Properties

 Label 2.225.8t7.a.a Dimension $2$ Group $C_8:C_2$ Conductor $225$ Root number not computed Indicator $0$

Related objects

Basic invariants

 Dimension: $2$ Group: $C_8:C_2$ Conductor: $$225$$$$\medspace = 3^{2} \cdot 5^{2}$$ Artin stem field: Galois closure of 8.4.56953125.1 Galois orbit size: $2$ Smallest permutation container: $C_8:C_2$ Parity: odd Determinant: 1.5.4t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{-3}, \sqrt{5})$$

Defining polynomial

 $f(x)$ $=$ $$x^{8} - x^{7} + 2x^{6} + 2x^{5} - 5x^{4} + 13x^{3} - 13x^{2} + x + 1$$ x^8 - x^7 + 2*x^6 + 2*x^5 - 5*x^4 + 13*x^3 - 13*x^2 + x + 1 .

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$14 + 36\cdot 61 + 7\cdot 61^{2} + 26\cdot 61^{3} + 33\cdot 61^{4} +O(61^{5})$$ 14 + 36*61 + 7*61^2 + 26*61^3 + 33*61^4+O(61^5) $r_{ 2 }$ $=$ $$21 + 47\cdot 61 + 14\cdot 61^{2} + 3\cdot 61^{3} + 18\cdot 61^{4} +O(61^{5})$$ 21 + 47*61 + 14*61^2 + 3*61^3 + 18*61^4+O(61^5) $r_{ 3 }$ $=$ $$24 + 3\cdot 61 + 60\cdot 61^{2} + 3\cdot 61^{3} + 43\cdot 61^{4} +O(61^{5})$$ 24 + 3*61 + 60*61^2 + 3*61^3 + 43*61^4+O(61^5) $r_{ 4 }$ $=$ $$37 + 17\cdot 61 + 12\cdot 61^{2} + 59\cdot 61^{3} + 42\cdot 61^{4} +O(61^{5})$$ 37 + 17*61 + 12*61^2 + 59*61^3 + 42*61^4+O(61^5) $r_{ 5 }$ $=$ $$44 + 33\cdot 61 + 37\cdot 61^{2} + 20\cdot 61^{3} + 26\cdot 61^{4} +O(61^{5})$$ 44 + 33*61 + 37*61^2 + 20*61^3 + 26*61^4+O(61^5) $r_{ 6 }$ $=$ $$52 + 61 + 60\cdot 61^{2} + 24\cdot 61^{3} + 23\cdot 61^{4} +O(61^{5})$$ 52 + 61 + 60*61^2 + 24*61^3 + 23*61^4+O(61^5) $r_{ 7 }$ $=$ $$56 + 28\cdot 61 + 52\cdot 61^{2} + 16\cdot 61^{3} + 38\cdot 61^{4} +O(61^{5})$$ 56 + 28*61 + 52*61^2 + 16*61^3 + 38*61^4+O(61^5) $r_{ 8 }$ $=$ $$58 + 13\cdot 61 + 60\cdot 61^{2} + 27\cdot 61^{3} + 18\cdot 61^{4} +O(61^{5})$$ 58 + 13*61 + 60*61^2 + 27*61^3 + 18*61^4+O(61^5)

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.