Properties

Label 2.7851.4t3.c.a
Dimension $2$
Group $D_{4}$
Conductor $7851$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(7851\)\(\medspace = 3 \cdot 2617 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.20546067.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.7851.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{2617})\)

Defining polynomial

$f(x)$$=$ \( x^{4} + 43x^{2} - 192 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 8.

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