Properties

Label 1.5_7.4t1.1c2
Dimension 1
Group $C_4$
Conductor $ 5 \cdot 7 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$35= 5 \cdot 7 $
Artin number field: Splitting field of $f= x^{4} - x^{3} - 9 x^{2} + 9 x + 11 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Even
Corresponding Dirichlet character: \(\chi_{35}(13,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 11 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 6\cdot 11 + 4\cdot 11^{3} + 10\cdot 11^{4} +O\left(11^{ 5 }\right) \\ r_{ 2 } &= 1 + 7\cdot 11 + 8\cdot 11^{2} + 2\cdot 11^{3} + 6\cdot 11^{4} +O\left(11^{ 5 }\right) \\ r_{ 3 } &= 3 + 7\cdot 11 + 2\cdot 11^{2} + 10\cdot 11^{4} +O\left(11^{ 5 }\right) \\ r_{ 4 } &= 8 + 11 + 10\cdot 11^{2} + 3\cdot 11^{3} + 6\cdot 11^{4} +O\left(11^{ 5 }\right) \\ \end{aligned}\]

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.