# Properties

 Label 45.199...000.110.a.a Dimension $45$ Group $M_{11}$ Conductor $1.992\times 10^{111}$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $45$ Group: $M_{11}$ Conductor: $$199\!\cdots\!000$$$$\medspace = 2^{102} \cdot 3^{40} \cdot 5^{88}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 11.3.6561000000000000000000.1 Galois orbit size: $1$ Smallest permutation container: 110 Parity: even Determinant: 1.1.1t1.a.a Projective image: $M_{11}$ Projective stem field: Galois closure of 11.3.6561000000000000000000.1

## Defining polynomial

 $f(x)$ $=$ $$x^{11} - 2 x^{10} - 5 x^{9} - 50 x^{8} + 70 x^{7} + 232 x^{6} + 796 x^{5} - 1400 x^{4} - 5075 x^{3} - 10950 x^{2} + 2805 x + 90$$ x^11 - 2*x^10 - 5*x^9 - 50*x^8 + 70*x^7 + 232*x^6 + 796*x^5 - 1400*x^4 - 5075*x^3 - 10950*x^2 + 2805*x + 90 .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $$x^{5} + 4x + 11$$

Roots:
 $r_{ 1 }$ $=$ $$7 a^{4} + 3 a^{3} + 9 a^{2} + 6 a + 6 + \left(a^{4} + 6 a^{3} + 10 a^{2} + 6 a + 7\right)\cdot 13 + \left(11 a^{4} + 7 a^{3} + 2 a^{2} + 11 a + 10\right)\cdot 13^{2} + \left(6 a^{4} + 10 a^{3} + 12 a^{2} + 8 a + 12\right)\cdot 13^{3} + \left(4 a^{4} + 10 a^{3} + 8 a^{2} + 5\right)\cdot 13^{4} + \left(2 a^{4} + 6 a^{3} + a^{2} + 7 a + 12\right)\cdot 13^{5} + \left(11 a^{4} + 11 a^{3} + 9 a^{2} + 4\right)\cdot 13^{6} + \left(2 a^{4} + a^{3} + 9 a^{2} + 7 a + 8\right)\cdot 13^{7} + \left(12 a^{4} + 2 a^{3} + 6 a^{2} + a + 8\right)\cdot 13^{8} + \left(2 a^{4} + 2 a^{3} + 3 a^{2} + 10 a + 2\right)\cdot 13^{9} +O(13^{10})$$ 7*a^4 + 3*a^3 + 9*a^2 + 6*a + 6 + (a^4 + 6*a^3 + 10*a^2 + 6*a + 7)*13 + (11*a^4 + 7*a^3 + 2*a^2 + 11*a + 10)*13^2 + (6*a^4 + 10*a^3 + 12*a^2 + 8*a + 12)*13^3 + (4*a^4 + 10*a^3 + 8*a^2 + 5)*13^4 + (2*a^4 + 6*a^3 + a^2 + 7*a + 12)*13^5 + (11*a^4 + 11*a^3 + 9*a^2 + 4)*13^6 + (2*a^4 + a^3 + 9*a^2 + 7*a + 8)*13^7 + (12*a^4 + 2*a^3 + 6*a^2 + a + 8)*13^8 + (2*a^4 + 2*a^3 + 3*a^2 + 10*a + 2)*13^9+O(13^10) $r_{ 2 }$ $=$ $$a^{4} + 2 a^{3} + 9 a^{2} + 4 a + 3 + \left(5 a^{4} + 3 a^{3} + 7 a^{2} + 6 a + 8\right)\cdot 13 + \left(6 a^{4} + 8 a^{3} + 9 a^{2} + a\right)\cdot 13^{2} + \left(3 a^{4} + 10 a^{3} + 12 a^{2} + 11 a + 10\right)\cdot 13^{3} + \left(10 a^{4} + 3 a^{3} + 2\right)\cdot 13^{4} + \left(11 a^{3} + 11 a^{2} + 5 a + 9\right)\cdot 13^{5} + \left(8 a^{4} + 4 a^{3} + a^{2} + 2 a + 2\right)\cdot 13^{6} + \left(5 a^{4} + 7 a^{3} + a + 5\right)\cdot 13^{7} + \left(12 a^{4} + 11 a^{3} + a^{2} + 4 a + 6\right)\cdot 13^{8} + \left(10 a^{4} + a^{3} + 6 a^{2} + 2 a\right)\cdot 13^{9} +O(13^{10})$$ a^4 + 2*a^3 + 9*a^2 + 4*a + 3 + (5*a^4 + 3*a^3 + 7*a^2 + 6*a + 8)*13 + (6*a^4 + 8*a^3 + 9*a^2 + a)*13^2 + (3*a^4 + 10*a^3 + 12*a^2 + 11*a + 10)*13^3 + (10*a^4 + 3*a^3 + 2)*13^4 + (11*a^3 + 11*a^2 + 5*a + 9)*13^5 + (8*a^4 + 4*a^3 + a^2 + 2*a + 2)*13^6 + (5*a^4 + 7*a^3 + a + 5)*13^7 + (12*a^4 + 11*a^3 + a^2 + 4*a + 6)*13^8 + (10*a^4 + a^3 + 6*a^2 + 2*a)*13^9+O(13^10) $r_{ 3 }$ $=$ $$3 a^{4} + 6 a^{3} + 5 a^{2} + 12 a + 1 + \left(12 a^{4} + 6 a^{3} + 6 a^{2} + 11 a\right)\cdot 13 + \left(11 a^{4} + 10 a^{3} + 9 a^{2} + 8 a + 8\right)\cdot 13^{2} + \left(6 a^{4} + 9 a^{3} + 7 a^{2} + 11 a + 2\right)\cdot 13^{3} + \left(12 a^{4} + a^{3} + 6 a^{2} + 3 a + 8\right)\cdot 13^{4} + \left(9 a^{4} + a^{3} + 4 a^{2} + 7 a + 5\right)\cdot 13^{5} + \left(4 a^{4} + 6 a^{3} + 10 a + 10\right)\cdot 13^{6} + \left(8 a^{4} + 9 a^{3} + 5 a^{2} + 9 a + 7\right)\cdot 13^{7} + \left(4 a^{4} + 8 a^{3} + 3 a^{2} + 2 a + 2\right)\cdot 13^{8} + \left(7 a^{4} + a^{3} + 7 a^{2} + 10 a + 1\right)\cdot 13^{9} +O(13^{10})$$ 3*a^4 + 6*a^3 + 5*a^2 + 12*a + 1 + (12*a^4 + 6*a^3 + 6*a^2 + 11*a)*13 + (11*a^4 + 10*a^3 + 9*a^2 + 8*a + 8)*13^2 + (6*a^4 + 9*a^3 + 7*a^2 + 11*a + 2)*13^3 + (12*a^4 + a^3 + 6*a^2 + 3*a + 8)*13^4 + (9*a^4 + a^3 + 4*a^2 + 7*a + 5)*13^5 + (4*a^4 + 6*a^3 + 10*a + 10)*13^6 + (8*a^4 + 9*a^3 + 5*a^2 + 9*a + 7)*13^7 + (4*a^4 + 8*a^3 + 3*a^2 + 2*a + 2)*13^8 + (7*a^4 + a^3 + 7*a^2 + 10*a + 1)*13^9+O(13^10) $r_{ 4 }$ $=$ $$a^{4} + 8 a^{3} + 