Properties

Label 2.5491.8t7.a.a
Dimension $2$
Group $C_8:C_2$
Conductor $5491$
Root number not computed
Indicator $0$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $2$
Group: $C_8:C_2$
Conductor: \(5491\)\(\medspace = 17^{2} \cdot 19 \)
Artin stem field: Galois closure of 8.4.148132260953.1
Galois orbit size: $2$
Smallest permutation container: $C_8:C_2$
Parity: odd
Determinant: 1.323.4t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{17}, \sqrt{-19})\)

Defining polynomial

$f(x)$$=$ \( x^{8} - x^{7} - 7x^{6} + 6x^{5} - 19x^{4} + 24x^{3} + 58x^{2} + 106x - 67 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 9.

Roots:
$r_{ 1 }$ $=$ \( 3 + 34\cdot 47 + 33\cdot 47^{3} + 8\cdot 47^{4} + 6\cdot 47^{5} + 23\cdot 47^{6} + 28\cdot 47^{8} +O(47^{9})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 4 + 37\cdot 47 + 38\cdot 47^{2} + 20\cdot 47^{4} + 38\cdot 47^{5} + 34\cdot 47^{6} + 12\cdot 47^{7} + 12\cdot 47^{8} +O(47^{9})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 13 + 5\cdot 47 + 42\cdot 47^{2} + 31\cdot 47^{3} + 15\cdot 47^{4} + 6\cdot 47^{5} + 3\cdot 47^{7} + 44\cdot 47^{8} +O(47^{9})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 23 + 12\cdot 47 + 45\cdot 47^{2} + 45\cdot 47^{3} + 32\cdot 47^{4} + 19\cdot 47^{5} + 17\cdot 47^{6} + 5\cdot 47^{7} + 46\cdot 47^{8} +O(47^{9})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 24 + 4\cdot 47 + 13\cdot 47^{2} + 16\cdot 47^{3} + 37\cdot 47^{4} + 29\cdot 47^{5} + 20\cdot 47^{6} + 42\cdot 47^{7} + 46\cdot 47^{8} +O(47^{9})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 35 + 37\cdot 47 + 2\cdot 47^{2} + 46\cdot 47^{3} + 12\cdot 47^{4} + 35\cdot 47^{5} + 21\cdot 47^{6} + 10\cdot 47^{7} + 19\cdot 47^{8} +O(47^{9})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 42 + 4\cdot 47 + 22\cdot 47^{2} + 4\cdot 47^{3} + 15\cdot 47^{4} + 18\cdot 47^{5} + 7\cdot 47^{6} + 24\cdot 47^{7} + 4\cdot 47^{8} +O(47^{9})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 45 + 4\cdot 47 + 23\cdot 47^{2} + 9\cdot 47^{3} + 45\cdot 47^{4} + 33\cdot 47^{5} + 15\cdot 47^{6} + 42\cdot 47^{7} + 33\cdot 47^{8} +O(47^{9})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,5)(2,3)(4,6)(7,8)$$-2$
$2$$2$$(2,3)(4,6)$$0$
$1$$4$$(1,8,5,7)(2,6,3,4)$$2 \zeta_{4}$
$1$$4$$(1,7,5,8)(2,4,3,6)$$-2 \zeta_{4}$
$2$$4$$(1,8,5,7)(2,4,3,6)$$0$
$2$$8$$(1,6,8,3,5,4,7,2)$$0$
$2$$8$$(1,3,7,6,5,2,8,4)$$0$
$2$$8$$(1,6,7,2,5,4,8,3)$$0$
$2$$8$$(1,2,8,6,5,3,7,4)$$0$

The blue line marks the conjugacy class containing complex conjugation.