Properties

Label 2.3e2_5e2_61.8t7.2c2
Dimension 2
Group $C_8:C_2$
Conductor $ 3^{2} \cdot 5^{2} \cdot 61 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

Learn more about

Basic invariants

Dimension:$2$
Group:$C_8:C_2$
Conductor:$13725= 3^{2} \cdot 5^{2} \cdot 61 $
Artin number field: Splitting field of $f= x^{8} - x^{7} - 13 x^{6} - 13 x^{5} - 20 x^{4} + 358 x^{3} + 227 x^{2} - 539 x + 991 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_8:C_2$
Parity: Odd
Determinant: 1.5_61.4t1.3c2

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 271 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 9 + 117\cdot 271 + 190\cdot 271^{2} + 24\cdot 271^{3} + 53\cdot 271^{4} +O\left(271^{ 5 }\right) \\ r_{ 2 } &= 14 + 108\cdot 271 + 236\cdot 271^{2} + 18\cdot 271^{3} + 177\cdot 271^{4} +O\left(271^{ 5 }\right) \\ r_{ 3 } &= 63 + 208\cdot 271 + 142\cdot 271^{2} + 4\cdot 271^{3} + 75\cdot 271^{4} +O\left(271^{ 5 }\right) \\ r_{ 4 } &= 121 + 94\cdot 271 + 129\cdot 271^{2} + 234\cdot 271^{3} + 137\cdot 271^{4} +O\left(271^{ 5 }\right) \\ r_{ 5 } &= 154 + 121\cdot 271 + 68\cdot 271^{2} + 136\cdot 271^{3} + 138\cdot 271^{4} +O\left(271^{ 5 }\right) \\ r_{ 6 } &= 237 + 61\cdot 271 + 147\cdot 271^{2} + 187\cdot 271^{3} + 34\cdot 271^{4} +O\left(271^{ 5 }\right) \\ r_{ 7 } &= 241 + 225\cdot 271 + 183\cdot 271^{2} + 260\cdot 271^{3} + 150\cdot 271^{4} +O\left(271^{ 5 }\right) \\ r_{ 8 } &= 246 + 146\cdot 271 + 256\cdot 271^{2} + 216\cdot 271^{3} + 45\cdot 271^{4} +O\left(271^{ 5 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

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