4 a^{2} + 7 a + 3 + \left(8 a^{4} + 7 a^{3} + 4 a^{2} + 12 a + 2\right)\cdot 13 + \left(4 a^{4} + 8 a^{3} + 7 a^{2} + 8 a\right)\cdot 13^{2} + \left(2 a^{4} + 4 a^{3} + 3 a + 9\right)\cdot 13^{3} + \left(11 a^{4} + 10 a^{3} + 10 a^{2} + 4 a + 5\right)\cdot 13^{4} + \left(2 a^{4} + 6 a^{3} + 10 a + 5\right)\cdot 13^{5} + \left(6 a^{4} + 6 a^{3} + 7 a + 12\right)\cdot 13^{6} + \left(7 a^{4} + 10 a^{3} + 10 a^{2} + 8 a + 5\right)\cdot 13^{7} + \left(a^{4} + 5 a^{3} + 8 a^{2} + 2 a + 5\right)\cdot 13^{8} + \left(8 a^{4} + 12 a^{3} + 10 a^{2} + 11 a + 4\right)\cdot 13^{9} +O(13^{10})$$ a^4 + 8*a^3 + 4*a^2 + 7*a + 3 + (8*a^4 + 7*a^3 + 4*a^2 + 12*a + 2)*13 + (4*a^4 + 8*a^3 + 7*a^2 + 8*a)*13^2 + (2*a^4 + 4*a^3 + 3*a + 9)*13^3 + (11*a^4 + 10*a^3 + 10*a^2 + 4*a + 5)*13^4 + (2*a^4 + 6*a^3 + 10*a + 5)*13^5 + (6*a^4 + 6*a^3 + 7*a + 12)*13^6 + (7*a^4 + 10*a^3 + 10*a^2 + 8*a + 5)*13^7 + (a^4 + 5*a^3 + 8*a^2 + 2*a + 5)*13^8 + (8*a^4 + 12*a^3 + 10*a^2 + 11*a + 4)*13^9+O(13^10) $r_{ 5 }$ $=$ $$4 a^{3} + 3 a^{2} + 4 a + 7 + \left(5 a^{4} + 3 a^{3} + 2 a^{2} + 5\right)\cdot 13 + \left(7 a^{4} + 9 a^{3} + 4 a + 6\right)\cdot 13^{2} + \left(9 a^{4} + 7 a^{2} + 12 a + 8\right)\cdot 13^{3} + \left(5 a^{4} + 4 a^{2} + 2 a + 4\right)\cdot 13^{4} + \left(5 a^{4} + 5 a^{3} + 6 a^{2} + a + 9\right)\cdot 13^{5} + \left(7 a^{3} + 8 a^{2} + 12 a + 1\right)\cdot 13^{6} + \left(9 a^{4} + a^{3} + 3 a^{2} + 7 a + 2\right)\cdot 13^{7} + \left(5 a^{4} + 12 a^{3} + 8 a^{2} + 6 a + 11\right)\cdot 13^{8} + \left(11 a^{4} + 10 a^{3} + 9 a^{2} + 8 a + 3\right)\cdot 13^{9} +O(13^{10})$$ 4*a^3 + 3*a^2 + 4*a + 7 + (5*a^4 + 3*a^3 + 2*a^2 + 5)*13 + (7*a^4 + 9*a^3 + 4*a + 6)*13^2 + (9*a^4 + 7*a^2 + 12*a + 8)*13^3 + (5*a^4 + 4*a^2 + 2*a + 4)*13^4 + (5*a^4 + 5*a^3 + 6*a^2 + a + 9)*13^5 + (7*a^3 + 8*a^2 + 12*a + 1)*13^6 + (9*a^4 + a^3 + 3*a^2 + 7*a + 2)*13^7 + (5*a^4 + 12*a^3 + 8*a^2 + 6*a + 11)*13^8 + (11*a^4 + 10*a^3 + 9*a^2 + 8*a + 3)*13^9+O(13^10) $r_{ 6 }$ $=$ $$6 a^{4} + a^{3} + 8 a^{2} + 6 a + 8 + \left(9 a^{4} + 5 a^{3} + 6 a^{2} + 5 a + 1\right)\cdot 13 + \left(9 a^{4} + 6 a^{3} + 6 a^{2} + 12 a + 6\right)\cdot 13^{2} + \left(9 a^{4} + 3 a^{3} + 11 a^{2} + 10 a + 6\right)\cdot 13^{3} + \left(5 a^{4} + 9 a^{3} + 12\right)\cdot 13^{4} + \left(8 a^{4} + 12 a^{3} + 9 a^{2} + 2 a + 5\right)\cdot 13^{5} + \left(9 a^{4} + 6 a^{3} + 5 a^{2} + 7 a + 10\right)\cdot 13^{6} + \left(12 a^{4} + 2 a^{3} + 3 a^{2} + 5 a + 8\right)\cdot 13^{7} + \left(6 a^{4} + 12 a^{3} + 8 a^{2} + a + 12\right)\cdot 13^{8} + \left(8 a^{3} + 5 a^{2} + 9 a + 7\right)\cdot 13^{9} +O(13^{10})$$ 6*a^4 + a^3 + 8*a^2 + 6*a + 8 + (9*a^4 + 5*a^3 + 6*a^2 + 5*a + 1)*13 + (9*a^4 + 6*a^3 + 6*a^2 + 12*a + 6)*13^2 + (9*a^4 + 3*a^3 + 11*a^2 + 10*a + 6)*13^3 + (5*a^4 + 9*a^3 + 12)*13^4 + (8*a^4 + 12*a^3 + 9*a^2 + 2*a + 5)*13^5 + (9*a^4 + 6*a^3 + 5*a^2 + 7*a + 10)*13^6 + (12*a^4 + 2*a^3 + 3*a^2 + 5*a + 8)*13^7 + (6*a^4 + 12*a^3 + 8*a^2 + a + 12)*13^8 + (8*a^3 + 5*a^2 + 9*a + 7)*13^9+O(13^10) $r_{ 7 }$ $=$ $$7 + 13 + 3\cdot 13^{2} + 12\cdot 13^{3} + 10\cdot 13^{4} + 6\cdot 13^{5} + 9\cdot 13^{6} + 2\cdot 13^{7} + 7\cdot 13^{8} + 11\cdot 13^{9} +O(13^{10})$$ 7 + 13 + 3*13^2 + 12*13^3 + 10*13^4 + 6*13^5 + 9*13^6 + 2*13^7 + 7*13^8 + 11*13^9+O(13^10) $r_{ 8 }$ $=$ $$10 a^{4} + 12 a^{3} + a^{2} + 11 a + \left(10 a^{4} + 4 a^{3} + a + 11\right)\cdot 13 + \left(11 a^{4} + 5 a^{3} + 7 a^{2} + 2 a + 12\right)\cdot 13^{2} + \left(5 a^{4} + a^{3} + 8 a + 1\right)\cdot 13^{3} + \left(10 a^{4} + 4 a^{3} + 5 a^{2} + 4 a + 4\right)\cdot 13^{4} + \left(12 a^{4} + 4 a^{2} + 8 a + 4\right)\cdot 13^{5} + \left(12 a^{4} + 7 a^{3} + 2 a^{2} + 8 a\right)\cdot 13^{6} + \left(5 a^{4} + 10 a^{3} + 4 a^{2} + 8 a + 8\right)\cdot 13^{7} + \left(9 a^{4} + 3 a^{3} + 12 a^{2} + 7\right)\cdot 13^{8} + \left(3 a^{4} + 2 a^{3} + 12 a^{2} + a + 2\right)\cdot 13^{9} +O(13^{10})$$ 10*a^4 + 12*a^3 + a^2 + 11*a + (10*a^4 + 4*a^3 + a + 11)*13 + (11*a^4 + 5*a^3 + 7*a^2 + 2*a + 12)*13^2 + (5*a^4 + a^3 + 8*a + 1)*13^3 + (10*a^4 + 4*a^3 + 5*a^2 + 4*a + 4)*13^4 + (12*a^4 + 4*a^2 + 8*a + 4)*13^5 + (12*a^4 + 7*a^3 + 2*a^2 + 8*a)*13^6 + (5*a^4 + 10*a^3 + 4*a^2 + 8*a + 8)*13^7 + (9*a^4 + 3*a^3 + 12*a^2 + 7)*13^8 + (3*a^4 + 2*a^3 + 12*a^2 + a + 2)*13^9+O(13^10) $r_{ 9 }$ $=$ $$3 a^{4} + 6 a^{3} + 2 a^{2} + a + 12 + \left(12 a^{4} + 7 a^{3} + 12 a^{2} + 4 a + 12\right)\cdot 13 + \left(2 a^{4} + 3 a^{3} + 9 a^{2} + 12 a + 12\right)\cdot 13^{2} + \left(4 a^{4} + 6 a^{3} + 4 a^{2} + 2 a + 1\right)\cdot 13^{3} + \left(3 a^{4} + 7 a^{3} + 5 a^{2} + 11 a + 9\right)\cdot 13^{4} + \left(10 a^{4} + 2 a^{3} + 6 a^{2} + 7 a + 5\right)\cdot 13^{5} + \left(4 a^{4} + 5 a^{3} + a^{2} + 8 a + 10\right)\cdot 13^{6} + \left(2 a^{4} + 6 a^{3} + 2 a^{2} + 8 a + 7\right)\cdot 13^{7} + \left(2 a^{4} + 3 a^{3} + 3 a^{2} + 10 a + 12\right)\cdot 13^{8} + \left(11 a^{4} + 6 a^{3} + 10 a^{2} + 8\right)\cdot 13^{9} +O(13^{10})$$ 3*a^4 + 6*a^3 + 2*a^2 + a + 12 + (12*a^4 + 7*a^3 + 12*a^2 + 4*a + 12)*13 + (2*a^4 + 3*a^3 + 9*a^2 + 12*a + 12)*13^2 + (4*a^4 + 6*a^3 + 4*a^2 + 2*a + 1)*13^3 + (3*a^4 + 7*a^3 + 5*a^2 + 11*a + 9)*13^4 + (10*a^4 + 2*a^3 + 6*a^2 + 7*a + 5)*13^5 + (4*a^4 + 5*a^3 + a^2 + 8*a + 10)*13^6 + (2*a^4 + 6*a^3 + 2*a^2 + 8*a + 7)*13^7 + (2*a^4 + 3*a^3 + 3*a^2 + 10*a + 12)*13^8 + (11*a^4 + 6*a^3 + 10*a^2 + 8)*13^9+O(13^10) $r_{ 10 }$ $=$ $$2 a^{4} + 9 a^{3} + a^{2} + 4 a + 1 + \left(11 a^{4} + 2 a^{3} + 9 a^{2} + 11 a + 12\right)\cdot 13 + \left(10 a^{4} + 8 a^{3} + 3 a^{2} + 5 a + 1\right)\cdot 13^{2} + \left(12 a^{4} + 2 a^{2} + 5 a + 6\right)\cdot 13^{3} + \left(8 a^{4} + 5 a^{3} + 10 a + 6\right)\cdot 13^{4} + \left(2 a^{4} + 5 a^{3} + 12 a^{2} + 2 a + 7\right)\cdot 13^{5} + \left(11 a^{4} + 5 a^{3} + 8 a^{2} + 5 a + 7\right)\cdot 13^{6} + \left(a^{4} + 11 a^{2} + 11 a + 3\right)\cdot 13^{7} + \left(6 a^{4} + 7 a^{3} + 10 a^{2} + a + 12\right)\cdot 13^{8} + \left(12 a^{4} + 4 a^{2} + a + 7\right)\cdot 13^{9} +O(13^{10})$$ 2*a^4 + 9*a^3 + a^2 + 4*a + 1 + (11*a^4 + 2*a^3 + 9*a^2 + 11*a + 12)*13 + (10*a^4 + 8*a^3 + 