Properties

Label 2.3639.5t2.a.a
Dimension $2$
Group $D_{5}$
Conductor $3639$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{5}$
Conductor: \(3639\)\(\medspace = 3 \cdot 1213 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 5.1.13242321.1
Galois orbit size: $2$
Smallest permutation container: $D_{5}$
Parity: odd
Determinant: 1.3639.2t1.a.a
Projective image: $D_5$
Projective stem field: 5.1.13242321.1

Defining polynomial

$f(x)$$=$\(x^{5} - x^{4} + x^{3} + 12 x^{2} + 21 x + 9\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \(x^{2} + 7 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 5 a + 3 + \left(6 a + 9\right)\cdot 11 + \left(8 a + 10\right)\cdot 11^{2} + 7\cdot 11^{3} + \left(4 a + 4\right)\cdot 11^{4} + \left(8 a + 5\right)\cdot 11^{5} +O(11^{6})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 8 + 3\cdot 11^{2} + 2\cdot 11^{3} + 8\cdot 11^{4} + 8\cdot 11^{5} +O(11^{6})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 3 a + 5 + \left(9 a + 1\right)\cdot 11 + \left(4 a + 7\right)\cdot 11^{2} + \left(2 a + 7\right)\cdot 11^{3} + \left(4 a + 6\right)\cdot 11^{5} +O(11^{6})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 6 a + 1 + \left(4 a + 8\right)\cdot 11 + \left(2 a + 5\right)\cdot 11^{2} + \left(10 a + 2\right)\cdot 11^{3} + \left(6 a + 9\right)\cdot 11^{4} + \left(2 a + 1\right)\cdot 11^{5} +O(11^{6})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 8 a + 6 + \left(a + 2\right)\cdot 11 + \left(6 a + 6\right)\cdot 11^{2} + \left(8 a + 1\right)\cdot 11^{3} + \left(10 a + 10\right)\cdot 11^{4} + \left(6 a + 10\right)\cdot 11^{5} +O(11^{6})\)  Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

Cycle notation
$(1,2)(4,5)$
$(1,5)(2,3)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character value
$1$$1$$()$$2$
$5$$2$$(1,5)(2,3)$$0$
$2$$5$$(1,4,5,2,3)$$\zeta_{5}^{3} + \zeta_{5}^{2}$
$2$$5$$(1,5,3,4,2)$$-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$

The blue line marks the conjugacy class containing complex conjugation.