Properties

Label 2.3600.8t11.c.b
Dimension $2$
Group $Q_8:C_2$
Conductor $3600$
Root number not computed
Indicator $0$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $2$
Group: $Q_8:C_2$
Conductor: \(3600\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Artin stem field: Galois closure of 8.0.2916000000.5
Galois orbit size: $2$
Smallest permutation container: $Q_8:C_2$
Parity: odd
Determinant: 1.20.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\zeta_{12})\)

Defining polynomial

$f(x)$$=$ \( x^{8} - 15x^{6} + 90x^{4} - 225x^{2} + 225 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 6.

Roots:
$r_{ 1 }$ $=$ \( 2 + 30\cdot 61 + 7\cdot 61^{2} + 56\cdot 61^{3} + 3\cdot 61^{4} + 33\cdot 61^{5} +O(61^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 18 + 59\cdot 61 + 46\cdot 61^{2} + 20\cdot 61^{3} + 34\cdot 61^{4} + 38\cdot 61^{5} +O(61^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 19 + 10\cdot 61 + 50\cdot 61^{2} + 58\cdot 61^{3} + 19\cdot 61^{4} + 22\cdot 61^{5} +O(61^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 27 + 47\cdot 61 + 42\cdot 61^{2} + 26\cdot 61^{3} + 22\cdot 61^{4} + 54\cdot 61^{5} +O(61^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 34 + 13\cdot 61 + 18\cdot 61^{2} + 34\cdot 61^{3} + 38\cdot 61^{4} + 6\cdot 61^{5} +O(61^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 42 + 50\cdot 61 + 10\cdot 61^{2} + 2\cdot 61^{3} + 41\cdot 61^{4} + 38\cdot 61^{5} +O(61^{6})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 43 + 61 + 14\cdot 61^{2} + 40\cdot 61^{3} + 26\cdot 61^{4} + 22\cdot 61^{5} +O(61^{6})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 59 + 30\cdot 61 + 53\cdot 61^{2} + 4\cdot 61^{3} + 57\cdot 61^{4} + 27\cdot 61^{5} +O(61^{6})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.