Properties

Label 20.115...864.70.a.a
Dimension $20$
Group $S_7$
Conductor $1.150\times 10^{50}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $20$
Group: $S_7$
Conductor: \(115\!\cdots\!864\)\(\medspace = 2^{36} \cdot 8363^{10}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.535232.1
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 7.1.535232.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 337 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 337 }$: \( x^{2} + 332x + 10 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 320 a + 196 + \left(81 a + 78\right)\cdot 337 + \left(102 a + 288\right)\cdot 337^{2} + \left(222 a + 60\right)\cdot 337^{3} + \left(322 a + 212\right)\cdot 337^{4} +O(337^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 114 a + 195 + \left(129 a + 208\right)\cdot 337 + \left(292 a + 305\right)\cdot 337^{2} + \left(144 a + 58\right)\cdot 337^{3} + \left(256 a + 57\right)\cdot 337^{4} +O(337^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 17 a + 111 + \left(255 a + 168\right)\cdot 337 + \left(234 a + 43\right)\cdot 337^{2} + \left(114 a + 59\right)\cdot 337^{3} + \left(14 a + 255\right)\cdot 337^{4} +O(337^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 293 + 63\cdot 337 + 144\cdot 337^{2} + 331\cdot 337^{3} + 317\cdot 337^{4} +O(337^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 223 a + 91 + \left(207 a + 67\right)\cdot 337 + \left(44 a + 290\right)\cdot 337^{2} + \left(192 a + 153\right)\cdot 337^{3} + \left(80 a + 183\right)\cdot 337^{4} +O(337^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 208 a + 217 + \left(294 a + 253\right)\cdot 337 + \left(115 a + 332\right)\cdot 337^{2} + \left(43 a + 122\right)\cdot 337^{3} + \left(248 a + 236\right)\cdot 337^{4} +O(337^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 129 a + 246 + \left(42 a + 170\right)\cdot 337 + \left(221 a + 280\right)\cdot 337^{2} + \left(293 a + 223\right)\cdot 337^{3} + \left(88 a + 85\right)\cdot 337^{4} +O(337^{5})\) 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.