Properties

Label 14.207...296.15t47.a.a
Dimension $14$
Group $A_7$
Conductor $2.071\times 10^{27}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $14$
Group: $A_7$
Conductor: \(207\!\cdots\!296\)\(\medspace = 2^{12} \cdot 7^{6} \cdot 73^{10} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.3.16711744.1
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 7.3.16711744.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 227 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 227 }$: \( x^{2} + 220x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 128 + 168\cdot 227 + 137\cdot 227^{2} + 129\cdot 227^{3} + 178\cdot 227^{4} +O(227^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 55 + 149\cdot 227 + 170\cdot 227^{2} + 56\cdot 227^{3} + 186\cdot 227^{4} +O(227^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 15 + 157\cdot 227 + 55\cdot 227^{2} + 95\cdot 227^{3} + 2\cdot 227^{4} +O(227^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 77 a + 157 + \left(121 a + 116\right)\cdot 227 + \left(167 a + 170\right)\cdot 227^{2} + \left(3 a + 69\right)\cdot 227^{3} + \left(33 a + 29\right)\cdot 227^{4} +O(227^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 150 a + 15 + \left(105 a + 208\right)\cdot 227 + \left(59 a + 86\right)\cdot 227^{2} + \left(223 a + 155\right)\cdot 227^{3} + \left(193 a + 29\right)\cdot 227^{4} +O(227^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 190 a + 59 + \left(137 a + 7\right)\cdot 227 + \left(131 a + 92\right)\cdot 227^{2} + \left(194 a + 39\right)\cdot 227^{3} + \left(133 a + 210\right)\cdot 227^{4} +O(227^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 37 a + 27 + \left(89 a + 101\right)\cdot 227 + \left(95 a + 194\right)\cdot 227^{2} + \left(32 a + 134\right)\cdot 227^{3} + \left(93 a + 44\right)\cdot 227^{4} +O(227^{5})\) 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$$1$$()$$14$
$105$$2$$(1,2)(3,4)$$2$
$70$$3$$(1,2,3)$$-1$
$280$$3$$(1,2,3)(4,5,6)$$2$
$630$$4$$(1,2,3,4)(5,6)$$0$
$504$$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.