Properties

Label 2.225.8t7.a.a
Dimension $2$
Group $C_8:C_2$
Conductor $225$
Root number not computed
Indicator $0$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $2$
Group: $C_8:C_2$
Conductor: \(225\)\(\medspace = 3^{2} \cdot 5^{2}\)
Artin stem field: Galois closure of 8.4.56953125.1
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)$$=$ \( x^{8} - x^{7} + 2x^{6} + 2x^{5} - 5x^{4} + 13x^{3} - 13x^{2} + x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 14 + 36\cdot 61 + 7\cdot 61^{2} + 26\cdot 61^{3} + 33\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 21 + 47\cdot 61 + 14\cdot 61^{2} + 3\cdot 61^{3} + 18\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 24 + 3\cdot 61 + 60\cdot 61^{2} + 3\cdot 61^{3} + 43\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 37 + 17\cdot 61 + 12\cdot 61^{2} + 59\cdot 61^{3} + 42\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 44 + 33\cdot 61 + 37\cdot 61^{2} + 20\cdot 61^{3} + 26\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 52 + 61 + 60\cdot 61^{2} + 24\cdot 61^{3} + 23\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 56 + 28\cdot 61 + 52\cdot 61^{2} + 16\cdot 61^{3} + 38\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 58 + 13\cdot 61 + 60\cdot 61^{2} + 27\cdot 61^{3} + 18\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.