Properties

Label 2.401.5t2.a.a
Dimension $2$
Group $D_{5}$
Conductor $401$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $2$
Group: $D_{5}$
Conductor: \(401\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.5.160801.1
Galois orbit size: $2$
Smallest permutation container: $D_{5}$
Parity: even
Determinant: 1.401.2t1.a.a
Projective image: $D_5$
Projective stem field: Galois closure of 5.5.160801.1

Defining polynomial

$f(x)$$=$ \( x^{5} - x^{4} - 5x^{3} + 4x^{2} + 3x - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{2} + 12x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 5 a + 12 + \left(6 a + 6\right)\cdot 13 + \left(4 a + 11\right)\cdot 13^{2} + \left(11 a + 5\right)\cdot 13^{3} + \left(a + 3\right)\cdot 13^{4} +O(13^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 11 a + 4 + \left(3 a + 3\right)\cdot 13 + \left(12 a + 5\right)\cdot 13^{2} + 5\cdot 13^{3} + \left(6 a + 11\right)\cdot 13^{4} +O(13^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 2 a + 2 + \left(9 a + 9\right)\cdot 13 + \left(12 a + 7\right)\cdot 13^{3} + \left(6 a + 3\right)\cdot 13^{4} +O(13^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 5 + 11\cdot 13 + 11\cdot 13^{2} + 7\cdot 13^{3} +O(13^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 8 a + 4 + \left(6 a + 8\right)\cdot 13 + \left(8 a + 9\right)\cdot 13^{2} + \left(a + 12\right)\cdot 13^{3} + \left(11 a + 6\right)\cdot 13^{4} +O(13^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.