Properties

Label 3.97969.7t3.a.b
Dimension $3$
Group $C_7:C_3$
Conductor $97969$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $3$
Group: $C_7:C_3$
Conductor: \(97969\)\(\medspace = 313^{2} \)
Artin stem field: Galois closure of 7.7.9597924961.1
Galois orbit size: $2$
Smallest permutation container: $C_7:C_3$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $C_7:C_3$
Projective stem field: Galois closure of 7.7.9597924961.1

Defining polynomial

$f(x)$$=$ \( x^{7} - x^{6} - 15x^{5} + 20x^{4} + 33x^{3} - 22x^{2} - 32x - 8 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{3} + 2x + 9 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.