Properties

Label 4.3e2_5e2_89.6t13.1c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 3^{2} \cdot 5^{2} \cdot 89 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$20025= 3^{2} \cdot 5^{2} \cdot 89 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + 4 x^{4} - 4 x^{3} + 2 x^{2} + x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.89.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{2} + 69 x + 7 $
Roots:
$r_{ 1 }$ $=$ $ 7 + 58\cdot 71 + 14\cdot 71^{2} + 19\cdot 71^{3} + 12\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 12 a + 52 + \left(48 a + 4\right)\cdot 71 + \left(32 a + 13\right)\cdot 71^{2} + \left(48 a + 17\right)\cdot 71^{3} + \left(60 a + 68\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 27 a + 41 + \left(22 a + 68\right)\cdot 71 + \left(54 a + 55\right)\cdot 71^{2} + \left(15 a + 1\right)\cdot 71^{3} + \left(10 a + 27\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 44 a + 24 + \left(48 a + 15\right)\cdot 71 + 16 a\cdot 71^{2} + \left(55 a + 50\right)\cdot 71^{3} + \left(60 a + 31\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 15 + 48\cdot 71 + 27\cdot 71^{2} + 43\cdot 71^{3} + 3\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 59 a + 5 + \left(22 a + 18\right)\cdot 71 + \left(38 a + 30\right)\cdot 71^{2} + \left(22 a + 10\right)\cdot 71^{3} + \left(10 a + 70\right)\cdot 71^{4} +O\left(71^{ 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.