.1777.2t1.a.a
Projective image: $S_5$
Projective stem field: 6.2.5611284433.1

Defining polynomial

$f(x)$$=$\(x^{6} - 3 x^{5} + 8 x^{4} + 45 x^{3} - 211 x^{2} + 293 x - 121\)  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:
$r_{ 1 }$ $=$ \( 11 a + 4 + \left(20 a + 9\right)\cdot 23 + \left(19 a + 19\right)\cdot 23^{2} + 5\cdot 23^{3} + \left(20 a + 10\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 17 + 19\cdot 23 + 20\cdot 23^{2} + 17\cdot 23^{3} + 19\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 12 a + 3 + \left(2 a + 16\right)\cdot 23 + \left(3 a + 15\right)\cdot 23^{2} + \left(22 a + 10\right)\cdot 23^{3} + \left(2 a + 3\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 10 a + 8 + \left(20 a + 19\right)\cdot 23 + \left(19 a + 10\right)\cdot 23^{2} + \left(4 a + 16\right)\cdot 23^{3} + \left(13 a + 16\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 13 a + 5 + \left(2 a + 4\right)\cdot 23 + \left(3 a + 7\right)\cdot 23^{2} + \left(18 a + 6\right)\cdot 23^{3} + \left(9 a + 15\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 12 + 18\cdot 23^{2} + 11\cdot 23^{3} + 3\cdot 23^{4} +O(23^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.