Properties

Label 21.132...249.84.a.a
Dimension $21$
Group $S_7$
Conductor $1.330\times 10^{42}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $21$
Group: $S_7$
Conductor: \(132\!\cdots\!249\)\(\medspace = 23^{10} \cdot 709^{10}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.5.11561663.1
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 7.5.11561663.1

Defining polynomial

$f(x)$$=$ \( x^{7} - x^{6} - 3x^{5} + 5x^{4} + 2x^{3} - 8x^{2} - 2x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 311 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 311 }$: \( x^{2} + 310x + 17 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 247 a + 33 + \left(16 a + 221\right)\cdot 311 + \left(23 a + 182\right)\cdot 311^{2} + \left(118 a + 249\right)\cdot 311^{3} + \left(206 a + 242\right)\cdot 311^{4} +O(311^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 27 + 256\cdot 311 + 198\cdot 311^{2} + 92\cdot 311^{3} + 86\cdot 311^{4} +O(311^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 105 + 264\cdot 311 + 16\cdot 311^{2} + 213\cdot 311^{3} + 152\cdot 311^{4} +O(311^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 5 + 126\cdot 311 + 126\cdot 311^{2} + 251\cdot 311^{3} + 32\cdot 311^{4} +O(311^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 259 a + 268 + \left(294 a + 174\right)\cdot 311 + \left(22 a + 245\right)\cdot 311^{2} + \left(90 a + 12\right)\cdot 311^{3} + \left(173 a + 2\right)\cdot 311^{4} +O(311^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 64 a + 280 + \left(294 a + 301\right)\cdot 311 + \left(287 a + 188\right)\cdot 311^{2} + \left(192 a + 33\right)\cdot 311^{3} + \left(104 a + 20\right)\cdot 311^{4} +O(311^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 52 a + 216 + \left(16 a + 210\right)\cdot 311 + \left(288 a + 284\right)\cdot 311^{2} + \left(220 a + 79\right)\cdot 311^{3} + \left(137 a + 85\right)\cdot 311^{4} +O(311^{5})\) 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.