# Properties

 Label 1.91.4t1.a.a Dimension $1$ Group $C_4$ Conductor $91$ Root number not computed Indicator $0$

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## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$91$$$$\medspace = 7 \cdot 13$$ Artin field: Galois closure of 4.4.107653.1 Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: even Dirichlet character: $$\chi_{91}(83,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} - x^{3} - 24x^{2} - 22x + 29$$ x^4 - x^3 - 24*x^2 - 22*x + 29 .

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

Roots:
 $r_{ 1 }$ $=$ $$4\cdot 29 + 18\cdot 29^{2} + 22\cdot 29^{3} + 23\cdot 29^{4} +O(29^{5})$$ 4*29 + 18*29^2 + 22*29^3 + 23*29^4+O(29^5) $r_{ 2 }$ $=$ $$10 + 5\cdot 29 + 11\cdot 29^{2} + 15\cdot 29^{3} + 16\cdot 29^{4} +O(29^{5})$$ 10 + 5*29 + 11*29^2 + 15*29^3 + 16*29^4+O(29^5) $r_{ 3 }$ $=$ $$21 + 16\cdot 29 + 21\cdot 29^{2} + 23\cdot 29^{3} + 22\cdot 29^{4} +O(29^{5})$$ 21 + 16*29 + 21*29^2 + 23*29^3 + 22*29^4+O(29^5) $r_{ 4 }$ $=$ $$28 + 2\cdot 29 + 7\cdot 29^{2} + 25\cdot 29^{3} + 23\cdot 29^{4} +O(29^{5})$$ 28 + 2*29 + 7*29^2 + 25*29^3 + 23*29^4+O(29^5)

## 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

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

The blue line marks the conjugacy class containing complex conjugation.