Properties

Label 2.3_109.4t3.2c1
Dimension 2
Group $D_{4}$
Conductor $ 3 \cdot 109 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$327= 3 \cdot 109 $
Artin number field: Splitting field of $f= x^{4} - x^{3} - 4 x^{2} + 4 x + 7 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.3_109.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 7 }$ to precision 6.
Roots:
$r_{ 1 }$ $=$ $ 5\cdot 7 + 5\cdot 7^{2} + 6\cdot 7^{4} + 2\cdot 7^{5} +O\left(7^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 1 + 5\cdot 7 + 5\cdot 7^{2} + 6\cdot 7^{3} +O\left(7^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 2 + 6\cdot 7 + 4\cdot 7^{3} + 6\cdot 7^{4} + 7^{5} +O\left(7^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 5 + 4\cdot 7 + 7^{2} + 2\cdot 7^{3} + 2\cdot 7^{5} +O\left(7^{ 6 }\right)$

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.