Properties

Label 2.38656.4t3.a.a
Dimension $2$
Group $D_{4}$
Conductor $38656$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(38656\)\(\medspace = 2^{8} \cdot 151 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 4.2.309248.4
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.151.2t1.a.a
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{2}, \sqrt{-151})\)

Defining polynomial

$f(x)$$=$\(x^{4} + 12 x^{2} - 302\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 7.

Roots:
$r_{ 1 }$ $=$ \( 1 + 14\cdot 17^{2} + 17^{3} + 2\cdot 17^{4} + 3\cdot 17^{5} + 5\cdot 17^{6} +O(17^{7})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 2 + 4\cdot 17 + 10\cdot 17^{2} + 8\cdot 17^{3} + 5\cdot 17^{4} + 8\cdot 17^{5} + 7\cdot 17^{6} +O(17^{7})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 15 + 12\cdot 17 + 6\cdot 17^{2} + 8\cdot 17^{3} + 11\cdot 17^{4} + 8\cdot 17^{5} + 9\cdot 17^{6} +O(17^{7})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 16 + 16\cdot 17 + 2\cdot 17^{2} + 15\cdot 17^{3} + 14\cdot 17^{4} + 13\cdot 17^{5} + 11\cdot 17^{6} +O(17^{7})\)  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.