Properties

Label 35.257...144.126.a.a
Dimension $35$
Group $S_7$
Conductor $2.570\times 10^{65}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $35$
Group: $S_7$
Conductor: \(257\!\cdots\!144\)\(\medspace = 2^{100} \cdot 3^{74}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.60466176.1
Galois orbit size: $1$
Smallest permutation container: 126
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.60466176.1

Defining polynomial

$f(x)$$=$ \( x^{7} - 2x^{6} + 3x^{5} + x^{4} - 5x^{3} + 12x^{2} - 7x + 5 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{3} + x + 14 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 3 + 8\cdot 17 + 14\cdot 17^{2} + 17^{3} + 11\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 16 a^{2} + 15 + \left(14 a^{2} + 11 a + 7\right)\cdot 17 + \left(3 a^{2} + 15 a + 16\right)\cdot 17^{2} + \left(2 a^{2} + 2\right)\cdot 17^{3} + \left(14 a^{2} + 16 a + 5\right)\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 12 a^{2} + 7 a + 9 + \left(3 a^{2} + 10 a + 7\right)\cdot 17 + \left(15 a^{2} + 7 a + 8\right)\cdot 17^{2} + \left(4 a^{2} + 12 a + 12\right)\cdot 17^{3} + \left(16 a^{2} + 15 a + 5\right)\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 10 a^{2} + 11 a + 2 + \left(15 a^{2} + 7 a + 4\right)\cdot 17 + \left(2 a^{2} + 3 a\right)\cdot 17^{2} + \left(13 a^{2} + 15 a + 1\right)\cdot 17^{3} + \left(11 a^{2} + 14\right)\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 8 a^{2} + 7 a + 4 + \left(15 a^{2} + 11 a + 8\right)\cdot 17 + \left(7 a^{2} + 2 a + 13\right)\cdot 17^{2} + \left(7 a^{2} + 16 a\right)\cdot 17^{3} + \left(14 a^{2} + 11\right)\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 10 a^{2} + 10 a + 11 + \left(3 a^{2} + 11 a + 11\right)\cdot 17 + \left(5 a^{2} + 15 a + 11\right)\cdot 17^{2} + \left(7 a^{2} + 16 a\right)\cdot 17^{3} + \left(5 a^{2} + 16 a + 5\right)\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 12 a^{2} + 16 a + 9 + \left(14 a^{2} + 15 a + 3\right)\cdot 17 + \left(15 a^{2} + 5 a + 3\right)\cdot 17^{2} + \left(15 a^{2} + 6 a + 14\right)\cdot 17^{3} + \left(5 a^{2} + 15\right)\cdot 17^{4} +O(17^{5})\) 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$$1$$()$$35$
$21$$2$$(1,2)$$-5$
$105$$2$$(1,2)(3,4)(5,6)$$-1$
$105$$2$$(1,2)(3,4)$$-1$
$70$$3$$(1,2,3)$$-1$
$280$$3$$(1,2,3)(4,5,6)$$-1$
$210$$4$$(1,2,3,4)$$1$
$630$$4$$(1,2,3,4)(5,6)$$1$
$504$$5$$(1,2,3,4,5)$$0$
$210$$6$$(1,2,3)(4,5)(6,7)$$-1$
$420$$6$$(1,2,3)(4,5)$$1$
$840$$6$$(1,2,3,4,5,6)$$-1$
$720$$7$$(1,2,3,4,5,6,7)$$0$
$504$$10$$(1,2,3,4,5)(6,7)$$0$
$420$$12$$(1,2,3,4)(5,6,7)$$1$

The blue line marks the conjugacy class containing complex conjugation.