Properties

Label 2.11025.8t7.a.a
Dimension $2$
Group $C_8:C_2$
Conductor $11025$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_8:C_2$
Conductor: \(11025\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 7^{2} \)
Artin stem field: Galois closure of 8.4.136744453125.1
Galois orbit size: $2$
Smallest permutation container: $C_8:C_2$
Parity: odd
Determinant: 1.5.4t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{5})\)

Defining polynomial

$f(x)$$=$ \( x^{8} - 2x^{7} - 2x^{6} + 31x^{5} - 65x^{4} - 239x^{3} + 58x^{2} + 868x + 961 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 181 }$ to precision 8.

Roots:
$r_{ 1 }$ $=$ \( 52 + 73\cdot 181 + 23\cdot 181^{2} + 118\cdot 181^{3} + 98\cdot 181^{4} + 43\cdot 181^{5} + 33\cdot 181^{6} + 101\cdot 181^{7} +O(181^{8})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 58 + 2\cdot 181 + 113\cdot 181^{2} + 177\cdot 181^{3} + 170\cdot 181^{4} + 158\cdot 181^{5} + 102\cdot 181^{6} + 105\cdot 181^{7} +O(181^{8})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 69 + 95\cdot 181 + 116\cdot 181^{2} + 167\cdot 181^{3} + 77\cdot 181^{4} + 124\cdot 181^{5} + 95\cdot 181^{6} + 40\cdot 181^{7} +O(181^{8})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 80 + 178\cdot 181 + 45\cdot 181^{2} + 93\cdot 181^{3} + 127\cdot 181^{4} + 93\cdot 181^{5} + 25\cdot 181^{6} + 98\cdot 181^{7} +O(181^{8})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 90 + 174\cdot 181 + 47\cdot 181^{2} + 179\cdot 181^{3} + 63\cdot 181^{4} + 35\cdot 181^{5} + 86\cdot 181^{6} + 83\cdot 181^{7} +O(181^{8})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 121 + 72\cdot 181 + 135\cdot 181^{2} + 117\cdot 181^{3} + 96\cdot 181^{4} + 76\cdot 181^{5} + 77\cdot 181^{6} + 26\cdot 181^{7} +O(181^{8})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 124 + 94\cdot 181 + 151\cdot 181^{2} + 102\cdot 181^{3} + 92\cdot 181^{4} + 108\cdot 181^{5} + 154\cdot 181^{6} + 139\cdot 181^{7} +O(181^{8})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 132 + 32\cdot 181 + 90\cdot 181^{2} + 129\cdot 181^{3} + 176\cdot 181^{4} + 82\cdot 181^{5} + 148\cdot 181^{6} + 128\cdot 181^{7} +O(181^{8})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.