Properties

Label 35.253...375.70.a.a
Dimension $35$
Group $S_7$
Conductor $2.531\times 10^{78}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $35$
Group: $S_7$
Conductor: \(253\!\cdots\!375\)\(\medspace = 5^{24} \cdot 37^{15} \cdot 347^{15}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.320975.1
Galois orbit size: $1$
Smallest permutation container: 70
Parity: odd
Determinant: 1.12839.2t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.320975.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: \( x^{2} + 49x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 44 a + 35 + \left(25 a + 4\right)\cdot 53 + \left(49 a + 10\right)\cdot 53^{2} + \left(11 a + 34\right)\cdot 53^{3} + 35\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 14 + 21\cdot 53 + 16\cdot 53^{2} + 29\cdot 53^{3} + 35\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 9 a + 52 + \left(27 a + 10\right)\cdot 53 + \left(3 a + 23\right)\cdot 53^{2} + \left(41 a + 32\right)\cdot 53^{3} + \left(52 a + 24\right)\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 52 a + 51 + \left(41 a + 51\right)\cdot 53 + \left(20 a + 39\right)\cdot 53^{2} + \left(31 a + 47\right)\cdot 53^{3} + \left(25 a + 16\right)\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 31 + 2\cdot 53 + 52\cdot 53^{2} + 10\cdot 53^{3} + 26\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( a + 47 + \left(11 a + 8\right)\cdot 53 + \left(32 a + 28\right)\cdot 53^{2} + \left(21 a + 46\right)\cdot 53^{3} + \left(27 a + 34\right)\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 36 + 5\cdot 53 + 42\cdot 53^{2} + 10\cdot 53^{3} + 38\cdot 53^{4} +O(53^{5})\) 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.