Properties

Label 4.3e5_79.6t13.1c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 3^{5} \cdot 79 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$19197= 3^{5} \cdot 79 $
Artin number field: Splitting field of $f= x^{6} - 3 x^{4} - x^{3} + 3 x^{2} + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.3_79.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 27 + 20\cdot 31 + 29\cdot 31^{2} + 10\cdot 31^{3} + 16\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 24 a + 9 + \left(28 a + 19\right)\cdot 31 + \left(12 a + 17\right)\cdot 31^{2} + \left(6 a + 25\right)\cdot 31^{3} + \left(16 a + 9\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 16 + 31 + 8\cdot 31^{2} + 10\cdot 31^{3} + 19\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 6 a + 17 + \left(8 a + 9\right)\cdot 31 + 15 a\cdot 31^{2} + \left(28 a + 5\right)\cdot 31^{3} + \left(5 a + 14\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 7 a + 26 + \left(2 a + 21\right)\cdot 31 + \left(18 a + 14\right)\cdot 31^{2} + \left(24 a + 25\right)\cdot 31^{3} + \left(14 a + 4\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 25 a + 29 + \left(22 a + 19\right)\cdot 31 + \left(15 a + 22\right)\cdot 31^{2} + \left(2 a + 15\right)\cdot 31^{3} + \left(25 a + 28\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$4$
$6$$2$$(1,3)(2,4)(5,6)$$0$
$6$$2$$(2,5)$$2$
$9$$2$$(2,5)(4,6)$$0$
$4$$3$$(1,2,5)$$1$
$4$$3$$(1,2,5)(3,4,6)$$-2$
$18$$4$$(1,3)(2,6,5,4)$$0$
$12$$6$$(1,4,2,6,5,3)$$0$
$12$$6$$(2,5)(3,4,6)$$-1$
The blue line marks the conjugacy class containing complex conjugation.