Properties

Label 2.1859.7t2.a.a
Dimension $2$
Group $D_{7}$
Conductor $1859$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{7}$
Conductor: \(1859\)\(\medspace = 11 \cdot 13^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.6424482779.1
Galois orbit size: $3$
Smallest permutation container: $D_{7}$
Parity: odd
Determinant: 1.11.2t1.a.a
Projective image: $D_7$
Projective stem field: Galois closure of 7.1.6424482779.1

Defining polynomial

$f(x)$$=$ \( x^{7} - 3x^{6} + 2x^{5} - 7x^{4} + 42x^{3} - 3x^{2} - 267x + 344 \) Copy content 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 $\mathbb{Q}_{ 17 }$: \( x^{2} + 16x + 3 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.