Properties

Label 1.80.4t1.a.b
Dimension $1$
Group $C_4$
Conductor $80$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_4$
Conductor: \(80\)\(\medspace = 2^{4} \cdot 5 \)
Artin number field: Galois closure of 4.4.51200.1
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: even
Dirichlet character: \(\chi_{80}(29,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$$ x^{4} - 20 x^{2} + 50 $.

The roots of $f$ are computed in $\Q_{ 7 }$ to precision 7.

Roots:
$r_{ 1 }$ $=$ $ 2 + 2\cdot 7 + 5\cdot 7^{2} + 2\cdot 7^{3} + 5\cdot 7^{6} +O\left(7^{ 7 }\right)$
$r_{ 2 }$ $=$ $ 3 + 4\cdot 7^{2} + 5\cdot 7^{4} + 4\cdot 7^{5} + 2\cdot 7^{6} +O\left(7^{ 7 }\right)$
$r_{ 3 }$ $=$ $ 4 + 6\cdot 7 + 2\cdot 7^{2} + 6\cdot 7^{3} + 7^{4} + 2\cdot 7^{5} + 4\cdot 7^{6} +O\left(7^{ 7 }\right)$
$r_{ 4 }$ $=$ $ 5 + 4\cdot 7 + 7^{2} + 4\cdot 7^{3} + 6\cdot 7^{4} + 6\cdot 7^{5} + 7^{6} +O\left(7^{ 7 }\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.