Properties

Label 2.3e2_7e2_11e2.8t5.2c1
Dimension 2
Group $Q_8$
Conductor $ 3^{2} \cdot 7^{2} \cdot 11^{2}$
Root number 1
Frobenius-Schur indicator -1

Related objects

Learn more about

Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$53361= 3^{2} \cdot 7^{2} \cdot 11^{2} $
Artin number field: Splitting field of $f= x^{8} - x^{7} + 50 x^{6} + 71 x^{5} + 529 x^{4} + 2173 x^{3} + 842 x^{2} + 5545 x + 17623 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $Q_8$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 6 + 22\cdot 67 + 21\cdot 67^{2} + 22\cdot 67^{3} + 56\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 14 + 4\cdot 67 + 3\cdot 67^{2} + 14\cdot 67^{3} + 4\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 17 + 49\cdot 67 + 36\cdot 67^{2} + 12\cdot 67^{3} + 38\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 18 + 14\cdot 67 + 51\cdot 67^{2} + 19\cdot 67^{3} + 51\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 21 + 30\cdot 67 + 51\cdot 67^{2} + 62\cdot 67^{3} + 17\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 34 + 25\cdot 67 + 41\cdot 67^{2} + 51\cdot 67^{3} + 22\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 38 + 33\cdot 67 + 7\cdot 67^{2} + 41\cdot 67^{3} + 58\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 54 + 21\cdot 67 + 55\cdot 67^{2} + 43\cdot 67^{3} + 18\cdot 67^{4} +O\left(67^{ 5 }\right)$

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

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

Character values on conjugacy classes

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