Properties

Label 1.6001.4t1.a.a
Dimension $1$
Group $C_4$
Conductor $6001$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_4$
Conductor: \(6001\)\(\medspace = 17 \cdot 353 \)
Artin number field: Galois closure of 4.4.216108018001.1
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: even
Dirichlet character: \(\chi_{6001}(395,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$$ x^{4} - x^{3} - 2250 x^{2} + 12377 x + 116621 $.

The roots of $f$ are computed in $\Q_{ 37 }$ to precision 6.

Roots:
$r_{ 1 }$ $=$ $ 6 + 9\cdot 37 + 24\cdot 37^{2} + 15\cdot 37^{3} + 37^{4} + 20\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 8 + 20\cdot 37 + 17\cdot 37^{2} + 5\cdot 37^{3} + 10\cdot 37^{4} + 34\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 27 + 2\cdot 37 + 8\cdot 37^{2} + 19\cdot 37^{4} + 8\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 34 + 4\cdot 37 + 24\cdot 37^{2} + 15\cdot 37^{3} + 6\cdot 37^{4} + 11\cdot 37^{5} +O\left(37^{ 6 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

Cycle notation
$(1,2,3,4)$
$(1,3)(2,4)$

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$$4$$(1,2,3,4)$$\zeta_{4}$
$1$$4$$(1,4,3,2)$$-\zeta_{4}$

The blue line marks the conjugacy class containing complex conjugation.