Properties

Label 4.5e3_1069e2.12t34.1c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 5^{3} \cdot 1069^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$142845125= 5^{3} \cdot 1069^{2} $
Artin number field: Splitting field of $f= x^{6} + 2 x^{4} - x^{3} + x^{2} - x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 12T34
Parity: Even
Determinant: 1.5.2t1.1c1

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:
$r_{ 1 }$ $=$ $ 8 a + 4 + \left(14 a + 18\right)\cdot 29 + \left(18 a + 19\right)\cdot 29^{2} + \left(3 a + 28\right)\cdot 29^{3} + \left(4 a + 9\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 4 a + 27 + \left(5 a + 14\right)\cdot 29 + \left(12 a + 14\right)\cdot 29^{2} + \left(8 a + 21\right)\cdot 29^{3} + \left(12 a + 21\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 21 a + 15 + \left(14 a + 23\right)\cdot 29 + \left(10 a + 10\right)\cdot 29^{2} + \left(25 a + 28\right)\cdot 29^{3} + \left(24 a + 26\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 25 a + 18 + \left(23 a + 7\right)\cdot 29 + \left(16 a + 12\right)\cdot 29^{2} + \left(20 a + 22\right)\cdot 29^{3} + \left(16 a + 16\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 13 + 6\cdot 29 + 2\cdot 29^{2} + 14\cdot 29^{3} + 19\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 10 + 16\cdot 29 + 27\cdot 29^{2} + 21\cdot 29^{4} +O\left(29^{ 5 }\right)$

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

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

Character values on conjugacy classes

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