Properties

Label 1.15.4t1.a.b
Dimension $1$
Group $C_4$
Conductor $15$
Root number not computed
Indicator $0$

Related objects

Learn more about

Basic invariants

Dimension: $1$
Group: $C_4$
Conductor: \(15\)\(\medspace = 3 \cdot 5 \)
Artin field: \(\Q(\zeta_{15})^+\)
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: even
Dirichlet character: \(\chi_{15}(2,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{4} - x^{3} - 4 x^{2} + 4 x + 1\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 4 + 4\cdot 29 + 21\cdot 29^{3} + 5\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 14 + 3\cdot 29 + 17\cdot 29^{2} + 23\cdot 29^{3} + 10\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 20 + 3\cdot 29 + 24\cdot 29^{2} + 8\cdot 29^{3} + 8\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 21 + 17\cdot 29 + 16\cdot 29^{2} + 4\cdot 29^{3} + 4\cdot 29^{4} +O(29^{5})\)  Toggle raw display

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.