Properties

Label 4.24489.6t13.b.a
Dimension $4$
Group $C_3^2:D_4$
Conductor $24489$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $C_3^2:D_4$
Conductor: \(24489\)\(\medspace = 3^{3} \cdot 907 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.0.73467.1
Galois orbit size: $1$
Smallest permutation container: $C_3^2:D_4$
Parity: even
Determinant: 1.2721.2t1.a.a
Projective image: $S_3\wr C_2$
Projective field: Galois closure of 6.0.73467.1

Defining polynomial

$f(x)$$=$$ x^{6} - x^{5} + 3 x^{4} - 4 x^{3} + 4 x^{2} - 3 x + 3 $.

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} + 18 x + 2 $

Roots:
$r_{ 1 }$ $=$ $ 12 a + 5 + \left(5 a + 7\right)\cdot 19 + \left(3 a + 9\right)\cdot 19^{2} + \left(3 a + 1\right)\cdot 19^{3} + \left(9 a + 16\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 9 + 14\cdot 19 + 13\cdot 19^{2} + 15\cdot 19^{3} + 17\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 9 a + \left(14 a + 18\right)\cdot 19 + \left(14 a + 16\right)\cdot 19^{2} + \left(8 a + 10\right)\cdot 19^{3} + \left(18 a + 12\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 7 a + 17 + 13 a\cdot 19 + \left(15 a + 7\right)\cdot 19^{2} + \left(15 a + 1\right)\cdot 19^{3} + \left(9 a + 3\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 10 a + 9 + \left(4 a + 4\right)\cdot 19 + \left(4 a + 17\right)\cdot 19^{2} + \left(10 a + 4\right)\cdot 19^{3} + 3\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 18 + 11\cdot 19 + 11\cdot 19^{2} + 3\cdot 19^{3} + 4\cdot 19^{4} +O\left(19^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$4$
$6$$2$$(1,3)(2,5)(4,6)$$0$
$6$$2$$(2,4)$$2$
$9$$2$$(2,4)(5,6)$$0$
$4$$3$$(1,2,4)$$1$
$4$$3$$(1,2,4)(3,5,6)$$-2$
$18$$4$$(1,3)(2,6,4,5)$$0$
$12$$6$$(1,5,2,6,4,3)$$0$
$12$$6$$(2,4)(3,5,6)$$-1$

The blue line marks the conjugacy class containing complex conjugation.