Properties

Label 1.1015.4t1.c.b
Dimension $1$
Group $C_4$
Conductor $1015$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_4$
Conductor: \(1015\)\(\medspace = 5 \cdot 7 \cdot 29 \)
Artin field: Galois closure of 4.0.149382625.2
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: odd
Dirichlet character: \(\chi_{1015}(713,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{4} - x^{3} + 236x^{2} + 9x + 18351 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 4 + 2\cdot 17 + 8\cdot 17^{2} + 7\cdot 17^{3} +O(17^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 8 + 8\cdot 17 + 14\cdot 17^{2} + 4\cdot 17^{3} + 7\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 11 + 3\cdot 17 + 11\cdot 17^{2} + 7\cdot 17^{3} + 7\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 12 + 2\cdot 17 + 14\cdot 17^{3} + 17^{4} +O(17^{5})\) Copy content Toggle raw display

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

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

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

The blue line marks the conjugacy class containing complex conjugation.