Properties

Label 10.965...361.70.a
Dimension $10$
Group $A_7$
Conductor $9.656\times 10^{26}$
Indicator $0$

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Basic invariants

Dimension:$10$
Group:$A_7$
Conductor:\(965\!\cdots\!361\)\(\medspace = 149^{6} \cdot 211^{6} \)
Artin number field: Galois closure of 7.7.988410721.1
Galois orbit size: $2$
Smallest permutation container: 70
Parity: even
Projective image: $A_7$
Projective field: Galois closure of 7.7.988410721.1

Galois action

Roots of defining polynomial

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:
$r_{ 1 }$ $=$ \( 44 + 46\cdot 53 + 14\cdot 53^{2} + 25\cdot 53^{3} + 50\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 5 a + 28 + \left(18 a + 26\right)\cdot 53 + \left(41 a + 19\right)\cdot 53^{2} + \left(29 a + 22\right)\cdot 53^{3} + \left(14 a + 17\right)\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 49 + 42\cdot 53 + 16\cdot 53^{2} + 6\cdot 53^{3} + 22\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 48 a + 48 + \left(34 a + 40\right)\cdot 53 + \left(11 a + 7\right)\cdot 53^{2} + \left(23 a + 47\right)\cdot 53^{3} + \left(38 a + 45\right)\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 16 a + 30 + \left(13 a + 11\right)\cdot 53 + \left(26 a + 41\right)\cdot 53^{2} + \left(33 a + 29\right)\cdot 53^{3} + \left(36 a + 3\right)\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 37 a + 41 + \left(39 a + 48\right)\cdot 53 + \left(26 a + 26\right)\cdot 53^{2} + \left(19 a + 31\right)\cdot 53^{3} + \left(16 a + 10\right)\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 27 + 47\cdot 53 + 31\cdot 53^{2} + 49\cdot 53^{3} + 8\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

Cycle notation
$(3,4,5,6,7)$
$(1,2,3)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 7 }$ Character values
$c1$ $c2$
$1$ $1$ $()$ $10$ $10$
$105$ $2$ $(1,2)(3,4)$ $-2$ $-2$
$70$ $3$ $(1,2,3)$ $1$ $1$
$280$ $3$ $(1,2,3)(4,5,6)$ $1$ $1$
$630$ $4$ $(1,2,3,4)(5,6)$ $0$ $0$
$504$ $5$ $(1,2,3,4,5)$ $0$ $0$
$210$ $6$ $(1,2,3)(4,5)(6,7)$ $1$ $1$
$360$ $7$ $(1,2,3,4,5,6,7)$ $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$
$360$ $7$ $(1,3,4,5,6,7,2)$ $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$
The blue line marks the conjugacy class containing complex conjugation.