Properties

Label 1.273.4t1.a.a
Dimension $1$
Group $C_4$
Conductor $273$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_4$
Conductor: \(273\)\(\medspace = 3 \cdot 7 \cdot 13 \)
Artin field: Galois closure of 4.0.968877.2
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: odd
Dirichlet character: \(\chi_{273}(125,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{4} - x^{3} + 67x^{2} + 69x + 393 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 1 + 8\cdot 23^{2} + 16\cdot 23^{3} + 16\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 12 + 10\cdot 23 + 20\cdot 23^{2} + 6\cdot 23^{3} + 19\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 14 + 10\cdot 23 + 20\cdot 23^{3} + 17\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 20 + 23 + 17\cdot 23^{2} + 2\cdot 23^{3} + 15\cdot 23^{4} +O(23^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.