Properties

Label 2.2e4_3e2.4t3.3c1
Dimension 2
Group $D_4$
Conductor $ 2^{4} \cdot 3^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_4$
Conductor:$144= 2^{4} \cdot 3^{2} $
Artin number field: Splitting field of $f= x^{8} - 2 x^{7} + 2 x^{6} - 2 x^{5} + 7 x^{4} - 10 x^{3} + 8 x^{2} - 4 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.2e2.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 10 + 32\cdot 37 + 27\cdot 37^{2} + 11\cdot 37^{3} + 22\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 14 + 35\cdot 37 + 27\cdot 37^{2} + 22\cdot 37^{3} + 34\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 19 + 2\cdot 37 + 9\cdot 37^{2} + 9\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 21 + 31\cdot 37 + 15\cdot 37^{2} + 34\cdot 37^{3} + 4\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 22 + 6\cdot 37 + 31\cdot 37^{2} + 2\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 31 + 10\cdot 37 + 14\cdot 37^{2} + 6\cdot 37^{3} + 4\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 34 + 37 + 34\cdot 37^{2} + 17\cdot 37^{3} + 22\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 36 + 26\cdot 37 + 24\cdot 37^{2} + 16\cdot 37^{3} + 11\cdot 37^{4} +O\left(37^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation
$(1,2)(3,5)(4,7)(6,8)$
$(1,3,6,4)(2,7,8,5)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,6)(2,8)(3,4)(5,7)$$-2$
$2$$2$$(1,2)(3,5)(4,7)(6,8)$$0$
$2$$2$$(1,7)(2,3)(4,8)(5,6)$$0$
$2$$4$$(1,3,6,4)(2,7,8,5)$$0$
The blue line marks the conjugacy class containing complex conjugation.