Properties

Label 2.5_61.4t3.2c1
Dimension 2
Group $D_{4}$
Conductor $ 5 \cdot 61 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$305= 5 \cdot 61 $
Artin number field: Splitting field of $f= x^{4} - 2 x^{3} + 6 x^{2} - 5 x + 5 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Even
Determinant: 1.5_61.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 2 + 7\cdot 19 + 12\cdot 19^{2} + 17\cdot 19^{3} + 17\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 4 + 5\cdot 19 + 11\cdot 19^{2} + 14\cdot 19^{3} + 17\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 16 + 13\cdot 19 + 7\cdot 19^{2} + 4\cdot 19^{3} + 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 18 + 11\cdot 19 + 6\cdot 19^{2} + 19^{3} + 19^{4} +O\left(19^{ 5 }\right)$

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

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

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.