Properties

Label 2.2e3_7.4t3.2c1
Dimension 2
Group $D_{4}$
Conductor $ 2^{3} \cdot 7 $
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$56= 2^{3} \cdot 7 $
Artin number field: Splitting field of $f= x^{4} - x^{3} + x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.2e3_7.2t1.2c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 5 + 2\cdot 23 + 17\cdot 23^{2} + 22\cdot 23^{3} + 21\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 9 + 19\cdot 23 + 16\cdot 23^{2} + 2\cdot 23^{3} + 7\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 12 + 9\cdot 23 + 13\cdot 23^{2} + 12\cdot 23^{3} + 6\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 21 + 14\cdot 23 + 21\cdot 23^{2} + 7\cdot 23^{3} + 10\cdot 23^{4} +O\left(23^{ 5 }\right)$

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

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

Character values on conjugacy classes

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