Properties

Label 4.5e2_839.6t13.1c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 5^{2} \cdot 839 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$20975= 5^{2} \cdot 839 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} - x^{4} + x^{3} + 2 x^{2} + x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Odd
Determinant: 1.839.2t1.1c1

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

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