Normalized defining polynomial

\( x^{11} + 2x^{9} - x^{8} - x^{7} + 2x^{6} - 4x^{5} + 2x^{4} - 2x^{3} - 2x^{2} + 2x - 1 \) x^11 + 2*x^9 - x^8 - x^7 + 2*x^6 - 4*x^5 + 2*x^4 - 2*x^3 - 2*x^2 + 2*x - 1

Invariants

Degree:		$11$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[3, 4]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(121321855232\) 121321855232 \(\medspace = 2^{8}\cdot 2927\cdot 161911\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(10.18\)	sage: (K.disc().abs())^(1./K.degree())   gp: abs(K.disc)^(1/poldegree(K.pol))   magma: Abs(Discriminant(OK))^(1/Degree(K));   oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:		$2^{2}2927^{1/2}161911^{1/2}\approx 87078.2174369687$
Ramified primes:		\(2\), \(2927\), \(161911\) 2, 2927, 161911	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		$\Q(\sqrt{473913497}$)
$\card{ \Aut(K/\Q) }$:		$1$	sage: K.automorphisms()   magma: Automorphisms(K);   oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{47}a^{10}-\frac{16}{47}a^{9}+\frac{23}{47}a^{8}+\frac{7}{47}a^{7}-\frac{19}{47}a^{6}-\frac{23}{47}a^{5}-\frac{12}{47}a^{4}+\frac{6}{47}a^{3}-\frac{4}{47}a^{2}+\frac{15}{47}a-\frac{3}{47}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, 1/47*a^10 - 16/47*a^9 + 23/47*a^8 + 7/47*a^7 - 19/47*a^6 - 23/47*a^5 - 12/47*a^4 + 6/47*a^3 - 4/47*a^2 + 15/47*a - 3/47

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$6$	sage: UK.rank()   gp: K.fu   magma: UnitRank(K);   oscar: rank(UK)
Torsion generator:		\( -1 \) -1  (order $2$)	sage: UK.torsion_generator()   gp: K.tu[2]   magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);   oscar: torsion_units_generator(OK)
Fundamental units:		$a^{10}+2a^{8}-a^{7}-a^{6}+2a^{5}-4a^{4}+2a^{3}-2a^{2}-2a+2$, $\frac{49}{47}a^{10}+\frac{15}{47}a^{9}+\frac{93}{47}a^{8}-\frac{33}{47}a^{7}-\frac{85}{47}a^{6}+\frac{48}{47}a^{5}-\frac{165}{47}a^{4}+\frac{59}{47}a^{3}-\frac{55}{47}a^{2}-\frac{64}{47}a+\frac{41}{47}$, $\frac{20}{47}a^{10}+\frac{9}{47}a^{9}+\frac{37}{47}a^{8}-\frac{1}{47}a^{7}-\frac{51}{47}a^{6}+\frac{10}{47}a^{5}-\frac{99}{47}a^{4}-\frac{21}{47}a^{3}+\frac{14}{47}a^{2}-\frac{29}{47}a+\frac{34}{47}$, $\frac{15}{47}a^{10}-\frac{5}{47}a^{9}+\frac{16}{47}a^{8}-\frac{36}{47}a^{7}-\frac{50}{47}a^{6}+\frac{31}{47}a^{5}-\frac{86}{47}a^{4}+\frac{43}{47}a^{3}-\frac{13}{47}a^{2}-\frac{10}{47}a+\frac{96}{47}$, $\frac{33}{47}a^{10}-\frac{11}{47}a^{9}+\frac{54}{47}a^{8}-\frac{51}{47}a^{7}-\frac{63}{47}a^{6}+\frac{87}{47}a^{5}-\frac{161}{47}a^{4}+\frac{104}{47}a^{3}+\frac{9}{47}a^{2}-\frac{69}{47}a+\frac{89}{47}$, $\frac{52}{47}a^{10}+\frac{14}{47}a^{9}+\frac{115}{47}a^{8}-\frac{12}{47}a^{7}-\frac{48}{47}a^{6}+\frac{73}{47}a^{5}-\frac{201}{47}a^{4}+\frac{30}{47}a^{3}-\frac{67}{47}a^{2}-\frac{113}{47}a+\frac{32}{47}$ a^10 + 2*a^8 - a^7 - a^6 + 2*a^5 - 4*a^4 + 2*a^3 - 2*a^2 - 2*a + 2, 49/47*a^10 + 15/47*a^9 + 93/47*a^8 - 33/47*a^7 - 85/47*a^6 + 48/47*a^5 - 165/47*a^4 + 59/47*a^3 - 55/47*a^2 - 64/47*a + 41/47, 20/47*a^10 + 9/47*a^9 + 37/47*a^8 - 1/47*a^7 - 51/47*a^6 + 10/47*a^5 - 99/47*a^4 - 21/47*a^3 + 14/47*a^2 - 29/47*a + 34/47, 15/47*a^10 - 5/47*a^9 + 16/47*a^8 - 36/47*a^7 - 50/47*a^6 + 31/47*a^5 - 86/47*a^4 + 43/47*a^3 - 13/47*a^2 - 10/47*a + 96/47, 33/47*a^10 - 11/47*a^9 + 54/47*a^8 - 51/47*a^7 - 63/47*a^6 + 87/47*a^5 - 161/47*a^4 + 104/47*a^3 + 9/47*a^2 - 69/47*a + 89/47, 52/47*a^10 + 14/47*a^9 + 115/47*a^8 - 12/47*a^7 - 48/47*a^6 + 73/47*a^5 - 201/47*a^4 + 30/47*a^3 - 67/47*a^2 - 113/47*a + 32/47	sage: UK.fundamental_units()   gp: K.fu   magma: [K|fUK(g): g in Generators(UK)];   oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:		\( 18.3096406338 \)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{4}\cdot 18.3096406338 \cdot 1}{2\cdot\sqrt{121321855232}}\cr\approx \mathstrut & 0.327710052648 \end{aligned}\]

Galois group

$S_{11}$ (as 11T8):

Intermediate fields

Sibling fields

Frobenius cycle types

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
\(2\) 2	2.4.8.2	$x^{4} + 2 x^{2} + 4 x + 2$	$4$	$1$	$8$	$C_2^2$	$[2, 3]$
	2.7.0.1	$x^{7} + x + 1$	$1$	$7$	$0$	$C_7$	$[\ ]^{7}$
\(2927\) 2927	$\Q_{2927}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{2927}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{2927}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $4$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
\(161911\) 161911	$\Q_{161911}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $8$	$1$	$8$	$0$	$C_8$	$[\ ]^{8}$