Properties

Label 4.2e4_1289.6t13.1c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 2^{4} \cdot 1289 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$20624= 2^{4} \cdot 1289 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{4} - 2 x^{3} + 2 x^{2} + 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.1289.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 2 + 15\cdot 29 + 16\cdot 29^{2} + 3\cdot 29^{3} + 16\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 18 + 14\cdot 29 + 26\cdot 29^{2} + 23\cdot 29^{3} + 24\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 16 a + 17 + \left(22 a + 16\right)\cdot 29 + \left(8 a + 24\right)\cdot 29^{2} + \left(16 a + 19\right)\cdot 29^{3} + \left(a + 10\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 13 a + 10 + \left(6 a + 26\right)\cdot 29 + \left(20 a + 16\right)\cdot 29^{2} + \left(12 a + 5\right)\cdot 29^{3} + \left(27 a + 2\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 26 a + 13 + \left(9 a + 24\right)\cdot 29 + \left(5 a + 21\right)\cdot 29^{2} + \left(22 a + 7\right)\cdot 29^{3} + \left(11 a + 27\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 3 a + 27 + \left(19 a + 18\right)\cdot 29 + \left(23 a + 9\right)\cdot 29^{2} + \left(6 a + 26\right)\cdot 29^{3} + \left(17 a + 5\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$

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

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

Character values on conjugacy classes

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