Properties

Label 1.5_61.4t1.3c2
Dimension 1
Group $C_4$
Conductor $ 5 \cdot 61 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$305= 5 \cdot 61 $
Artin number field: Splitting field of $f= x^{4} - x^{3} + 76 x^{2} - 76 x + 1201 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{305}(182,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 3 + 10\cdot 29 + 27\cdot 29^{2} + 27\cdot 29^{3} + 6\cdot 29^{4} +O\left(29^{ 5 }\right) \\ r_{ 2 } &= 16 + 26\cdot 29 + 12\cdot 29^{2} + 26\cdot 29^{3} + 11\cdot 29^{4} +O\left(29^{ 5 }\right) \\ r_{ 3 } &= 19 + 23\cdot 29 + 20\cdot 29^{2} + 29^{3} + 3\cdot 29^{4} +O\left(29^{ 5 }\right) \\ r_{ 4 } &= 21 + 26\cdot 29 + 25\cdot 29^{2} + 29^{3} + 7\cdot 29^{4} +O\left(29^{ 5 }\right) \\ \end{aligned}\]

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.