Properties

Label 1.13.4t1.1c2
Dimension 1
Group $C_4$
Conductor $ 13 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$13 $
Artin number field: Splitting field of $f= x^{4} - x^{3} + 2 x^{2} + 4 x + 3 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{13}(5,\cdot)\)

Galois action

Roots of defining polynomial

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\left(29^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 15 + 25\cdot 29 + 4\cdot 29^{2} + 19\cdot 29^{3} + 6\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 17 + 28\cdot 29 + 20\cdot 29^{2} + 6\cdot 29^{3} + 6\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 24 + 12\cdot 29 + 24\cdot 29^{2} + 18\cdot 29^{3} + 4\cdot 29^{4} +O\left(29^{ 5 }\right)$

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.