Properties

 Label 35.193...736.126.a.a Dimension $35$ Group $S_7$ Conductor $1.935\times 10^{60}$ Root number $1$ Indicator $1$

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Basic invariants

 Dimension: $35$ Group: $S_7$ Conductor: $$193\!\cdots\!736$$$$\medspace = 2^{102} \cdot 3^{62}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 7.1.161243136.1 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 7.1.161243136.1

Defining polynomial

 $f(x)$ $=$ $$x^{7} - 2x^{6} + 6x^{5} - 10x^{4} + 14x^{3} - 12x^{2} + 4x + 4$$ x^7 - 2*x^6 + 6*x^5 - 10*x^4 + 14*x^3 - 12*x^2 + 4*x + 4 .

The roots of $f$ are computed in an extension of $\Q_{ 389 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 389 }$: $$x^{2} + 379x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$46 + 318\cdot 389 + 188\cdot 389^{2} + 303\cdot 389^{3} + 325\cdot 389^{4} +O(389^{5})$$ 46 + 318*389 + 188*389^2 + 303*389^3 + 325*389^4+O(389^5) $r_{ 2 }$ $=$ $$226 + 81\cdot 389 + 2\cdot 389^{2} + 134\cdot 389^{3} + 3\cdot 389^{4} +O(389^{5})$$ 226 + 81*389 + 2*389^2 + 134*389^3 + 3*389^4+O(389^5) $r_{ 3 }$ $=$ $$78 a + 124 + \left(217 a + 258\right)\cdot 389 + \left(350 a + 367\right)\cdot 389^{2} + \left(324 a + 32\right)\cdot 389^{3} + \left(237 a + 251\right)\cdot 389^{4} +O(389^{5})$$ 78*a + 124 + (217*a + 258)*389 + (350*a + 367)*389^2 + (324*a + 32)*389^3 + (237*a + 251)*389^4+O(389^5) $r_{ 4 }$ $=$ $$311 a + 126 + \left(171 a + 18\right)\cdot 389 + \left(38 a + 155\right)\cdot 389^{2} + \left(64 a + 208\right)\cdot 389^{3} + \left(151 a + 359\right)\cdot 389^{4} +O(389^{5})$$ 311*a + 126 + (171*a + 18)*389 + (38*a + 155)*389^2 + (64*a + 208)*389^3 + (151*a + 359)*389^4+O(389^5) $r_{ 5 }$ $=$ $$339 a + 174 + \left(340 a + 227\right)\cdot 389 + \left(254 a + 363\right)\cdot 389^{2} + \left(49 a + 186\right)\cdot 389^{3} + \left(334 a + 140\right)\cdot 389^{4} +O(389^{5})$$ 339*a + 174 + (340*a + 227)*389 + (254*a + 363)*389^2 + (49*a + 186)*389^3 + (334*a + 140)*389^4+O(389^5) $r_{ 6 }$ $=$ $$21 + 78\cdot 389 + 241\cdot 389^{2} + 261\cdot 389^{3} + 155\cdot 389^{4} +O(389^{5})$$ 21 + 78*389 + 241*389^2 + 261*389^3 + 155*389^4+O(389^5) $r_{ 7 }$ $=$ $$50 a + 63 + \left(48 a + 185\right)\cdot 389 + \left(134 a + 237\right)\cdot 389^{2} + \left(339 a + 39\right)\cdot 389^{3} + \left(54 a + 320\right)\cdot 389^{4} +O(389^{5})$$ 50*a + 63 + (48*a + 185)*389 + (134*a + 237)*389^2 + (339*a + 39)*389^3 + (54*a + 320)*389^4+O(389^5)

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

 Size Order Action 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.