Properties

Label 1.7_1429.3t1.2c1
Dimension 1
Group $C_3$
Conductor $ 7 \cdot 1429 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_3$
Conductor:$10003= 7 \cdot 1429 $
Artin number field: Splitting field of $f= x^{3} - x^{2} - 3334 x + 39271 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_3$
Parity: Even
Corresponding Dirichlet character: \(\chi_{10003}(4951,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 6.
Roots:
$r_{ 1 }$ $=$ $ 5 + 7\cdot 23 + 11\cdot 23^{2} + 9\cdot 23^{3} + 6\cdot 23^{4} + 14\cdot 23^{5} +O\left(23^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 9 + 10\cdot 23 + 12\cdot 23^{2} + 21\cdot 23^{3} + 8\cdot 23^{4} + 17\cdot 23^{5} +O\left(23^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 10 + 5\cdot 23 + 22\cdot 23^{2} + 14\cdot 23^{3} + 7\cdot 23^{4} + 14\cdot 23^{5} +O\left(23^{ 6 }\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 value
$1$$1$$()$$1$
$1$$3$$(1,2,3)$$\zeta_{3}$
$1$$3$$(1,3,2)$$-\zeta_{3} - 1$
The blue line marks the conjugacy class containing complex conjugation.