Properties

Label 4.3e2_1489.6t13.2c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 3^{2} \cdot 1489 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$13401= 3^{2} \cdot 1489 $
Artin number field: Splitting field of $f= x^{6} - x^{5} - 3 x^{4} + 4 x^{3} + 3 x^{2} - 4 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.1489.2t1.1c1

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

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

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

Character values on conjugacy classes

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