Properties

Label 4.2e6_13_29.6t13.1c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 2^{6} \cdot 13 \cdot 29 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$24128= 2^{6} \cdot 13 \cdot 29 $
Artin number field: Splitting field of $f= x^{6} + x^{4} - 2 x^{3} - x^{2} + 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.13_29.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 16 + 13\cdot 53^{2} + 25\cdot 53^{3} + 13\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 28 a + 4 + \left(35 a + 44\right)\cdot 53 + \left(40 a + 40\right)\cdot 53^{2} + \left(44 a + 52\right)\cdot 53^{3} + \left(11 a + 16\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 7 a + 4 + \left(33 a + 48\right)\cdot 53 + \left(21 a + 2\right)\cdot 53^{2} + \left(14 a + 18\right)\cdot 53^{3} + \left(37 a + 38\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 25 a + 10 + \left(17 a + 52\right)\cdot 53 + \left(12 a + 8\right)\cdot 53^{2} + \left(8 a + 32\right)\cdot 53^{3} + \left(41 a + 19\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 40 + 52\cdot 53 + 36\cdot 53^{2} + 29\cdot 53^{3} + 3\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 46 a + 32 + \left(19 a + 14\right)\cdot 53 + \left(31 a + 3\right)\cdot 53^{2} + \left(38 a + 1\right)\cdot 53^{3} + \left(15 a + 14\right)\cdot 53^{4} +O\left(53^{ 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.