Properties

Label 4.820125.6t10.d.a
Dimension $4$
Group $C_3^2:C_4$
Conductor $820125$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $C_3^2:C_4$
Conductor: \(820125\)\(\medspace = 3^{8} \cdot 5^{3}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.2.4100625.2
Galois orbit size: $1$
Smallest permutation container: $C_3^2:C_4$
Parity: even
Determinant: 1.5.2t1.a.a
Projective image: $C_3:S_3.C_2$
Projective field: Galois closure of 6.2.4100625.2

Defining polynomial

$f(x)$$=$$ x^{6} + 6 x^{4} - 3 x^{3} + 9 x^{2} - 9 x + 1 $.

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 13.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{2} + 18 x + 2 $

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.