Properties

Label 2.3_7e2_13_19e2.4t3.1c1
Dimension 2
Group $D_{4}$
Conductor $ 3 \cdot 7^{2} \cdot 13 \cdot 19^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$689871= 3 \cdot 7^{2} \cdot 13 \cdot 19^{2} $
Artin number field: Splitting field of $f= x^{4} - x^{3} + 32 x^{2} - 232 x - 3233 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.3_13.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 43 }$ to precision 6.
Roots:
$r_{ 1 }$ $=$ $ 27 + 35\cdot 43 + 33\cdot 43^{2} + 40\cdot 43^{3} + 13\cdot 43^{4} + 42\cdot 43^{5} +O\left(43^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 28 + 5\cdot 43 + 7\cdot 43^{2} + 26\cdot 43^{3} + 10\cdot 43^{4} + 7\cdot 43^{5} +O\left(43^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 33 + 14\cdot 43 + 40\cdot 43^{2} + 29\cdot 43^{3} + 20\cdot 43^{4} + 43^{5} +O\left(43^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 42 + 29\cdot 43 + 4\cdot 43^{2} + 32\cdot 43^{3} + 40\cdot 43^{4} + 34\cdot 43^{5} +O\left(43^{ 6 }\right)$

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

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

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.