Properties

Label 2.1595.4t3.d.a
Dimension 2
Group $D_{4}$
Conductor $ 5 \cdot 11 \cdot 29 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$1595= 5 \cdot 11 \cdot 29 $
Artin number field: Splitting field of 4.2.7975.1 defined by $f= x^{4} - x^{3} + 3 x^{2} - 12 x - 16 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.1595.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{5}, \sqrt{-319})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 33 + 11\cdot 61 + 38\cdot 61^{2} + 35\cdot 61^{3} + 34\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 46 + 14\cdot 61 + 5\cdot 61^{2} + 43\cdot 61^{3} + 25\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 52 + 52\cdot 61 + 4\cdot 61^{2} + 42\cdot 61^{3} + 34\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 53 + 42\cdot 61 + 12\cdot 61^{2} + 61^{3} + 27\cdot 61^{4} +O\left(61^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 4 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,2)(3,4)$$-2$
$2$$2$$(1,3)(2,4)$$0$
$2$$2$$(1,2)$$0$
$2$$4$$(1,4,2,3)$$0$
The blue line marks the conjugacy class containing complex conjugation.