Properties

Label 4.3e2_19_67.6t13.2c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 3^{2} \cdot 19 \cdot 67 $
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$11457= 3^{2} \cdot 19 \cdot 67 $
Artin number field: Splitting field of $f= x^{6} - x^{5} - 2 x^{3} + 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.19_67.2t1.1c1

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:
$r_{ 1 }$ $=$ $ 9 a + 23 + \left(15 a + 19\right)\cdot 31 + \left(21 a + 17\right)\cdot 31^{2} + \left(26 a + 14\right)\cdot 31^{3} + \left(a + 22\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 22 a + 10 + \left(15 a + 10\right)\cdot 31 + \left(9 a + 14\right)\cdot 31^{2} + \left(4 a + 15\right)\cdot 31^{3} + \left(29 a + 30\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 24 + 17\cdot 31 + 5\cdot 31^{2} + 23\cdot 31^{3} + 21\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 2 a + 25 + \left(11 a + 15\right)\cdot 31 + \left(9 a + 21\right)\cdot 31^{2} + \left(25 a + 17\right)\cdot 31^{3} + \left(11 a + 28\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 14 + 24\cdot 31 + 4\cdot 31^{2} + 25\cdot 31^{3} + 24\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 29 a + 29 + \left(19 a + 4\right)\cdot 31 + \left(21 a + 29\right)\cdot 31^{2} + \left(5 a + 27\right)\cdot 31^{3} + \left(19 a + 26\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$

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

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

Character values on conjugacy classes

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