Properties

Label 1.13.4t1.a.a
Dimension $1$
Group $C_4$
Conductor $13$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_4$
Conductor: \(13\)
Artin field: 4.0.2197.1
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: odd
Dirichlet character: \(\chi_{13}(8,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{4} - x^{3} + 2 x^{2} + 4 x + 3\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 3 + 20\cdot 29 + 7\cdot 29^{2} + 13\cdot 29^{3} + 11\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 15 + 25\cdot 29 + 4\cdot 29^{2} + 19\cdot 29^{3} + 6\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 17 + 28\cdot 29 + 20\cdot 29^{2} + 6\cdot 29^{3} + 6\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 24 + 12\cdot 29 + 24\cdot 29^{2} + 18\cdot 29^{3} + 4\cdot 29^{4} +O(29^{5})\)  Toggle raw display

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

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

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$$4$$(1,2,3,4)$$\zeta_{4}$
$1$$4$$(1,4,3,2)$$-\zeta_{4}$

The blue line marks the conjugacy class containing complex conjugation.