Properties

Label 6.5_53_1327.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 5 \cdot 53 \cdot 1327 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$351655= 5 \cdot 53 \cdot 1327 $
Artin number field: Splitting field of $f= x^{7} - x^{6} - 2 x^{4} + x^{3} + x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.5_53_1327.2t1.1c1

Galois action

Roots of defining polynomial

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\left(659^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 307 + 109\cdot 659 + 175\cdot 659^{2} + 260\cdot 659^{3} + 652\cdot 659^{4} +O\left(659^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 312 + 7\cdot 659 + 100\cdot 659^{2} + 302\cdot 659^{3} + 90\cdot 659^{4} +O\left(659^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 357 + 397\cdot 659 + 594\cdot 659^{2} + 483\cdot 659^{3} + 535\cdot 659^{4} +O\left(659^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 457 + 9\cdot 659 + 2\cdot 659^{2} + 251\cdot 659^{3} + 433\cdot 659^{4} +O\left(659^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 480 + 621\cdot 659 + 17\cdot 659^{2} + 489\cdot 659^{3} + 569\cdot 659^{4} +O\left(659^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 526 + 148\cdot 659 + 276\cdot 659^{2} + 20\cdot 659^{3} + 262\cdot 659^{4} +O\left(659^{ 5 }\right)$

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$$1$$()$$6$
$21$$2$$(1,2)$$4$
$105$$2$$(1,2)(3,4)(5,6)$$0$
$105$$2$$(1,2)(3,4)$$2$
$70$$3$$(1,2,3)$$3$
$280$$3$$(1,2,3)(4,5,6)$$0$
$210$$4$$(1,2,3,4)$$2$
$630$$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$
$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)$$-1$
$504$$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.