3*a^2 + 5*a + 1)*13^2 + (12*a^4 + 2*a^2 + 5*a + 6)*13^3 + (8*a^4 + 5*a^3 + 10*a + 6)*13^4 + (2*a^4 + 5*a^3 + 12*a^2 + 2*a + 7)*13^5 + (11*a^4 + 5*a^3 + 8*a^2 + 5*a + 7)*13^6 + (a^4 + 11*a^2 + 11*a + 3)*13^7 + (6*a^4 + 7*a^3 + 10*a^2 + a + 12)*13^8 + (12*a^4 + 4*a^2 + a + 7)*13^9+O(13^10) $r_{ 11 }$ $=$ $$6 a^{4} + a^{3} + 10 a^{2} + 10 a + 6 + \left(2 a^{4} + 5 a^{3} + 5 a^{2} + 4 a + 2\right)\cdot 13 + \left(a^{4} + 10 a^{3} + 8 a^{2} + 10 a + 2\right)\cdot 13^{2} + \left(3 a^{4} + 3 a^{3} + 5 a^{2} + 2 a + 6\right)\cdot 13^{3} + \left(5 a^{4} + 12 a^{3} + 9 a^{2} + 12 a + 7\right)\cdot 13^{4} + \left(9 a^{4} + 12 a^{3} + 8 a^{2} + 12 a + 5\right)\cdot 13^{5} + \left(8 a^{4} + 3 a^{3} + a + 7\right)\cdot 13^{6} + \left(8 a^{4} + a^{3} + 2 a^{2} + 9 a + 4\right)\cdot 13^{7} + \left(3 a^{4} + 11 a^{3} + 2 a^{2} + 6 a + 4\right)\cdot 13^{8} + \left(9 a^{4} + 4 a^{3} + 7 a^{2} + 10 a\right)\cdot 13^{9} +O(13^{10})$$ 6*a^4 + a^3 + 10*a^2 + 10*a + 6 + (2*a^4 + 5*a^3 + 5*a^2 + 4*a + 2)*13 + (a^4 + 10*a^3 + 8*a^2 + 10*a + 2)*13^2 + (3*a^4 + 3*a^3 + 5*a^2 + 2*a + 6)*13^3 + (5*a^4 + 12*a^3 + 9*a^2 + 12*a + 7)*13^4 + (9*a^4 + 12*a^3 + 8*a^2 + 12*a + 5)*13^5 + (8*a^4 + 3*a^3 + a + 7)*13^6 + (8*a^4 + a^3 + 2*a^2 + 9*a + 4)*13^7 + (3*a^4 + 11*a^3 + 2*a^2 + 6*a + 4)*13^8 + (9*a^4 + 4*a^3 + 7*a^2 + 10*a)*13^9+O(13^10)

## Generators of the action on the roots $r_1, \ldots, r_{ 11 }$

 Cycle notation $(2,3)(4,5)(6,7)(10,11)$ $(1,3,8,4)(5,9,11,7)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 11 }$ Character value $1$ $1$ $()$ $45$ $165$ $2$ $(3,8)(4,6)(5,10)(7,11)$ $-3$ $440$ $3$ $(2,9,10)(3,6,8)(4,11,7)$ $0$ $990$ $4$ $(3,7,8,11)(4,5,6,10)$ $1$ $1584$ $5$ $(1,3,6,2,11)(4,10,5,8,7)$ $0$ $1320$ $6$ $(1,5,3)(2,8,10,4,6,11)(7,9)$ $0$ $990$ $8$ $(2,9)(3,10,7,4,8,5,11,6)$ $-1$ $990$ $8$ $(2,9)(3,5,7,6,8,10,11,4)$ $-1$ $720$ $11$ $(1,6,8,4,3,10,2,5,7,11,9)$ $1$ $720$ $11$ $(1,8,3,2,7,9,6,4,10,5,11)$ $1$

The blue line marks the conjugacy class containing complex conjugation.