# Properties

 Label 2.11025.8t5.b.a Dimension $2$ Group $Q_8$ Conductor $11025$ Root number $-1$ Indicator $-1$

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## Basic invariants

 Dimension: $2$ Group: $Q_8$ Conductor: $$11025$$$$\medspace = 3^{2} \cdot 5^{2} \cdot 7^{2}$$ Frobenius-Schur indicator: $-1$ Root number: $-1$ Artin field: Galois closure of 8.0.1340095640625.1 Galois orbit size: $1$ Smallest permutation container: $Q_8$ Parity: even Determinant: 1.1.1t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{5}, \sqrt{21})$$

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 3x^{7} + 22x^{6} - 60x^{5} + 201x^{4} - 450x^{3} + 1528x^{2} - 3069x + 4561$$ x^8 - 3*x^7 + 22*x^6 - 60*x^5 + 201*x^4 - 450*x^3 + 1528*x^2 - 3069*x + 4561 .

The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$13 + 43\cdot 79 + 24\cdot 79^{2} + 4\cdot 79^{3} + 13\cdot 79^{4} +O(79^{5})$$ 13 + 43*79 + 24*79^2 + 4*79^3 + 13*79^4+O(79^5) $r_{ 2 }$ $=$ $$14 + 4\cdot 79 + 56\cdot 79^{2} + 36\cdot 79^{3} + 63\cdot 79^{4} +O(79^{5})$$ 14 + 4*79 + 56*79^2 + 36*79^3 + 63*79^4+O(79^5) $r_{ 3 }$ $=$ $$24 + 15\cdot 79 + 18\cdot 79^{2} + 67\cdot 79^{4} +O(79^{5})$$ 24 + 15*79 + 18*79^2 + 67*79^4+O(79^5) $r_{ 4 }$ $=$ $$29 + 24\cdot 79 + 33\cdot 79^{2} + 45\cdot 79^{3} + 78\cdot 79^{4} +O(79^{5})$$ 29 + 24*79 + 33*79^2 + 45*79^3 + 78*79^4+O(79^5) $r_{ 5 }$ $=$ $$49 + 37\cdot 79 + 72\cdot 79^{2} + 50\cdot 79^{3} + 56\cdot 79^{4} +O(79^{5})$$ 49 + 37*79 + 72*79^2 + 50*79^3 + 56*79^4+O(79^5) $r_{ 6 }$ $=$ $$54 + 51\cdot 79 + 20\cdot 79^{2} + 25\cdot 79^{3} + 11\cdot 79^{4} +O(79^{5})$$ 54 + 51*79 + 20*79^2 + 25*79^3 + 11*79^4+O(79^5) $r_{ 7 }$ $=$ $$63 + 30\cdot 79 + 79^{2} + 12\cdot 79^{3} + 15\cdot 79^{4} +O(79^{5})$$ 63 + 30*79 + 79^2 + 12*79^3 + 15*79^4+O(79^5) $r_{ 8 }$ $=$ $$73 + 29\cdot 79 + 10\cdot 79^{2} + 62\cdot 79^{3} + 10\cdot 79^{4} +O(79^{5})$$ 73 + 29*79 + 10*79^2 + 62*79^3 + 10*79^4+O(79^5)

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,4)(2,6)(3,5)(7,8)$ $-2$ $2$ $4$ $(1,8,4,7)(2,3,6,5)$ $0$ $2$ $4$ $(1,2,4,6)(3,8,5,7)$ $0$ $2$ $4$ $(1,5,4,3)(2,8,6,7)$ $0$

The blue line marks the conjugacy class containing complex conjugation.