Properties

Label 5.467943424.10t13.d.a
Dimension $5$
Group $\PGL(2,5)$
Conductor $467943424$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $\PGL(2,5)$
Conductor: \(467943424\)\(\medspace = 2^{14} \cdot 13^{4} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.58492928.4
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 6.0.58492928.4

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{5} - 3x^{4} - 6x^{3} - x^{2} + 23x + 27 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{2} + 18x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 17 a + 15 + \left(8 a + 11\right)\cdot 19 + \left(17 a + 11\right)\cdot 19^{2} + \left(7 a + 9\right)\cdot 19^{3} + \left(13 a + 7\right)\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 2 a + 13 + \left(10 a + 3\right)\cdot 19 + \left(a + 1\right)\cdot 19^{2} + 11 a\cdot 19^{3} + \left(5 a + 13\right)\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 15 + 6\cdot 19 + 2\cdot 19^{2} + 6\cdot 19^{3} + 10\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 11 + 19 + 14\cdot 19^{3} + 3\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 16 a + 13 + \left(3 a + 3\right)\cdot 19 + \left(11 a + 17\right)\cdot 19^{2} + \left(11 a + 3\right)\cdot 19^{3} + \left(2 a + 6\right)\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 3 a + 10 + \left(15 a + 10\right)\cdot 19 + \left(7 a + 5\right)\cdot 19^{2} + \left(7 a + 4\right)\cdot 19^{3} + \left(16 a + 16\right)\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.