Properties

Label 1.97.4t1.1c2
Dimension 1
Group $C_4$
Conductor $ 97 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$97 $
Artin number field: Splitting field of $f= x^{4} - x^{3} - 36 x^{2} - 91 x - 61 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Even
Corresponding Dirichlet character: \(\chi_{97}(22,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 10 + 14\cdot 43 + 27\cdot 43^{2} + 8\cdot 43^{3} + 8\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 22 + 21\cdot 43 + 41\cdot 43^{2} + 41\cdot 43^{3} + 42\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 23 + 24\cdot 43 + 22\cdot 43^{2} + 23\cdot 43^{3} + 10\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 32 + 25\cdot 43 + 37\cdot 43^{2} + 11\cdot 43^{3} + 24\cdot 43^{4} +O\left(43^{ 5 }\right)$

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

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

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$$4$$(1,4,3,2)$$-\zeta_{4}$
$1$$4$$(1,2,3,4)$$\zeta_{4}$
The blue line marks the conjugacy class containing complex conjugation.