Properties

Label 1.3_17_197.2t1.1
Dimension 1
Group $C_2$
Conductor $ 3 \cdot 17 \cdot 197 $
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$1$
Group:$C_2$
Conductor:$10047= 3 \cdot 17 \cdot 197 $
Artin number field: Splitting field of $f= x^{2} - x + 2512 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_2$
Parity: Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 9 + 11\cdot 19 + 12\cdot 19^{2} + 8\cdot 19^{3} +O\left(19^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 11 + 7\cdot 19 + 6\cdot 19^{2} + 10\cdot 19^{3} + 18\cdot 19^{4} +O\left(19^{ 5 }\right)$

Generators of the action on the roots $ r_{ 1 }, r_{ 2 } $

Cycle notation
$(1,2)$

Character values on conjugacy classes

SizeOrderAction on $ r_{ 1 }, r_{ 2 } $ Character values
$c1$
$1$ $1$ $()$ $1$
$1$ $2$ $(1,2)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.