Properties

Label 4.33792250337.10t12.b.a
Dimension $4$
Group $\PGL(2,5)$
Conductor $33792250337$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $\PGL(2,5)$
Conductor: \(33792250337\)\(\medspace = 53^{3} \cdot 61^{3}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 6.2.33792250337.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Determinant: 1.3233.2t1.a.a
Projective image: $S_5$
Projective stem field: 6.2.33792250337.1

Defining polynomial

$f(x)$$=$\(x^{6} - x^{5} - 19 x^{4} + 58 x^{3} + 220 x^{2} - 112 x - 741\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \(x^{2} + 12 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 13 + 8\cdot 13^{2} + 10\cdot 13^{3} + 12\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 5 a + 12 + \left(6 a + 8\right)\cdot 13 + \left(3 a + 6\right)\cdot 13^{2} + \left(12 a + 3\right)\cdot 13^{3} + \left(7 a + 4\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 4 + 2\cdot 13 + 8\cdot 13^{2} + 12\cdot 13^{3} + 9\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 8 a + 6 + \left(3 a + 10\right)\cdot 13 + \left(7 a + 10\right)\cdot 13^{2} + \left(2 a + 8\right)\cdot 13^{3} + \left(4 a + 11\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 8 a + 4 + \left(6 a + 10\right)\cdot 13 + \left(9 a + 3\right)\cdot 13^{2} + 12\cdot 13^{3} + \left(5 a + 12\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 5 a + 1 + \left(9 a + 6\right)\cdot 13 + \left(5 a + 1\right)\cdot 13^{2} + \left(10 a + 4\right)\cdot 13^{3} + 8 a\cdot 13^{4} +O(13^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.