Properties

Label 1.37.3t1.a.b
Dimension $1$
Group $C_3$
Conductor $37$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_3$
Conductor: \(37\)
Artin field: Galois closure of 3.3.1369.1
Galois orbit size: $2$
Smallest permutation container: $C_3$
Parity: even
Dirichlet character: \(\chi_{37}(26,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{3} - x^{2} - 12x - 11 \) Copy content Toggle raw display .

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

Roots:
$r_{ 1 }$ $=$ \( 10\cdot 11 + 10\cdot 11^{2} + 9\cdot 11^{3} + 9\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 4 + 2\cdot 11 + 11^{2} + 3\cdot 11^{3} + 5\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 8 + 9\cdot 11 + 9\cdot 11^{2} + 8\cdot 11^{3} + 6\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display

Generators of the action on the roots $ r_{ 1 }, r_{ 2 }, r_{ 3 } $

Cycle notation
$(1,2,3)$

Character values on conjugacy classes

SizeOrderAction on $ r_{ 1 }, r_{ 2 }, r_{ 3 } $ Character value
$1$$1$$()$$1$
$1$$3$$(1,2,3)$$-\zeta_{3} - 1$
$1$$3$$(1,3,2)$$\zeta_{3}$

The blue line marks the conjugacy class containing complex conjugation.