Properties

Label 2.5_31.4t3.1c1
Dimension 2
Group $D_{4}$
Conductor $ 5 \cdot 31 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$155= 5 \cdot 31 $
Artin number field: Splitting field of $f= x^{4} - x^{3} - 3 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.5_31.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 14 + 13\cdot 41 + 3\cdot 41^{2} + 10\cdot 41^{3} + 25\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 17 + 13\cdot 41 + 14\cdot 41^{2} + 29\cdot 41^{3} + 29\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 21 + 5\cdot 41 + 18\cdot 41^{2} + 20\cdot 41^{3} + 28\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 31 + 8\cdot 41 + 5\cdot 41^{2} + 22\cdot 41^{3} + 39\cdot 41^{4} +O\left(41^{ 5 }\right)$

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

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

Character values on conjugacy classes

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