Properties

Label 1.3_337.4t1.1c2
Dimension 1
Group $C_4$
Conductor $ 3 \cdot 337 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$1011= 3 \cdot 337 $
Artin number field: Splitting field of $f= x^{4} - x^{3} + 211 x^{2} + 1032 x + 17628 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{1011}(863,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 6.
Roots:
$r_{ 1 }$ $=$ $ 7\cdot 13 + 5\cdot 13^{2} + 12\cdot 13^{3} + 6\cdot 13^{4} + 8\cdot 13^{5} +O\left(13^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 4 + 5\cdot 13 + 11\cdot 13^{2} + 8\cdot 13^{3} + 11\cdot 13^{4} + 10\cdot 13^{5} +O\left(13^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 11 + 9\cdot 13 + 2\cdot 13^{2} + 3\cdot 13^{3} + 2\cdot 13^{4} + 3\cdot 13^{5} +O\left(13^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 12 + 3\cdot 13 + 6\cdot 13^{2} + 13^{3} + 5\cdot 13^{4} + 3\cdot 13^{5} +O\left(13^{ 6 }\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.