Properties

Label 2.2312.8t7.b.a
Dimension $2$
Group $C_8:C_2$
Conductor $2312$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_8:C_2$
Conductor: \(2312\)\(\medspace = 2^{3} \cdot 17^{2} \)
Artin stem field: Galois closure of 8.0.26261675072.1
Galois orbit size: $2$
Smallest permutation container: $C_8:C_2$
Parity: even
Determinant: 1.136.4t1.a.b
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{2}, \sqrt{17})\)

Defining polynomial

$f(x)$$=$ \( x^{8} - 2x^{7} + 6x^{6} - 3x^{5} + 19x^{4} - 14x^{3} + 6x^{2} + 36x + 52 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 239 }$ to precision 6.

Roots:
$r_{ 1 }$ $=$ \( 19 + 103\cdot 239 + 135\cdot 239^{2} + 94\cdot 239^{3} + 168\cdot 239^{4} + 214\cdot 239^{5} +O(239^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 25 + 22\cdot 239 + 177\cdot 239^{2} + 211\cdot 239^{3} + 20\cdot 239^{4} + 206\cdot 239^{5} +O(239^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 87 + 41\cdot 239 + 15\cdot 239^{2} + 178\cdot 239^{3} + 223\cdot 239^{4} + 176\cdot 239^{5} +O(239^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 93 + 199\cdot 239 + 56\cdot 239^{2} + 56\cdot 239^{3} + 76\cdot 239^{4} + 168\cdot 239^{5} +O(239^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 149 + 57\cdot 239 + 11\cdot 239^{2} + 112\cdot 239^{3} + 46\cdot 239^{4} + 118\cdot 239^{5} +O(239^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 176 + 40\cdot 239 + 239^{2} + 208\cdot 239^{3} + 26\cdot 239^{4} + 51\cdot 239^{5} +O(239^{6})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 191 + 134\cdot 239 + 45\cdot 239^{2} + 119\cdot 239^{3} + 206\cdot 239^{4} + 43\cdot 239^{5} +O(239^{6})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 218 + 117\cdot 239 + 35\cdot 239^{2} + 215\cdot 239^{3} + 186\cdot 239^{4} + 215\cdot 239^{5} +O(239^{6})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,4)(2,3)(5,8)(6,7)$$-2$
$2$$2$$(5,8)(6,7)$$0$
$1$$4$$(1,2,4,3)(5,6,8,7)$$2 \zeta_{4}$
$1$$4$$(1,3,4,2)(5,7,8,6)$$-2 \zeta_{4}$
$2$$4$$(1,3,4,2)(5,6,8,7)$$0$
$2$$8$$(1,6,2,8,4,7,3,5)$$0$
$2$$8$$(1,8,3,6,4,5,2,7)$$0$
$2$$8$$(1,8,2,7,4,5,3,6)$$0$
$2$$8$$(1,7,3,8,4,6,2,5)$$0$

The blue line marks the conjugacy class containing complex conjugation.