Properties

Label 2.3743.7t2.a.a
Dimension $2$
Group $D_{7}$
Conductor $3743$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{7}$
Conductor: \(3743\)\(\medspace = 19 \cdot 197 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 7.1.52439613407.1
Galois orbit size: $3$
Smallest permutation container: $D_{7}$
Parity: odd
Determinant: 1.3743.2t1.a.a
Projective image: $D_7$
Projective stem field: 7.1.52439613407.1

Defining polynomial

$f(x)$$=$\(x^{7} - 2 x^{6} + 11 x^{5} - 26 x^{4} + 45 x^{3} - 60 x^{2} + 39 x + 11\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \(x^{2} + 16 x + 3\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 3 a + \left(9 a + 4\right)\cdot 17 + \left(6 a + 8\right)\cdot 17^{2} + \left(2 a + 16\right)\cdot 17^{3} + \left(12 a + 8\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 14 a + 3 + \left(7 a + 10\right)\cdot 17 + \left(10 a + 5\right)\cdot 17^{2} + \left(14 a + 12\right)\cdot 17^{3} + \left(4 a + 1\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 4 a + 12 + \left(9 a + 13\right)\cdot 17 + \left(7 a + 8\right)\cdot 17^{2} + \left(7 a + 15\right)\cdot 17^{3} + \left(2 a + 14\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 14 + 11\cdot 17 + 3\cdot 17^{2} + 6\cdot 17^{3} + 11\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 13 a + 16 + \left(7 a + 1\right)\cdot 17 + \left(9 a + 7\right)\cdot 17^{2} + \left(9 a + 15\right)\cdot 17^{3} + \left(14 a + 9\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 8 a + \left(8 a + 13\right)\cdot 17 + \left(14 a + 5\right)\cdot 17^{2} + \left(14 a + 9\right)\cdot 17^{3} + \left(14 a + 10\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 9 a + 8 + \left(8 a + 13\right)\cdot 17 + \left(2 a + 11\right)\cdot 17^{2} + \left(2 a + 9\right)\cdot 17^{3} + \left(2 a + 10\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.