Properties

Label 1.1013.4t1.1c2
Dimension 1
Group $C_4$
Conductor $ 1013 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

Learn more about

Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$1013 $
Artin number field: Splitting field of $f= x^{4} - x^{3} + 127 x^{2} + 2849 x + 26983 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{1013}(45,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 3 + 4\cdot 19 + 6\cdot 19^{2} + 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 4 + 15\cdot 19 + 14\cdot 19^{2} + 18\cdot 19^{3} +O\left(19^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 14 + 10\cdot 19^{2} + 19^{3} + 12\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 18 + 17\cdot 19 + 6\cdot 19^{2} + 17\cdot 19^{3} + 4\cdot 19^{4} +O\left(19^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 4 }$ Character value
$1$$1$$()$$1$
$1$$2$$(1,3)(2,4)$$-1$
$1$$4$$(1,2,3,4)$$-\zeta_{4}$
$1$$4$$(1,4,3,2)$$\zeta_{4}$
The blue line marks the conjugacy class containing complex conjugation.