Properties

Label 1.7_13.4t1.1
Dimension 1
Group $C_4$
Conductor $ 7 \cdot 13 $
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$91= 7 \cdot 13 $
Artin number field: Splitting field of $f= x^{4} - x^{3} - 24 x^{2} - 22 x + 29 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 4\cdot 29 + 18\cdot 29^{2} + 22\cdot 29^{3} + 23\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 10 + 5\cdot 29 + 11\cdot 29^{2} + 15\cdot 29^{3} + 16\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 21 + 16\cdot 29 + 21\cdot 29^{2} + 23\cdot 29^{3} + 22\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 28 + 2\cdot 29 + 7\cdot 29^{2} + 25\cdot 29^{3} + 23\cdot 29^{4} +O\left(29^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 4 }$ Character values
$c1$ $c2$
$1$ $1$ $()$ $1$ $1$
$1$ $2$ $(1,2)(3,4)$ $-1$ $-1$
$1$ $4$ $(1,4,2,3)$ $\zeta_{4}$ $-\zeta_{4}$
$1$ $4$ $(1,3,2,4)$ $-\zeta_{4}$ $\zeta_{4}$
The blue line marks the conjugacy class containing complex conjugation.