Properties

Label 2.2e6_5.4t3.2c1
Dimension 2
Group $D_{4}$
Conductor $ 2^{6} \cdot 5 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$320= 2^{6} \cdot 5 $
Artin number field: Splitting field of $f= x^{4} - 4 x^{2} + 5 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.2e2_5.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 14 + 9\cdot 61 + 3\cdot 61^{2} + 36\cdot 61^{3} + 46\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 28 + 17\cdot 61 + 46\cdot 61^{2} + 35\cdot 61^{3} + 4\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 33 + 43\cdot 61 + 14\cdot 61^{2} + 25\cdot 61^{3} + 56\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 47 + 51\cdot 61 + 57\cdot 61^{2} + 24\cdot 61^{3} + 14\cdot 61^{4} +O\left(61^{ 5 }\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.