Properties

Label 2.3e2_5e2_31.8t7.1c1
Dimension 2
Group $C_8:C_2$
Conductor $ 3^{2} \cdot 5^{2} \cdot 31 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$C_8:C_2$
Conductor:$6975= 3^{2} \cdot 5^{2} \cdot 31 $
Artin number field: Splitting field of $f= x^{8} - 3 x^{7} - 17 x^{6} + 39 x^{5} + 105 x^{4} - 126 x^{3} - 242 x^{2} + 27 x + 61 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_8:C_2$
Parity: Even
Determinant: 1.5_31.4t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 269 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 16 + 86\cdot 269 + 126\cdot 269^{2} + 86\cdot 269^{3} + 113\cdot 269^{4} +O\left(269^{ 5 }\right) \\ r_{ 2 } &= 26 + 45\cdot 269 + 147\cdot 269^{2} + 169\cdot 269^{3} + 210\cdot 269^{4} +O\left(269^{ 5 }\right) \\ r_{ 3 } &= 88 + 143\cdot 269 + 113\cdot 269^{2} + 253\cdot 269^{3} + 189\cdot 269^{4} +O\left(269^{ 5 }\right) \\ r_{ 4 } &= 159 + 153\cdot 269 + 192\cdot 269^{2} + 200\cdot 269^{3} + 155\cdot 269^{4} +O\left(269^{ 5 }\right) \\ r_{ 5 } &= 176 + 7\cdot 269 + 85\cdot 269^{2} + 110\cdot 269^{3} + 125\cdot 269^{4} +O\left(269^{ 5 }\right) \\ r_{ 6 } &= 188 + 144\cdot 269 + 110\cdot 269^{2} + 177\cdot 269^{3} + 174\cdot 269^{4} +O\left(269^{ 5 }\right) \\ r_{ 7 } &= 210 + 264\cdot 269 + 30\cdot 269^{2} + 135\cdot 269^{3} + 151\cdot 269^{4} +O\left(269^{ 5 }\right) \\ r_{ 8 } &= 216 + 230\cdot 269 + 212\cdot 269^{3} + 223\cdot 269^{4} +O\left(269^{ 5 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

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