Properties

Label 2.1600.8t7.c.a
Dimension 2
Group $C_8:C_2$
Conductor $ 2^{6} \cdot 5^{2}$
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$C_8:C_2$
Conductor:$1600= 2^{6} \cdot 5^{2} $
Artin number field: Splitting field of 8.4.5120000000.2 defined by $f= x^{8} - 15 x^{4} + 10 x^{2} + 5 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_8:C_2$
Parity: Odd
Determinant: 1.40.4t1.b.b
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-2}, \sqrt{5})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 7.
Roots:
$r_{ 1 }$ $=$ $ 18 + 24\cdot 59 + 25\cdot 59^{2} + 3\cdot 59^{3} + 46\cdot 59^{4} + 18\cdot 59^{5} + 14\cdot 59^{6} +O\left(59^{ 7 }\right)$
$r_{ 2 }$ $=$ $ 20 + 17\cdot 59 + 19\cdot 59^{2} + 6\cdot 59^{3} + 46\cdot 59^{4} + 10\cdot 59^{5} + 54\cdot 59^{6} +O\left(59^{ 7 }\right)$
$r_{ 3 }$ $=$ $ 23 + 45\cdot 59 + 21\cdot 59^{2} + 59^{3} + 41\cdot 59^{4} + 4\cdot 59^{5} + 40\cdot 59^{6} +O\left(59^{ 7 }\right)$
$r_{ 4 }$ $=$ $ 24 + 31\cdot 59 + 55\cdot 59^{2} + 56\cdot 59^{3} + 5\cdot 59^{4} + 44\cdot 59^{5} + 23\cdot 59^{6} +O\left(59^{ 7 }\right)$
$r_{ 5 }$ $=$ $ 35 + 27\cdot 59 + 3\cdot 59^{2} + 2\cdot 59^{3} + 53\cdot 59^{4} + 14\cdot 59^{5} + 35\cdot 59^{6} +O\left(59^{ 7 }\right)$
$r_{ 6 }$ $=$ $ 36 + 13\cdot 59 + 37\cdot 59^{2} + 57\cdot 59^{3} + 17\cdot 59^{4} + 54\cdot 59^{5} + 18\cdot 59^{6} +O\left(59^{ 7 }\right)$
$r_{ 7 }$ $=$ $ 39 + 41\cdot 59 + 39\cdot 59^{2} + 52\cdot 59^{3} + 12\cdot 59^{4} + 48\cdot 59^{5} + 4\cdot 59^{6} +O\left(59^{ 7 }\right)$
$r_{ 8 }$ $=$ $ 41 + 34\cdot 59 + 33\cdot 59^{2} + 55\cdot 59^{3} + 12\cdot 59^{4} + 40\cdot 59^{5} + 44\cdot 59^{6} +O\left(59^{ 7 }\right)$

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

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

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