# Properties

 Label 2.1225.8t7.a Dimension $2$ Group $C_8:C_2$ Conductor $1225$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $C_8:C_2$ Conductor: $$1225$$$$\medspace = 5^{2} \cdot 7^{2}$$ Artin number field: Galois closure of 8.4.9191328125.1 Galois orbit size: $2$ Smallest permutation container: $C_8:C_2$ Parity: odd Projective image: $C_2^2$ Projective field: $$\Q(\sqrt{5}, \sqrt{-7})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 281 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$59 + 20\cdot 281 + 4\cdot 281^{2} + 109\cdot 281^{3} + 26\cdot 281^{4} +O(281^{5})$$ $r_{ 2 }$ $=$ $$94 + 154\cdot 281 + 133\cdot 281^{2} + 251\cdot 281^{3} + 160\cdot 281^{4} +O(281^{5})$$ $r_{ 3 }$ $=$ $$125 + 163\cdot 281 + 152\cdot 281^{2} + 22\cdot 281^{3} + 108\cdot 281^{4} +O(281^{5})$$ $r_{ 4 }$ $=$ $$142 + 113\cdot 281 + 139\cdot 281^{2} + 240\cdot 281^{3} + 199\cdot 281^{4} +O(281^{5})$$ $r_{ 5 }$ $=$ $$151 + 253\cdot 281 + 31\cdot 281^{2} + 135\cdot 281^{3} + 244\cdot 281^{4} +O(281^{5})$$ $r_{ 6 }$ $=$ $$152 + 105\cdot 281 + 86\cdot 281^{2} + 4\cdot 281^{3} + 177\cdot 281^{4} +O(281^{5})$$ $r_{ 7 }$ $=$ $$172 + 245\cdot 281 + 234\cdot 281^{2} + 212\cdot 281^{3} + 125\cdot 281^{4} +O(281^{5})$$ $r_{ 8 }$ $=$ $$230 + 67\cdot 281 + 60\cdot 281^{2} + 148\cdot 281^{3} + 81\cdot 281^{4} +O(281^{5})$$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $2$ $2$ $1$ $2$ $(1,8)(2,4)(3,7)(5,6)$ $-2$ $-2$ $2$ $2$ $(3,7)(5,6)$ $0$ $0$ $1$ $4$ $(1,2,8,4)(3,5,7,6)$ $2 \zeta_{4}$ $-2 \zeta_{4}$ $1$ $4$ $(1,4,8,2)(3,6,7,5)$ $-2 \zeta_{4}$ $2 \zeta_{4}$ $2$ $4$ $(1,2,8,4)(3,6,7,5)$ $0$ $0$ $2$ $8$ $(1,6,2,3,8,5,4,7)$ $0$ $0$ $2$ $8$ $(1,3,4,6,8,7,2,5)$ $0$ $0$ $2$ $8$ $(1,6,4,7,8,5,2,3)$ $0$ $0$ $2$ $8$ $(1,7,2,6,8,3,4,5)$ $0$ $0$
The blue line marks the conjugacy class containing complex conjugation.