Properties

Label 1.35.4t1.a.b
Dimension $1$
Group $C_4$
Conductor $35$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_4$
Conductor: \(35\)\(\medspace = 5 \cdot 7 \)
Artin field: 4.4.6125.1
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: even
Dirichlet character: \(\chi_{35}(13,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{4} - x^{3} - 9 x^{2} + 9 x + 11\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 11 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 6\cdot 11 + 4\cdot 11^{3} + 10\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 1 + 7\cdot 11 + 8\cdot 11^{2} + 2\cdot 11^{3} + 6\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 3 + 7\cdot 11 + 2\cdot 11^{2} + 10\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 8 + 11 + 10\cdot 11^{2} + 3\cdot 11^{3} + 6\cdot 11^{4} +O(11^{5})\)  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.