Properties

Label 1.2e3_5.4t1.2c2
Dimension 1
Group $C_4$
Conductor $ 2^{3} \cdot 5 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$40= 2^{3} \cdot 5 $
Artin number field: Splitting field of $f= x^{4} + 10 x^{2} + 20 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{40}(37,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 2 + 19 + 12\cdot 19^{3} + 2\cdot 19^{4} +O\left(19^{ 5 }\right) \\ r_{ 2 } &= 9 + 9\cdot 19 + 15\cdot 19^{2} + 14\cdot 19^{3} + 8\cdot 19^{4} +O\left(19^{ 5 }\right) \\ r_{ 3 } &= 10 + 9\cdot 19 + 3\cdot 19^{2} + 4\cdot 19^{3} + 10\cdot 19^{4} +O\left(19^{ 5 }\right) \\ r_{ 4 } &= 17 + 17\cdot 19 + 18\cdot 19^{2} + 6\cdot 19^{3} + 16\cdot 19^{4} +O\left(19^{ 5 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

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