Properties

Label 2.2e6_3e2_5e2.8t7.8c1
Dimension 2
Group $C_8:C_2$
Conductor $ 2^{6} \cdot 3^{2} \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:$14400= 2^{6} \cdot 3^{2} \cdot 5^{2} $
Artin number field: Splitting field of $f= x^{8} + 30 x^{6} + 240 x^{4} + 720 x^{2} + 720 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_8:C_2$
Parity: Even
Determinant: 1.2e2_5.4t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 10.
Roots:
$r_{ 1 }$ $=$ $ 1 + 11\cdot 59 + 54\cdot 59^{2} + 4\cdot 59^{3} + 46\cdot 59^{4} + 36\cdot 59^{5} + 22\cdot 59^{6} + 8\cdot 59^{7} + 39\cdot 59^{8} + 49\cdot 59^{9} +O\left(59^{ 10 }\right)$
$r_{ 2 }$ $=$ $ 13 + 7\cdot 59 + 42\cdot 59^{2} + 22\cdot 59^{3} + 3\cdot 59^{4} + 44\cdot 59^{5} + 11\cdot 59^{6} + 24\cdot 59^{7} + 11\cdot 59^{8} + 28\cdot 59^{9} +O\left(59^{ 10 }\right)$
$r_{ 3 }$ $=$ $ 14 + 10\cdot 59^{2} + 51\cdot 59^{3} + 27\cdot 59^{4} + 15\cdot 59^{5} + 56\cdot 59^{6} + 34\cdot 59^{7} + 40\cdot 59^{8} + 3\cdot 59^{9} +O\left(59^{ 10 }\right)$
$r_{ 4 }$ $=$ $ 28 + 55\cdot 59 + 43\cdot 59^{2} + 7\cdot 59^{3} + 8\cdot 59^{4} + 26\cdot 59^{5} + 55\cdot 59^{6} + 27\cdot 59^{7} + 45\cdot 59^{8} + 30\cdot 59^{9} +O\left(59^{ 10 }\right)$
$r_{ 5 }$ $=$ $ 31 + 3\cdot 59 + 15\cdot 59^{2} + 51\cdot 59^{3} + 50\cdot 59^{4} + 32\cdot 59^{5} + 3\cdot 59^{6} + 31\cdot 59^{7} + 13\cdot 59^{8} + 28\cdot 59^{9} +O\left(59^{ 10 }\right)$
$r_{ 6 }$ $=$ $ 45 + 58\cdot 59 + 48\cdot 59^{2} + 7\cdot 59^{3} + 31\cdot 59^{4} + 43\cdot 59^{5} + 2\cdot 59^{6} + 24\cdot 59^{7} + 18\cdot 59^{8} + 55\cdot 59^{9} +O\left(59^{ 10 }\right)$
$r_{ 7 }$ $=$ $ 46 + 51\cdot 59 + 16\cdot 59^{2} + 36\cdot 59^{3} + 55\cdot 59^{4} + 14\cdot 59^{5} + 47\cdot 59^{6} + 34\cdot 59^{7} + 47\cdot 59^{8} + 30\cdot 59^{9} +O\left(59^{ 10 }\right)$
$r_{ 8 }$ $=$ $ 58 + 47\cdot 59 + 4\cdot 59^{2} + 54\cdot 59^{3} + 12\cdot 59^{4} + 22\cdot 59^{5} + 36\cdot 59^{6} + 50\cdot 59^{7} + 19\cdot 59^{8} + 9\cdot 59^{9} +O\left(59^{ 10 }\right)$

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

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

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