Properties

Label 2.1400.8t6.a
Dimension $2$
Group $D_{8}$
Conductor $1400$
Indicator $1$

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{8}$
Conductor:\(1400\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 7 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 8.0.19208000000.1
Galois orbit size: $2$
Smallest permutation container: $D_{8}$
Parity: odd
Projective image: $D_4$
Projective field: 4.0.9800.2

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 193 }$ to precision 6.
Roots:
$r_{ 1 }$ $=$ \( 26 + 125\cdot 193 + 123\cdot 193^{2} + 186\cdot 193^{3} + 120\cdot 193^{4} + 38\cdot 193^{5} +O(193^{6})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 37 + 146\cdot 193 + 183\cdot 193^{2} + 85\cdot 193^{3} + 36\cdot 193^{4} + 61\cdot 193^{5} +O(193^{6})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 49 + 51\cdot 193 + 157\cdot 193^{2} + 15\cdot 193^{3} + 90\cdot 193^{4} + 6\cdot 193^{5} +O(193^{6})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 51 + 61\cdot 193 + 67\cdot 193^{2} + 133\cdot 193^{3} + 5\cdot 193^{4} +O(193^{6})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 59 + 82\cdot 193 + 180\cdot 193^{2} + 114\cdot 193^{3} + 92\cdot 193^{4} + 92\cdot 193^{5} +O(193^{6})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 86 + 29\cdot 193 + 154\cdot 193^{2} + 175\cdot 193^{3} + 77\cdot 193^{4} + 77\cdot 193^{5} +O(193^{6})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 130 + 136\cdot 193 + 88\cdot 193^{2} + 83\cdot 193^{3} + 57\cdot 193^{4} + 36\cdot 193^{5} +O(193^{6})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 142 + 139\cdot 193 + 9\cdot 193^{2} + 169\cdot 193^{3} + 97\cdot 193^{4} + 73\cdot 193^{5} +O(193^{6})\)  Toggle raw display

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

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

Character values on conjugacy classes

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