Properties

Label 4.2e6_193.6t13.2c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 2^{6} \cdot 193 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$12352= 2^{6} \cdot 193 $
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: 1.193.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $ x^{2} + 96 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 26 a + 8 + \left(74 a + 34\right)\cdot 97 + \left(47 a + 22\right)\cdot 97^{2} + \left(83 a + 39\right)\cdot 97^{3} + \left(38 a + 10\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 52 a + 62 + \left(19 a + 3\right)\cdot 97 + \left(35 a + 55\right)\cdot 97^{2} + \left(55 a + 60\right)\cdot 97^{3} + \left(25 a + 29\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 93 + 66\cdot 97 + 79\cdot 97^{2} + 27\cdot 97^{3} + 71\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 77 + 35\cdot 97 + 67\cdot 97^{2} + 7\cdot 97^{3} + 20\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 45 a + 17 + \left(77 a + 68\right)\cdot 97 + \left(61 a + 70\right)\cdot 97^{2} + \left(41 a + 80\right)\cdot 97^{3} + \left(71 a + 96\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 71 a + 34 + \left(22 a + 82\right)\cdot 97 + \left(49 a + 92\right)\cdot 97^{2} + \left(13 a + 74\right)\cdot 97^{3} + \left(58 a + 62\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$

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

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

Character values on conjugacy classes

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