Properties

Label 2.71.7t2.a.a
Dimension $2$
Group $D_{7}$
Conductor $71$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{7}$
Conductor: \(71\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 7.1.357911.1
Galois orbit size: $3$
Smallest permutation container: $D_{7}$
Parity: odd
Determinant: 1.71.2t1.a.a
Projective image: $D_7$
Projective stem field: 7.1.357911.1

Defining polynomial

$f(x)$$=$\(x^{7} - x^{6} - x^{5} + x^{4} - x^{3} - x^{2} + 2 x + 1\)  Toggle raw display.

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.