Properties

Label 1.49.7t1.a.b
Dimension $1$
Group $C_7$
Conductor $49$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_7$
Conductor: \(49\)\(\medspace = 7^{2}\)
Artin field: 7.7.13841287201.1
Galois orbit size: $6$
Smallest permutation container: $C_7$
Parity: even
Dirichlet character: \(\chi_{49}(36,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{7} - 21 x^{5} - 21 x^{4} + 91 x^{3} + 112 x^{2} - 84 x - 97\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \(x^{7} + 4 x + 9\)  Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.