Properties

Label 1.101.4t1.1c2
Dimension 1
Group $C_4$
Conductor $ 101 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$101 $
Artin number field: Splitting field of $f= x^{4} - x^{3} + 13 x^{2} - 19 x + 361 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{101}(10,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 11 + 30\cdot 31 + 7\cdot 31^{2} + 15\cdot 31^{3} + 25\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 13 + 29\cdot 31 + 12\cdot 31^{2} + 18\cdot 31^{3} + 15\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 15 + 4\cdot 31 + 12\cdot 31^{2} + 9\cdot 31^{3} + 7\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 24 + 28\cdot 31 + 28\cdot 31^{2} + 18\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
$(1,4,2,3)$
$(1,2)(3,4)$

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