Properties

Label 1.304.4t1.b.a
Dimension $1$
Group $C_4$
Conductor $304$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_4$
Conductor: \(304\)\(\medspace = 2^{4} \cdot 19 \)
Artin field: 4.0.739328.2
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: odd
Dirichlet character: \(\chi_{304}(37,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{4} + 76 x^{2} + 722\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 6.

Roots:
$r_{ 1 }$ $=$ \( 1 + 7\cdot 17 + 4\cdot 17^{2} + 5\cdot 17^{3} + 13\cdot 17^{4} + 6\cdot 17^{5} +O(17^{6})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 5 + 15\cdot 17 + 15\cdot 17^{2} + 7\cdot 17^{3} + 8\cdot 17^{4} + 11\cdot 17^{5} +O(17^{6})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 12 + 17 + 17^{2} + 9\cdot 17^{3} + 8\cdot 17^{4} + 5\cdot 17^{5} +O(17^{6})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 16 + 9\cdot 17 + 12\cdot 17^{2} + 11\cdot 17^{3} + 3\cdot 17^{4} + 10\cdot 17^{5} +O(17^{6})\)  Toggle raw display

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.