Properties

Label 1.5_7_29.4t1.2c2
Dimension 1
Group $C_4$
Conductor $ 5 \cdot 7 \cdot 29 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$1015= 5 \cdot 7 \cdot 29 $
Artin number field: Splitting field of $f= x^{4} - x^{3} - 254 x^{2} + 254 x + 12751 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Even
Corresponding Dirichlet character: \(\chi_{1015}(202,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 38\cdot 41 + 35\cdot 41^{2} + 16\cdot 41^{3} + 19\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 1 + 21\cdot 41 + 19\cdot 41^{2} + 9\cdot 41^{3} + 29\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 7 + 25\cdot 41 + 24\cdot 41^{2} + 34\cdot 41^{3} + 8\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 34 + 38\cdot 41 + 41^{2} + 21\cdot 41^{3} + 24\cdot 41^{4} +O\left(41^{ 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.