Properties

Label 2.751.5t2.a.a
Dimension $2$
Group $D_{5}$
Conductor $751$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{5}$
Conductor: \(751\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 5.1.564001.1
Galois orbit size: $2$
Smallest permutation container: $D_{5}$
Parity: odd
Determinant: 1.751.2t1.a.a
Projective image: $D_5$
Projective stem field: 5.1.564001.1

Defining polynomial

$f(x)$$=$\(x^{5} - x^{3} - 7 x^{2} - x - 3\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: \(x^{2} + 6 x + 3\)  Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.