Properties

Label 1.5_23.4t1.1c1
Dimension 1
Group $C_4$
Conductor $ 5 \cdot 23 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$115= 5 \cdot 23 $
Artin number field: Splitting field of $f= x^{4} - x^{3} - 29 x^{2} + 29 x + 151 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Even
Corresponding Dirichlet character: \(\chi_{115}(22,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 3 + 16\cdot 31 + 31^{2} + 17\cdot 31^{3} + 20\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 10 + 21\cdot 31 + 10\cdot 31^{2} + 14\cdot 31^{3} + 9\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 24 + 20\cdot 31 + 2\cdot 31^{2} + 10\cdot 31^{3} + 18\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 26 + 3\cdot 31 + 16\cdot 31^{2} + 20\cdot 31^{3} + 13\cdot 31^{4} +O\left(31^{ 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$$(1,2)(3,4)$$-1$
$1$$4$$(1,3,2,4)$$\zeta_{4}$
$1$$4$$(1,4,2,3)$$-\zeta_{4}$
The blue line marks the conjugacy class containing complex conjugation.