Properties

Label 2.2e4_3_7.4t3.5c1
Dimension 2
Group $D_4$
Conductor $ 2^{4} \cdot 3 \cdot 7 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_4$
Conductor:$336= 2^{4} \cdot 3 \cdot 7 $
Artin number field: Splitting field of $f= x^{8} - 2 x^{7} + 8 x^{6} + 12 x^{5} + 11 x^{4} + 12 x^{3} + 8 x^{2} - 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.2e2_3_7.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 6.
Roots:
$r_{ 1 }$ $=$ $ 2 + 11\cdot 19 + 10\cdot 19^{3} + 17\cdot 19^{4} + 10\cdot 19^{5} +O\left(19^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 3 + 14\cdot 19 + 4\cdot 19^{2} + 4\cdot 19^{3} + 7\cdot 19^{4} + 7\cdot 19^{5} +O\left(19^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 4 + 2\cdot 19 + 7\cdot 19^{2} + 4\cdot 19^{3} + 16\cdot 19^{4} + 17\cdot 19^{5} +O\left(19^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 5 + 2\cdot 19 + 9\cdot 19^{2} + 5\cdot 19^{3} + 11\cdot 19^{4} + 16\cdot 19^{5} +O\left(19^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 6 + 17\cdot 19 + 14\cdot 19^{2} + 13\cdot 19^{3} + 16\cdot 19^{4} + 2\cdot 19^{5} +O\left(19^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 10 + 11\cdot 19 + 12\cdot 19^{2} + 13\cdot 19^{3} + 13\cdot 19^{5} +O\left(19^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 13 + 2\cdot 19 + 8\cdot 19^{2} + 4\cdot 19^{3} + 4\cdot 19^{4} + 9\cdot 19^{5} +O\left(19^{ 6 }\right)$
$r_{ 8 }$ $=$ $ 16 + 14\cdot 19 + 18\cdot 19^{2} + 2\cdot 19^{4} + 17\cdot 19^{5} +O\left(19^{ 6 }\right)$

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

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

Character values on conjugacy classes

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