Properties

Label 2.1400.8t11.b
Dimension $2$
Group $Q_8:C_2$
Conductor $1400$
Indicator $0$

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Basic invariants

Dimension:$2$
Group:$Q_8:C_2$
Conductor:\(1400\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 7 \)
Artin number field: Galois closure of 8.0.2401000000.1
Galois orbit size: $2$
Smallest permutation container: $Q_8:C_2$
Parity: odd
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{2}, \sqrt{-7})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 7 + 38\cdot 79 + 77\cdot 79^{2} + 17\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 15 + 9\cdot 79 + 71\cdot 79^{2} + 75\cdot 79^{3} + 77\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 17 + 53\cdot 79 + 35\cdot 79^{2} + 6\cdot 79^{3} + 12\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 23 + 67\cdot 79 + 27\cdot 79^{2} + 17\cdot 79^{3} + 7\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 26 + 66\cdot 79 + 26\cdot 79^{2} + 32\cdot 79^{3} + 75\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 28 + 16\cdot 79 + 6\cdot 79^{2} + 62\cdot 79^{3} + 56\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 48 + 2\cdot 79 + 24\cdot 79^{2} + 16\cdot 79^{3} + 17\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 74 + 62\cdot 79 + 46\cdot 79^{2} + 25\cdot 79^{3} + 52\cdot 79^{4} +O(79^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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