Properties

Label 2.151.7t2.a.b
Dimension $2$
Group $D_{7}$
Conductor $151$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{7}$
Conductor: \(151\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 7.1.3442951.1
Galois orbit size: $3$
Smallest permutation container: $D_{7}$
Parity: odd
Determinant: 1.151.2t1.a.a
Projective image: $D_7$
Projective stem field: 7.1.3442951.1

Defining polynomial

$f(x)$$=$\(x^{7} - x^{6} + x^{5} + 3 x^{3} - x^{2} + 3 x + 1\)  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 $\mathbb{Q}_{ 13 }$: \(x^{2} + 12 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 10 a + 9 + \left(5 a + 6\right)\cdot 13 + \left(2 a + 9\right)\cdot 13^{2} + \left(4 a + 5\right)\cdot 13^{3} + \left(4 a + 11\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 2 a + 12 + \left(3 a + 7\right)\cdot 13 + \left(12 a + 9\right)\cdot 13^{2} + \left(2 a + 6\right)\cdot 13^{3} + \left(11 a + 10\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 4 a + 6 + 2 a\cdot 13 + \left(12 a + 12\right)\cdot 13^{2} + \left(9 a + 11\right)\cdot 13^{3} + \left(8 a + 1\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 9 a + 10 + \left(10 a + 11\right)\cdot 13 + 8\cdot 13^{2} + \left(3 a + 9\right)\cdot 13^{3} + 4 a\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 9 + 10\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 11 a + 1 + \left(9 a + 9\right)\cdot 13 + 5\cdot 13^{2} + \left(10 a + 10\right)\cdot 13^{3} + \left(a + 5\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 3 a + 6 + \left(7 a + 2\right)\cdot 13 + \left(10 a + 6\right)\cdot 13^{2} + \left(8 a + 7\right)\cdot 13^{3} + \left(8 a + 11\right)\cdot 13^{4} +O(13^{5})\)  Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

Cycle notation
$(1,4)(2,5)(3,6)$
$(1,6)(3,5)(4,7)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 7 }$ Character value
$1$$1$$()$$2$
$7$$2$$(1,4)(2,5)(3,6)$$0$
$2$$7$$(1,7,4,6,5,2,3)$$\zeta_{7}^{4} + \zeta_{7}^{3}$
$2$$7$$(1,4,5,3,7,6,2)$$-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$
$2$$7$$(1,6,3,4,2,7,5)$$\zeta_{7}^{5} + \zeta_{7}^{2}$

The blue line marks the conjugacy class containing complex conjugation.