Properties

Label 21.109...601.84.a.a
Dimension $21$
Group $S_7$
Conductor $1.097\times 10^{53}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $21$
Group: $S_7$
Conductor: \(109\!\cdots\!601\)\(\medspace = 7^{16} \cdot 8951^{10}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.438599.1
Galois orbit size: $1$
Smallest permutation container: 84
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.438599.1

Defining polynomial

$f(x)$$=$ \( x^{7} - 2x^{6} + 2x^{5} - 2x^{4} + x^{3} + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$: \( x^{2} + 78x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 74 a + 6 + \left(24 a + 68\right)\cdot 79 + \left(20 a + 74\right)\cdot 79^{2} + \left(16 a + 31\right)\cdot 79^{3} + \left(38 a + 59\right)\cdot 79^{4} +O(79^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 27 + 12\cdot 79 + 23\cdot 79^{2} + 18\cdot 79^{3} + 10\cdot 79^{4} +O(79^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 5 a + 1 + \left(54 a + 19\right)\cdot 79 + \left(58 a + 70\right)\cdot 79^{2} + \left(62 a + 27\right)\cdot 79^{3} + \left(40 a + 2\right)\cdot 79^{4} +O(79^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 76 a + 44 + \left(68 a + 31\right)\cdot 79 + \left(41 a + 19\right)\cdot 79^{2} + \left(56 a + 43\right)\cdot 79^{3} + \left(a + 62\right)\cdot 79^{4} +O(79^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 74 + 41\cdot 79 + 30\cdot 79^{2} + 2\cdot 79^{3} + 73\cdot 79^{4} +O(79^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 46 + 39\cdot 79 + 26\cdot 79^{2} + 55\cdot 79^{3} + 21\cdot 79^{4} +O(79^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 3 a + 41 + \left(10 a + 24\right)\cdot 79 + \left(37 a + 71\right)\cdot 79^{2} + \left(22 a + 57\right)\cdot 79^{3} + \left(77 a + 7\right)\cdot 79^{4} +O(79^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 7 }$ Character value
$1$$1$$()$$21$
$21$$2$$(1,2)$$1$
$105$$2$$(1,2)(3,4)(5,6)$$-3$
$105$$2$$(1,2)(3,4)$$1$
$70$$3$$(1,2,3)$$-3$
$280$$3$$(1,2,3)(4,5,6)$$0$
$210$$4$$(1,2,3,4)$$-1$
$630$$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$
$420$$6$$(1,2,3)(4,5)$$1$
$840$$6$$(1,2,3,4,5,6)$$0$
$720$$7$$(1,2,3,4,5,6,7)$$0$
$504$$10$$(1,2,3,4,5)(6,7)$$1$
$420$$12$$(1,2,3,4)(5,6,7)$$-1$

The blue line marks the conjugacy class containing complex conjugation.