Properties

Label 2.248.4t3.c.a
Dimension $2$
Group $D_{4}$
Conductor $248$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(248\)\(\medspace = 2^{3} \cdot 31 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.1984.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.248.2t1.b.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{2}, \sqrt{-31})\)

Defining polynomial

$f(x)$$=$ \( x^{4} - 2x^{3} + x^{2} - 2 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 4 + 60\cdot 71 + 18\cdot 71^{2} + 46\cdot 71^{3} + 61\cdot 71^{4} +O(71^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 17 + 56\cdot 71 + 17\cdot 71^{2} + 64\cdot 71^{3} + 33\cdot 71^{4} +O(71^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 55 + 14\cdot 71 + 53\cdot 71^{2} + 6\cdot 71^{3} + 37\cdot 71^{4} +O(71^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 68 + 10\cdot 71 + 52\cdot 71^{2} + 24\cdot 71^{3} + 9\cdot 71^{4} +O(71^{5})\) Copy content Toggle raw display

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

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

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.