Properties

Label 4.1458000.6t10.c.a
Dimension $4$
Group $C_3^2:C_4$
Conductor $1458000$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $C_3^2:C_4$
Conductor: \(1458000\)\(\medspace = 2^{4} \cdot 3^{6} \cdot 5^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.7290000.2
Galois orbit size: $1$
Smallest permutation container: $C_3^2:C_4$
Parity: even
Determinant: 1.5.2t1.a.a
Projective image: $C_3^2:C_4$
Projective stem field: Galois closure of 6.2.7290000.2

Defining polynomial

$f(x)$$=$ \( x^{6} + 6x^{4} - 2x^{3} + 9x^{2} - 6x - 4 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{2} + 29x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 15 a + 11 + \left(5 a + 27\right)\cdot 31 + \left(13 a + 12\right)\cdot 31^{2} + 30 a\cdot 31^{3} + \left(17 a + 21\right)\cdot 31^{4} + \left(19 a + 1\right)\cdot 31^{5} + \left(21 a + 20\right)\cdot 31^{6} + \left(23 a + 3\right)\cdot 31^{7} + \left(12 a + 18\right)\cdot 31^{8} + \left(26 a + 16\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 30 a + 18 + \left(28 a + 8\right)\cdot 31 + \left(30 a + 1\right)\cdot 31^{2} + \left(19 a + 27\right)\cdot 31^{3} + \left(9 a + 24\right)\cdot 31^{4} + \left(17 a + 9\right)\cdot 31^{5} + \left(15 a + 30\right)\cdot 31^{6} + \left(24 a + 8\right)\cdot 31^{7} + \left(23 a + 21\right)\cdot 31^{8} + \left(21 a + 8\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( a + 16 + \left(2 a + 5\right)\cdot 31 + 3\cdot 31^{2} + \left(11 a + 5\right)\cdot 31^{3} + \left(21 a + 24\right)\cdot 31^{4} + \left(13 a + 3\right)\cdot 31^{5} + \left(15 a + 13\right)\cdot 31^{6} + \left(6 a + 11\right)\cdot 31^{7} + \left(7 a + 13\right)\cdot 31^{8} + \left(9 a + 28\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 16 a + 10 + \left(25 a + 23\right)\cdot 31 + \left(17 a + 2\right)\cdot 31^{2} + 17\cdot 31^{3} + \left(13 a + 26\right)\cdot 31^{4} + \left(11 a + 22\right)\cdot 31^{5} + \left(9 a + 12\right)\cdot 31^{6} + \left(7 a + 29\right)\cdot 31^{7} + \left(18 a + 19\right)\cdot 31^{8} + \left(4 a + 25\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 10 + 11\cdot 31 + 15\cdot 31^{2} + 13\cdot 31^{3} + 14\cdot 31^{4} + 6\cdot 31^{5} + 29\cdot 31^{6} + 28\cdot 31^{7} + 23\cdot 31^{8} + 19\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 28 + 16\cdot 31 + 26\cdot 31^{2} + 29\cdot 31^{3} + 12\cdot 31^{4} + 17\cdot 31^{5} + 18\cdot 31^{6} + 10\cdot 31^{7} + 27\cdot 31^{8} + 24\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.