Properties

Label 2.2e2_7e2.4t3.2c1
Dimension 2
Group $D_{4}$
Conductor $ 2^{2} \cdot 7^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$196= 2^{2} \cdot 7^{2} $
Artin number field: Splitting field of $f= x^{4} - x^{3} + 3 x^{2} - 4 x + 2 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.2e2.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 6 + 41\cdot 53 + 35\cdot 53^{2} + 21\cdot 53^{3} + 39\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 25 + 3\cdot 53 + 49\cdot 53^{3} + 22\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 33 + 2\cdot 53 + 9\cdot 53^{2} + 38\cdot 53^{3} + 9\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 43 + 5\cdot 53 + 8\cdot 53^{2} + 50\cdot 53^{3} + 33\cdot 53^{4} +O\left(53^{ 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.