Properties

Label 1.10003.3t1.a
Dimension $1$
Group $C_3$
Conductor $10003$
Indicator $0$

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Basic invariants

Dimension:$1$
Group:$C_3$
Conductor:\(10003\)\(\medspace = 7 \cdot 1429 \)
Artin number field: Galois closure of 3.3.100060009.1
Galois orbit size: $2$
Smallest permutation container: $C_3$
Parity: even
Projective image: $C_1$
Projective field: \(\Q\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 7.
Roots:
$r_{ 1 }$ $=$ $ 7\cdot 17 + 10\cdot 17^{2} + 13\cdot 17^{3} + 13\cdot 17^{4} + 16\cdot 17^{5} + 14\cdot 17^{6} +O\left(17^{ 7 }\right)$
$r_{ 2 }$ $=$ $ 2 + 12\cdot 17 + 15\cdot 17^{2} + 14\cdot 17^{3} + 12\cdot 17^{4} + 11\cdot 17^{5} + 6\cdot 17^{6} +O\left(17^{ 7 }\right)$
$r_{ 3 }$ $=$ $ 16 + 14\cdot 17 + 7\cdot 17^{2} + 5\cdot 17^{3} + 7\cdot 17^{4} + 5\cdot 17^{5} + 12\cdot 17^{6} +O\left(17^{ 7 }\right)$

Generators of the action on the roots $ r_{ 1 }, r_{ 2 }, r_{ 3 } $

Cycle notation
$(1,2,3)$

Character values on conjugacy classes

SizeOrderAction on $ r_{ 1 }, r_{ 2 }, r_{ 3 } $ Character values
$c1$ $c2$
$1$ $1$ $()$ $1$ $1$
$1$ $3$ $(1,2,3)$ $\zeta_{3}$ $-\zeta_{3} - 1$
$1$ $3$ $(1,3,2)$ $-\zeta_{3} - 1$ $\zeta_{3}$
The blue line marks the conjugacy class containing complex conjugation.