Properties

Label 2.3143.7t2.a.b
Dimension $2$
Group $D_{7}$
Conductor $3143$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{7}$
Conductor: \(3143\)\(\medspace = 7 \cdot 449 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 7.1.31047965207.1
Galois orbit size: $3$
Smallest permutation container: $D_{7}$
Parity: odd
Determinant: 1.3143.2t1.a.a
Projective image: $D_7$
Projective stem field: 7.1.31047965207.1

Defining polynomial

$f(x)$$=$\(x^{7} - 3 x^{6} - 4 x^{5} + 30 x^{4} - 24 x^{3} - 70 x^{2} + 112 x - 47\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \(x^{2} + 24 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 11 a + 6 + 10\cdot 29 + 9\cdot 29^{2} + \left(23 a + 7\right)\cdot 29^{3} + \left(13 a + 13\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 19 a + 9 + \left(6 a + 14\right)\cdot 29 + \left(19 a + 22\right)\cdot 29^{2} + \left(28 a + 22\right)\cdot 29^{3} + \left(7 a + 18\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( a + 8 + \left(27 a + 10\right)\cdot 29 + \left(19 a + 20\right)\cdot 29^{2} + \left(27 a + 10\right)\cdot 29^{3} + \left(13 a + 16\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 18 a + 3 + \left(28 a + 1\right)\cdot 29 + \left(28 a + 9\right)\cdot 29^{2} + \left(5 a + 6\right)\cdot 29^{3} + \left(15 a + 1\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 10 a + 17 + \left(22 a + 28\right)\cdot 29 + \left(9 a + 24\right)\cdot 29^{2} + 29^{3} + \left(21 a + 1\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 5 + 23\cdot 29 + 23\cdot 29^{2} + 24\cdot 29^{3} + 6\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 28 a + 13 + \left(a + 28\right)\cdot 29 + \left(9 a + 5\right)\cdot 29^{2} + \left(a + 13\right)\cdot 29^{3} + 15 a\cdot 29^{4} +O(29^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.