Properties

Label 1.3e2_1117.3t1.1c2
Dimension 1
Group $C_3$
Conductor $ 3^{2} \cdot 1117 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

Learn more about

Basic invariants

Dimension:$1$
Group:$C_3$
Conductor:$10053= 3^{2} \cdot 1117 $
Artin number field: Splitting field of $f= x^{3} - 3351 x - 51382 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_3$
Parity: Even
Corresponding Dirichlet character: \(\chi_{10053}(1237,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 6.
Roots:
$r_{ 1 }$ $=$ $ 3 + 2\cdot 19 + 9\cdot 19^{2} + 14\cdot 19^{3} + 16\cdot 19^{4} + 9\cdot 19^{5} +O\left(19^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 17 + 2\cdot 19 + 8\cdot 19^{2} + 14\cdot 19^{3} + 6\cdot 19^{4} + 5\cdot 19^{5} +O\left(19^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 18 + 13\cdot 19 + 19^{2} + 9\cdot 19^{3} + 14\cdot 19^{4} + 3\cdot 19^{5} +O\left(19^{ 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$
$1$$3$$(1,3,2)$$\zeta_{3}$
The blue line marks the conjugacy class containing complex conjugation.