# Properties

 Label 2.1280.8t17.c.a Dimension $2$ Group $C_4\wr C_2$ Conductor $1280$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $C_4\wr C_2$ Conductor: $$1280$$$$\medspace = 2^{8} \cdot 5$$ Artin stem field: Galois closure of 8.0.2097152000.7 Galois orbit size: $2$ Smallest permutation container: $C_4\wr C_2$ Parity: odd Determinant: 1.5.4t1.a.b Projective image: $D_4$ Projective stem field: Galois closure of 4.2.8000.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 4x^{4} + 20$$ x^8 - 4*x^4 + 20 .

The roots of $f$ are computed in $\Q_{ 421 }$ to precision 7.

Roots:
 $r_{ 1 }$ $=$ $$8 + 113\cdot 421 + 277\cdot 421^{2} + 130\cdot 421^{3} + 130\cdot 421^{4} + 194\cdot 421^{5} + 405\cdot 421^{6} +O(421^{7})$$ 8 + 113*421 + 277*421^2 + 130*421^3 + 130*421^4 + 194*421^5 + 405*421^6+O(421^7) $r_{ 2 }$ $=$ $$138 + 164\cdot 421 + 368\cdot 421^{2} + 365\cdot 421^{3} + 85\cdot 421^{4} + 53\cdot 421^{5} + 184\cdot 421^{6} +O(421^{7})$$ 138 + 164*421 + 368*421^2 + 365*421^3 + 85*421^4 + 53*421^5 + 184*421^6+O(421^7) $r_{ 3 }$ $=$ $$189 + 279\cdot 421 + 121\cdot 421^{2} + 145\cdot 421^{3} + 154\cdot 421^{5} + 9\cdot 421^{6} +O(421^{7})$$ 189 + 279*421 + 121*421^2 + 145*421^3 + 154*421^5 + 9*421^6+O(421^7) $r_{ 4 }$ $=$ $$208 + 73\cdot 421 + 19\cdot 421^{2} + 285\cdot 421^{3} + 286\cdot 421^{4} + 123\cdot 421^{5} + 313\cdot 421^{6} +O(421^{7})$$ 208 + 73*421 + 19*421^2 + 285*421^3 + 286*421^4 + 123*421^5 + 313*421^6+O(421^7) $r_{ 5 }$ $=$ $$213 + 347\cdot 421 + 401\cdot 421^{2} + 135\cdot 421^{3} + 134\cdot 421^{4} + 297\cdot 421^{5} + 107\cdot 421^{6} +O(421^{7})$$ 213 + 347*421 + 401*421^2 + 135*421^3 + 134*421^4 + 297*421^5 + 107*421^6+O(421^7) $r_{ 6 }$ $=$ $$232 + 141\cdot 421 + 299\cdot 421^{2} + 275\cdot 421^{3} + 420\cdot 421^{4} + 266\cdot 421^{5} + 411\cdot 421^{6} +O(421^{7})$$ 232 + 141*421 + 299*421^2 + 275*421^3 + 420*421^4 + 266*421^5 + 411*421^6+O(421^7) $r_{ 7 }$ $=$ $$283 + 256\cdot 421 + 52\cdot 421^{2} + 55\cdot 421^{3} + 335\cdot 421^{4} + 367\cdot 421^{5} + 236\cdot 421^{6} +O(421^{7})$$ 283 + 256*421 + 52*421^2 + 55*421^3 + 335*421^4 + 367*421^5 + 236*421^6+O(421^7) $r_{ 8 }$ $=$ $$413 + 307\cdot 421 + 143\cdot 421^{2} + 290\cdot 421^{3} + 290\cdot 421^{4} + 226\cdot 421^{5} + 15\cdot 421^{6} +O(421^{7})$$ 413 + 307*421 + 143*421^2 + 290*421^3 + 290*421^4 + 226*421^5 + 15*421^6+O(421^7)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.