Properties

Label 2.3960.8t11.h.a
Dimension $2$
Group $Q_8:C_2$
Conductor $3960$
Root number not computed
Indicator $0$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $2$
Group: $Q_8:C_2$
Conductor: \(3960\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
Artin stem field: Galois closure of 8.0.9032601600.3
Galois orbit size: $2$
Smallest permutation container: $Q_8:C_2$
Parity: odd
Determinant: 1.440.2t1.b.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-6}, \sqrt{-55})\)

Defining polynomial

$f(x)$$=$ \( x^{8} - 4x^{7} + 2x^{6} - 10x^{5} + 112x^{4} - 272x^{3} + 341x^{2} - 434x + 343 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 337 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 22 + 181\cdot 337 + 331\cdot 337^{2} + 143\cdot 337^{3} + 234\cdot 337^{4} +O(337^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 79 + 189\cdot 337 + 54\cdot 337^{2} + 15\cdot 337^{3} + 331\cdot 337^{4} +O(337^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 102 + 305\cdot 337 + 130\cdot 337^{2} + 210\cdot 337^{3} + 161\cdot 337^{4} +O(337^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 141 + 162\cdot 337 + 67\cdot 337^{2} + 165\cdot 337^{3} + 85\cdot 337^{4} +O(337^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 194 + 238\cdot 337 + 86\cdot 337^{2} + 118\cdot 337^{3} + 224\cdot 337^{4} +O(337^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 213 + 317\cdot 337 + 294\cdot 337^{2} + 61\cdot 337^{3} + 190\cdot 337^{4} +O(337^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 300 + 12\cdot 337 + 317\cdot 337^{2} + 302\cdot 337^{3} + 163\cdot 337^{4} +O(337^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 301 + 277\cdot 337 + 64\cdot 337^{2} + 330\cdot 337^{3} + 293\cdot 337^{4} +O(337^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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