Properties

Label 2.3e2_5e2_7e2.8t5.2c1
Dimension 2
Group $Q_8$
Conductor $ 3^{2} \cdot 5^{2} \cdot 7^{2}$
Root number -1
Frobenius-Schur indicator -1

Related objects

Learn more about

Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$11025= 3^{2} \cdot 5^{2} \cdot 7^{2} $
Artin number field: Splitting field of $f= x^{8} - 3 x^{7} + 22 x^{6} - 60 x^{5} + 201 x^{4} - 450 x^{3} + 1528 x^{2} - 3069 x + 4561 $ 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 $\Q_{ 79 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 13 + 43\cdot 79 + 24\cdot 79^{2} + 4\cdot 79^{3} + 13\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 14 + 4\cdot 79 + 56\cdot 79^{2} + 36\cdot 79^{3} + 63\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 24 + 15\cdot 79 + 18\cdot 79^{2} + 67\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 29 + 24\cdot 79 + 33\cdot 79^{2} + 45\cdot 79^{3} + 78\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 49 + 37\cdot 79 + 72\cdot 79^{2} + 50\cdot 79^{3} + 56\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 54 + 51\cdot 79 + 20\cdot 79^{2} + 25\cdot 79^{3} + 11\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 63 + 30\cdot 79 + 79^{2} + 12\cdot 79^{3} + 15\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 73 + 29\cdot 79 + 10\cdot 79^{2} + 62\cdot 79^{3} + 10\cdot 79^{4} +O\left(79^{ 5 }\right)$

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

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

Character values on conjugacy classes

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