Properties

Label 1.2e2_3_5_17.4t1.1c1
Dimension 1
Group $C_4$
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 17 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$1020= 2^{2} \cdot 3 \cdot 5 \cdot 17 $
Artin number field: Splitting field of $f= x^{4} + 255 x^{2} + 765 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{1020}(803,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 6.
Roots:
$r_{ 1 }$ $=$ $ 8 + 8\cdot 59 + 32\cdot 59^{2} + 46\cdot 59^{3} + 39\cdot 59^{4} + 8\cdot 59^{5} +O\left(59^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 25 + 16\cdot 59 + 9\cdot 59^{2} + 42\cdot 59^{3} + 4\cdot 59^{4} + 38\cdot 59^{5} +O\left(59^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 34 + 42\cdot 59 + 49\cdot 59^{2} + 16\cdot 59^{3} + 54\cdot 59^{4} + 20\cdot 59^{5} +O\left(59^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 51 + 50\cdot 59 + 26\cdot 59^{2} + 12\cdot 59^{3} + 19\cdot 59^{4} + 50\cdot 59^{5} +O\left(59^{ 6 }\right)$

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

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

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$$4$$(1,3,4,2)$$\zeta_{4}$
$1$$4$$(1,2,4,3)$$-\zeta_{4}$
The blue line marks the conjugacy class containing complex conjugation.