Properties

Label 1.73_137.4t1.1c1
Dimension 1
Group $C_4$
Conductor $ 73 \cdot 137 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$10001= 73 \cdot 137 $
Artin number field: Splitting field of $f= x^{4} - x^{3} - 3750 x^{2} - 79383 x - 249439 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Even
Corresponding Dirichlet character: \(\chi_{10001}(9079,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 29 }$ to precision 6.
Roots:
$r_{ 1 }$ $=$ $ 1 + 21\cdot 29 + 5\cdot 29^{2} + 29^{3} + 20\cdot 29^{4} + 25\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 2 + 2\cdot 29 + 5\cdot 29^{2} + 19\cdot 29^{3} + 4\cdot 29^{4} + 20\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 7 + 17\cdot 29 + 23\cdot 29^{2} + 10\cdot 29^{3} + 7\cdot 29^{4} + 28\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 20 + 17\cdot 29 + 23\cdot 29^{2} + 26\cdot 29^{3} + 25\cdot 29^{4} + 12\cdot 29^{5} +O\left(29^{ 6 }\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.