Properties

 Label 2.13456.4t3.c.a Dimension $2$ Group $D_{4}$ Conductor $13456$ Root number $1$ Indicator $1$

Related objects

Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$13456$$$$\medspace = 2^{4} \cdot 29^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 4.2.390224.1 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Determinant: 1.4.2t1.a.a Projective image: $C_2^2$ Projective field: $$\Q(i, \sqrt{29})$$

Defining polynomial

 $f(x)$ $=$ $$x^{4} - 29$$  .

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 6.

Roots:
 $r_{ 1 }$ $=$ $$2 + 11\cdot 13 + 12\cdot 13^{2} + 2\cdot 13^{3} + 8\cdot 13^{4} +O(13^{6})$$ $r_{ 2 }$ $=$ $$3 + 12\cdot 13 + 7\cdot 13^{2} + 12\cdot 13^{4} + 3\cdot 13^{5} +O(13^{6})$$ $r_{ 3 }$ $=$ $$10 + 5\cdot 13^{2} + 12\cdot 13^{3} + 9\cdot 13^{5} +O(13^{6})$$ $r_{ 4 }$ $=$ $$11 + 13 + 10\cdot 13^{3} + 4\cdot 13^{4} + 12\cdot 13^{5} +O(13^{6})$$

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

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

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,4)(2,3)$ $-2$ $2$ $2$ $(1,2)(3,4)$ $0$ $2$ $2$ $(1,4)$ $0$ $2$ $4$ $(1,3,4,2)$ $0$

The blue line marks the conjugacy class containing complex conjugation.