Properties

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

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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:
$r_{ 1 }$ $=$ \( 59 + 20\cdot 281 + 4\cdot 281^{2} + 109\cdot 281^{3} + 26\cdot 281^{4} +O(281^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 94 + 154\cdot 281 + 133\cdot 281^{2} + 251\cdot 281^{3} + 160\cdot 281^{4} +O(281^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 125 + 163\cdot 281 + 152\cdot 281^{2} + 22\cdot 281^{3} + 108\cdot 281^{4} +O(281^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 142 + 113\cdot 281 + 139\cdot 281^{2} + 240\cdot 281^{3} + 199\cdot 281^{4} +O(281^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 151 + 253\cdot 281 + 31\cdot 281^{2} + 135\cdot 281^{3} + 244\cdot 281^{4} +O(281^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 152 + 105\cdot 281 + 86\cdot 281^{2} + 4\cdot 281^{3} + 177\cdot 281^{4} +O(281^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 172 + 245\cdot 281 + 234\cdot 281^{2} + 212\cdot 281^{3} + 125\cdot 281^{4} +O(281^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 230 + 67\cdot 281 + 60\cdot 281^{2} + 148\cdot 281^{3} + 81\cdot 281^{4} +O(281^{5})\)  Toggle raw display

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

SizeOrderAction 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.