Properties

Label 2.2e2_7_11.4t3.4c1
Dimension 2
Group $D_{4}$
Conductor $ 2^{2} \cdot 7 \cdot 11 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$308= 2^{2} \cdot 7 \cdot 11 $
Artin number field: Splitting field of $f= x^{4} - x^{3} - 2 x^{2} + 2 x + 4 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.2e2_7_11.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 4 + 35\cdot 43 + 24\cdot 43^{2} + 39\cdot 43^{3} + 8\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 7 + 35\cdot 43 + 35\cdot 43^{2} + 17\cdot 43^{3} + 7\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 12 + 38\cdot 43 + 18\cdot 43^{2} + 4\cdot 43^{3} + 24\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 21 + 20\cdot 43 + 6\cdot 43^{2} + 24\cdot 43^{3} + 2\cdot 43^{4} +O\left(43^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

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

Character values on conjugacy classes

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