Properties

 Label 2.712.4t3.b Dimension $2$ Group $D_{4}$ Conductor $712$ Indicator $1$

Related objects

Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$712$$$$\medspace = 2^{3} \cdot 89$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 4.0.5696.2 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Projective image: $C_2^2$ Projective field: $$\Q(\sqrt{-2}, \sqrt{89})$$

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$8 + 21\cdot 97 + 21\cdot 97^{2} + 7\cdot 97^{3} + 49\cdot 97^{4} +O(97^{5})$$ $r_{ 2 }$ $=$ $$55 + 24\cdot 97 + 77\cdot 97^{2} + 68\cdot 97^{3} + 80\cdot 97^{4} +O(97^{5})$$ $r_{ 3 }$ $=$ $$60 + 12\cdot 97 + 25\cdot 97^{2} + 68\cdot 97^{3} + 96\cdot 97^{4} +O(97^{5})$$ $r_{ 4 }$ $=$ $$73 + 38\cdot 97 + 70\cdot 97^{2} + 49\cdot 97^{3} + 64\cdot 97^{4} +O(97^{5})$$

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

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

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character values $c1$ $1$ $1$ $()$ $2$ $1$ $2$ $(1,4)(2,3)$ $-2$ $2$ $2$ $(1,2)(3,4)$ $0$ $2$ $2$ $(1,4)$ $0$ $2$ $4$ $(1,3,4,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.