Properties

Label 1.2e4_7.4t1.2c2
Dimension 1
Group $C_4$
Conductor $ 2^{4} \cdot 7 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$112= 2^{4} \cdot 7 $
Artin number field: Splitting field of $f= x^{4} + 28 x^{2} + 98 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{112}(13,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 8 + 21\cdot 41 + 31\cdot 41^{2} + 4\cdot 41^{3} + 31\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 20 + 20\cdot 41 + 12\cdot 41^{2} + 5\cdot 41^{3} + 11\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 21 + 20\cdot 41 + 28\cdot 41^{2} + 35\cdot 41^{3} + 29\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 33 + 19\cdot 41 + 9\cdot 41^{2} + 36\cdot 41^{3} + 9\cdot 41^{4} +O\left(41^{ 5 }\right)$

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.