Properties

Label 2.1600.8t7.b
Dimension $2$
Group $C_8:C_2$
Conductor $1600$
Indicator $0$

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Basic invariants

Dimension:$2$
Group:$C_8:C_2$
Conductor:\(1600\)\(\medspace = 2^{6} \cdot 5^{2} \)
Artin number field: Galois closure of 8.4.5120000000.1
Galois orbit size: $2$
Smallest permutation container: $C_8:C_2$
Parity: odd
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-2}, \sqrt{5})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 9.
Roots:
$r_{ 1 }$ $=$ \( 1 + 5\cdot 19 + 13\cdot 19^{2} + 17\cdot 19^{3} + 18\cdot 19^{4} + 6\cdot 19^{5} + 11\cdot 19^{6} + 13\cdot 19^{7} + 6\cdot 19^{8} +O(19^{9})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 2 + 2\cdot 19 + 13\cdot 19^{2} + 6\cdot 19^{3} + 15\cdot 19^{4} + 17\cdot 19^{5} + 17\cdot 19^{6} + 15\cdot 19^{7} + 15\cdot 19^{8} +O(19^{9})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 4 + 12\cdot 19 + 12\cdot 19^{2} + 7\cdot 19^{3} + 19^{4} + 13\cdot 19^{5} + 3\cdot 19^{6} + 10\cdot 19^{7} + 19^{8} +O(19^{9})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 6 + 14\cdot 19 + 19^{2} + 13\cdot 19^{3} + 9\cdot 19^{4} + 7\cdot 19^{5} + 7\cdot 19^{7} + 17\cdot 19^{8} +O(19^{9})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 13 + 4\cdot 19 + 17\cdot 19^{2} + 5\cdot 19^{3} + 9\cdot 19^{4} + 11\cdot 19^{5} + 18\cdot 19^{6} + 11\cdot 19^{7} + 19^{8} +O(19^{9})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 15 + 6\cdot 19 + 6\cdot 19^{2} + 11\cdot 19^{3} + 17\cdot 19^{4} + 5\cdot 19^{5} + 15\cdot 19^{6} + 8\cdot 19^{7} + 17\cdot 19^{8} +O(19^{9})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 17 + 16\cdot 19 + 5\cdot 19^{2} + 12\cdot 19^{3} + 3\cdot 19^{4} + 19^{5} + 19^{6} + 3\cdot 19^{7} + 3\cdot 19^{8} +O(19^{9})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 18 + 13\cdot 19 + 5\cdot 19^{2} + 19^{3} + 12\cdot 19^{5} + 7\cdot 19^{6} + 5\cdot 19^{7} + 12\cdot 19^{8} +O(19^{9})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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