Properties

Label 20.100...769.70.a.a
Dimension $20$
Group $S_7$
Conductor $1.004\times 10^{53}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $20$
Group: $S_7$
Conductor: \(100\!\cdots\!769\)\(\medspace = 13^{12} \cdot 29^{10} \cdot 317^{10}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.3.1553617.1
Galois orbit size: $1$
Smallest permutation container: 70
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.3.1553617.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 449 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 449 }$: \( x^{2} + 444x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 7 + 54\cdot 449 + 279\cdot 449^{2} + 85\cdot 449^{3} + 421\cdot 449^{4} +O(449^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 365 a + 163 + \left(215 a + 222\right)\cdot 449 + \left(357 a + 216\right)\cdot 449^{2} + \left(56 a + 221\right)\cdot 449^{3} + \left(97 a + 297\right)\cdot 449^{4} +O(449^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 313 a + 267 + \left(60 a + 291\right)\cdot 449 + \left(246 a + 11\right)\cdot 449^{2} + \left(418 a + 88\right)\cdot 449^{3} + \left(73 a + 298\right)\cdot 449^{4} +O(449^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 136 a + 36 + \left(388 a + 282\right)\cdot 449 + \left(202 a + 283\right)\cdot 449^{2} + \left(30 a + 138\right)\cdot 449^{3} + \left(375 a + 249\right)\cdot 449^{4} +O(449^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 84 a + 192 + \left(233 a + 38\right)\cdot 449 + \left(91 a + 441\right)\cdot 449^{2} + \left(392 a + 147\right)\cdot 449^{3} + \left(351 a + 277\right)\cdot 449^{4} +O(449^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 378 + 196\cdot 449 + 202\cdot 449^{2} + 289\cdot 449^{3} + 198\cdot 449^{4} +O(449^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 305 + 261\cdot 449 + 361\cdot 449^{2} + 375\cdot 449^{3} + 53\cdot 449^{4} +O(449^{5})\) 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$$1$$()$$20$
$21$$2$$(1,2)$$0$
$105$$2$$(1,2)(3,4)(5,6)$$0$
$105$$2$$(1,2)(3,4)$$-4$
$70$$3$$(1,2,3)$$2$
$280$$3$$(1,2,3)(4,5,6)$$2$
$210$$4$$(1,2,3,4)$$0$
$630$$4$$(1,2,3,4)(5,6)$$0$
$504$$5$$(1,2,3,4,5)$$0$
$210$$6$$(1,2,3)(4,5)(6,7)$$2$
$420$$6$$(1,2,3)(4,5)$$0$
$840$$6$$(1,2,3,4,5,6)$$0$
$720$$7$$(1,2,3,4,5,6,7)$$-1$
$504$$10$$(1,2,3,4,5)(6,7)$$0$
$420$$12$$(1,2,3,4)(5,6,7)$$0$

The blue line marks the conjugacy class containing complex conjugation.