Properties

Label 2.1703.4t3.c.a
Dimension $2$
Group $D_{4}$
Conductor $1703$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(1703\)\(\medspace = 13 \cdot 131 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.22139.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.1703.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{13}, \sqrt{-131})\)

Defining polynomial

$f(x)$$=$ \( x^{4} - x^{3} - 7x^{2} - 11x - 9 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 10 + 88\cdot 101 + 24\cdot 101^{2} + 19\cdot 101^{3} + 95\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 53 + 75\cdot 101 + 56\cdot 101^{2} + 100\cdot 101^{3} + 20\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 66 + 48\cdot 101 + 20\cdot 101^{2} + 98\cdot 101^{3} + 12\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 74 + 90\cdot 101 + 99\cdot 101^{2} + 84\cdot 101^{3} + 72\cdot 101^{4} +O(101^{5})\) 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.