Properties

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

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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)$$=$\(x^{4} - 29\)  Toggle raw display.

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

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

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

SizeOrderAction 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.