Properties

Label 1.176.4t1.a.b
Dimension $1$
Group $C_4$
Conductor $176$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_4$
Conductor: \(176\)\(\medspace = 2^{4} \cdot 11 \)
Artin field: Galois closure of 4.4.247808.1
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: even
Dirichlet character: \(\chi_{176}(43,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{4} - 44x^{2} + 242 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 6.

Roots:
$r_{ 1 }$ $=$ \( 6 + 10\cdot 23 + 3\cdot 23^{2} + 15\cdot 23^{3} + 22\cdot 23^{4} + 13\cdot 23^{5} +O(23^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 10 + 3\cdot 23 + 5\cdot 23^{2} + 17\cdot 23^{3} + 20\cdot 23^{4} + 12\cdot 23^{5} +O(23^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 13 + 19\cdot 23 + 17\cdot 23^{2} + 5\cdot 23^{3} + 2\cdot 23^{4} + 10\cdot 23^{5} +O(23^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 17 + 12\cdot 23 + 19\cdot 23^{2} + 7\cdot 23^{3} + 9\cdot 23^{5} +O(23^{6})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

Cycle notation
$(1,4)(2,3)$
$(1,2,4,3)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.