Properties

Label 4.2e4_17_89.6t13.1c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 2^{4} \cdot 17 \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:$24208= 2^{4} \cdot 17 \cdot 89 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + 2 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: 1.17_89.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 26 + 21\cdot 37 + 19\cdot 37^{2} + 16\cdot 37^{3} + 25\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 9 + 8\cdot 37 + 26\cdot 37^{2} + 6\cdot 37^{3} + 24\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 20 a + \left(25 a + 2\right)\cdot 37 + \left(34 a + 11\right)\cdot 37^{2} + \left(17 a + 2\right)\cdot 37^{3} + \left(5 a + 2\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 17 a + 6 + \left(11 a + 10\right)\cdot 37 + \left(2 a + 13\right)\cdot 37^{2} + \left(19 a + 2\right)\cdot 37^{3} + \left(31 a + 6\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 28 a + 17 + \left(23 a + 19\right)\cdot 37 + \left(15 a + 19\right)\cdot 37^{2} + \left(26 a + 33\right)\cdot 37^{3} + \left(20 a + 16\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 9 a + 18 + \left(13 a + 12\right)\cdot 37 + \left(21 a + 21\right)\cdot 37^{2} + \left(10 a + 12\right)\cdot 37^{3} + \left(16 a + 36\right)\cdot 37^{4} +O\left(37^{ 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.