Properties

Label 1.5_7_29.4t1.1c2
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} + 236 x^{2} + 1024 x + 9216 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{1015}(307,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 8 + 27\cdot 47 + 21\cdot 47^{2} + 24\cdot 47^{3} + 45\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 15 + 25\cdot 47 + 40\cdot 47^{2} + 34\cdot 47^{3} + 40\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 32 + 14\cdot 47 + 8\cdot 47^{2} + 16\cdot 47^{3} + 29\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 40 + 26\cdot 47 + 23\cdot 47^{2} + 18\cdot 47^{3} + 25\cdot 47^{4} +O\left(47^{ 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.