Properties

Label 21.118...104.42t299.a.a
Dimension $21$
Group $A_7$
Conductor $1.186\times 10^{43}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $21$
Group: $A_7$
Conductor: \(118\!\cdots\!104\)\(\medspace = 2^{18} \cdot 3^{44} \cdot 11^{16}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.3.50808384.1
Galois orbit size: $1$
Smallest permutation container: $A_7$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_7$
Projective stem field: Galois closure of 7.3.50808384.1

Defining polynomial

$f(x)$$=$ \( x^{7} - 2x^{6} + 2x + 2 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: \( x^{2} + 102x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 21 a + 28 + \left(22 a + 5\right)\cdot 103 + \left(42 a + 60\right)\cdot 103^{2} + \left(13 a + 62\right)\cdot 103^{3} + \left(17 a + 70\right)\cdot 103^{4} +O(103^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 67 + 7\cdot 103 + 16\cdot 103^{2} + 44\cdot 103^{3} + 46\cdot 103^{4} +O(103^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 84 + 21\cdot 103 + 32\cdot 103^{2} + 43\cdot 103^{3} + 59\cdot 103^{4} +O(103^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 82 a + 49 + \left(80 a + 6\right)\cdot 103 + \left(60 a + 80\right)\cdot 103^{2} + \left(89 a + 33\right)\cdot 103^{3} + \left(85 a + 74\right)\cdot 103^{4} +O(103^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 66 + 36\cdot 103 + 56\cdot 103^{2} + 11\cdot 103^{3} + 73\cdot 103^{4} +O(103^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 77 a + 73 + \left(68 a + 16\right)\cdot 103 + \left(42 a + 45\right)\cdot 103^{2} + \left(20 a + 16\right)\cdot 103^{3} + \left(58 a + 25\right)\cdot 103^{4} +O(103^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 26 a + 47 + \left(34 a + 8\right)\cdot 103 + \left(60 a + 19\right)\cdot 103^{2} + \left(82 a + 97\right)\cdot 103^{3} + \left(44 a + 62\right)\cdot 103^{4} +O(103^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 7 }$ Character value
$1$$1$$()$$21$
$105$$2$$(1,2)(3,4)$$1$
$70$$3$$(1,2,3)$$-3$
$280$$3$$(1,2,3)(4,5,6)$$0$
$630$$4$$(1,2,3,4)(5,6)$$-1$
$504$$5$$(1,2,3,4,5)$$1$
$210$$6$$(1,2,3)(4,5)(6,7)$$1$
$360$$7$$(1,2,3,4,5,6,7)$$0$
$360$$7$$(1,3,4,5,6,7,2)$$0$

The blue line marks the conjugacy class containing complex conjugation.