Properties

Label 2.9225.8t7.a.a
Dimension $2$
Group $C_8:C_2$
Conductor $9225$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_8:C_2$
Conductor: \(9225\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 41 \)
Artin stem field: Galois closure of 8.4.95738203125.1
Galois orbit size: $2$
Smallest permutation container: $C_8:C_2$
Parity: odd
Determinant: 1.205.4t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{5}, \sqrt{-123})\)

Defining polynomial

$f(x)$$=$ \( x^{8} - 3x^{7} - 2x^{6} + 9x^{5} - 90x^{4} + 84x^{3} + 208x^{2} + 327x + 211 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 2 + 29\cdot 31^{2} + 2\cdot 31^{3} + 25\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 5 + 22\cdot 31 + 6\cdot 31^{2} + 25\cdot 31^{3} + 7\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 7 + 15\cdot 31 + 17\cdot 31^{2} + 20\cdot 31^{3} + 22\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 11 + 16\cdot 31 + 17\cdot 31^{2} + 3\cdot 31^{3} + 28\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 12 + 14\cdot 31 + 23\cdot 31^{2} + 10\cdot 31^{3} + 16\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 17 + 15\cdot 31 + 26\cdot 31^{2} + 22\cdot 31^{3} + 8\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 18 + 9\cdot 31 + 2\cdot 31^{2} + 2\cdot 31^{3} + 20\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 24 + 30\cdot 31 + 5\cdot 31^{3} + 26\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.