Properties

Label 14.152...625.21t38.a.a
Dimension $14$
Group $S_7$
Conductor $1.529\times 10^{22}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $14$
Group: $S_7$
Conductor: \(152\!\cdots\!625\)\(\medspace = 5^{4} \cdot 53^{4} \cdot 1327^{4}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.351655.1
Galois orbit size: $1$
Smallest permutation container: 21T38
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.351655.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 659 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 198 + 23\cdot 659 + 152\cdot 659^{2} + 170\cdot 659^{3} + 92\cdot 659^{4} +O(659^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 307 + 109\cdot 659 + 175\cdot 659^{2} + 260\cdot 659^{3} + 652\cdot 659^{4} +O(659^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 312 + 7\cdot 659 + 100\cdot 659^{2} + 302\cdot 659^{3} + 90\cdot 659^{4} +O(659^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 357 + 397\cdot 659 + 594\cdot 659^{2} + 483\cdot 659^{3} + 535\cdot 659^{4} +O(659^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 457 + 9\cdot 659 + 2\cdot 659^{2} + 251\cdot 659^{3} + 433\cdot 659^{4} +O(659^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 480 + 621\cdot 659 + 17\cdot 659^{2} + 489\cdot 659^{3} + 569\cdot 659^{4} +O(659^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 526 + 148\cdot 659 + 276\cdot 659^{2} + 20\cdot 659^{3} + 262\cdot 659^{4} +O(659^{5})\) 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$$1$$()$$14$
$21$$2$$(1,2)$$6$
$105$$2$$(1,2)(3,4)(5,6)$$2$
$105$$2$$(1,2)(3,4)$$2$
$70$$3$$(1,2,3)$$2$
$280$$3$$(1,2,3)(4,5,6)$$-1$
$210$$4$$(1,2,3,4)$$0$
$630$$4$$(1,2,3,4)(5,6)$$0$
$504$$5$$(1,2,3,4,5)$$-1$
$210$$6$$(1,2,3)(4,5)(6,7)$$2$
$420$$6$$(1,2,3)(4,5)$$0$
$840$$6$$(1,2,3,4,5,6)$$-1$
$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)$$0$

The blue line marks the conjugacy class containing complex conjugation.