Properties

Label 1.2e2_29.4t1.1c1
Dimension 1
Group $C_4$
Conductor $ 2^{2} \cdot 29 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

Learn more about

Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$116= 2^{2} \cdot 29 $
Artin number field: Splitting field of $f= x^{4} - 29 x^{2} + 29 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Even
Corresponding Dirichlet character: \(\chi_{116}(99,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 12 + 18\cdot 53 + 16\cdot 53^{2} + 35\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 16 + 31\cdot 53 + 34\cdot 53^{3} + 38\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 37 + 21\cdot 53 + 52\cdot 53^{2} + 18\cdot 53^{3} + 14\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 41 + 34\cdot 53 + 36\cdot 53^{2} + 52\cdot 53^{3} + 17\cdot 53^{4} +O\left(53^{ 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.