Properties

Label 1.10064.4t1.a.a
Dimension $1$
Group $C_4$
Conductor $10064$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_4$
Conductor: \(10064\)\(\medspace = 2^{4} \cdot 17 \cdot 37 \)
Artin field: Galois closure of 4.0.810272768.2
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: odd
Dirichlet character: \(\chi_{10064}(7547,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{4} + 2516x^{2} + 791282 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 8.

Roots:
$r_{ 1 }$ $=$ \( 1 + 13\cdot 23 + 5\cdot 23^{2} + 9\cdot 23^{3} + 2\cdot 23^{4} + 4\cdot 23^{5} + 7\cdot 23^{6} + 15\cdot 23^{7} +O(23^{8})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 6 + 2\cdot 23 + 11\cdot 23^{2} + 2\cdot 23^{3} + 23^{4} + 5\cdot 23^{5} + 21\cdot 23^{6} + 17\cdot 23^{7} +O(23^{8})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 17 + 20\cdot 23 + 11\cdot 23^{2} + 20\cdot 23^{3} + 21\cdot 23^{4} + 17\cdot 23^{5} + 23^{6} + 5\cdot 23^{7} +O(23^{8})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 22 + 9\cdot 23 + 17\cdot 23^{2} + 13\cdot 23^{3} + 20\cdot 23^{4} + 18\cdot 23^{5} + 15\cdot 23^{6} + 7\cdot 23^{7} +O(23^{8})\) 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.