Properties

Label 1.52.4t1.a.a
Dimension $1$
Group $C_4$
Conductor $52$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_4$
Conductor: \(52\)\(\medspace = 2^{2} \cdot 13 \)
Artin number field: Galois closure of 4.4.35152.1
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: even
Dirichlet character: \(\chi_{52}(31,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$$ x^{4} - 13 x^{2} + 13 $.

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ $ 2 + 8\cdot 23 + 13\cdot 23^{2} + 20\cdot 23^{3} + 4\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 3 + 10\cdot 23 + 17\cdot 23^{2} + 8\cdot 23^{3} + 20\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 20 + 12\cdot 23 + 5\cdot 23^{2} + 14\cdot 23^{3} + 2\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 21 + 14\cdot 23 + 9\cdot 23^{2} + 2\cdot 23^{3} + 18\cdot 23^{4} +O\left(23^{ 5 }\right)$

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.