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Normalized defining polynomial

\( x^{11} + 11x^{9} - 216513 \) x^11 + 11*x^9 - 216513

Invariants

Degree:		$11$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[1, 5]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(-162512240934979374477176655\) -162512240934979374477176655 \(\medspace = -\,3^{12}\cdot 5\cdot 11^{19}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(241.44\)	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:		$3^{31/18}5^{1/2}11^{199/110}\approx 1135.4496198838813$
Ramified primes:		\(3\), \(5\), \(11\) 3, 5, 11	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q(\sqrt{-55}) \)
$\card{ \Aut(K/\Q) }$:		$1$	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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{9}a^{4}+\frac{2}{9}a^{2}$, $\frac{1}{27}a^{5}+\frac{2}{27}a^{3}+\frac{1}{3}a$, $\frac{1}{81}a^{6}+\frac{2}{81}a^{4}-\frac{2}{9}a^{2}$, $\frac{1}{243}a^{7}+\frac{2}{243}a^{5}-\frac{2}{27}a^{3}-\frac{1}{3}a$, $\frac{1}{2187}a^{8}+\frac{11}{2187}a^{6}+\frac{1}{81}a^{5}+\frac{11}{81}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{6561}a^{9}+\frac{11}{6561}a^{7}+\frac{1}{243}a^{6}+\frac{11}{243}a^{4}+\frac{1}{9}a^{3}+\frac{2}{9}a$, $\frac{1}{98415}a^{10}-\frac{2}{32805}a^{9}+\frac{2}{98415}a^{8}+\frac{41}{32805}a^{7}-\frac{2}{10935}a^{6}+\frac{14}{3645}a^{5}+\frac{47}{1215}a^{4}-\frac{14}{405}a^{3}+\frac{8}{135}a^{2}+\frac{14}{45}a+\frac{7}{15}$ 1, a, a^2, 1/3*a^3 - 1/3*a, 1/9*a^4 + 2/9*a^2, 1/27*a^5 + 2/27*a^3 + 1/3*a, 1/81*a^6 + 2/81*a^4 - 2/9*a^2, 1/243*a^7 + 2/243*a^5 - 2/27*a^3 - 1/3*a, 1/2187*a^8 + 11/2187*a^6 + 1/81*a^5 + 11/81*a^3 + 1/3*a^2 - 1/3, 1/6561*a^9 + 11/6561*a^7 + 1/243*a^6 + 11/243*a^4 + 1/9*a^3 + 2/9*a, 1/98415*a^10 - 2/32805*a^9 + 2/98415*a^8 + 41/32805*a^7 - 2/10935*a^6 + 14/3645*a^5 + 47/1215*a^4 - 14/405*a^3 + 8/135*a^2 + 14/45*a + 7/15

Monogenic:		No
Index:		Not computed
Inessential primes:		$3$

Class group and class number

$C_{3}$, which has order $3$ (assuming GRH)

Unit group

Rank:		$5$	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:		$\frac{1}{19683}a^{10}+\frac{2}{19683}a^{8}+\frac{1}{729}a^{7}-\frac{11}{2187}a^{6}+\frac{2}{729}a^{5}+\frac{1}{27}a^{4}-\frac{11}{81}a^{3}+\frac{2}{27}a^{2}+a-\frac{14}{3}$, $\frac{166161325075}{6561}a^{10}-\frac{98773576097}{2187}a^{9}-\frac{125606170516}{6561}a^{8}+\frac{494500694048}{2187}a^{7}+\frac{64188571019}{729}a^{6}-\frac{728579225062}{243}a^{5}+\frac{16501580095}{81}a^{4}+36991264597a^{3}-\frac{218858353025}{9}a^{2}-\frac{1381210944052}{3}a+533567788456$, $\frac{18\!\cdots\!08}{98415}a^{10}-\frac{59954792757868}{1215}a^{9}+\frac{31\!\cdots\!91}{98415}a^{8}-\frac{27\!\cdots\!03}{3645}a^{7}+\frac{52\!\cdots\!73}{3645}a^{6}-\frac{63\!\cdots\!33}{3645}a^{5}-\frac{896871395613838}{405}a^{4}+\frac{390327917864194}{15}a^{3}-\frac{16\!\cdots\!96}{135}a^{2}+\frac{67\!\cdots\!49}{15}a-\frac{71\!\cdots\!33}{5}$, $\frac{10311512857948}{6561}a^{10}-\frac{8252865544889}{2187}a^{9}+\frac{163799809525355}{6561}a^{8}-\frac{114797452906183}{2187}a^{7}+\frac{178881696877805}{2187}a^{6}-\frac{4371161822185}{243}a^{5}-\frac{45516154448965}{81}a^{4}+\frac{257049106890149}{81}a^{3}-\frac{113193971169722}{9}a^{2}+\frac{126665492882459}{3}a-\frac{378727237275808}{3}$, $\frac{189880673}{6561}a^{10}-\frac{18637019983}{6561}a^{9}+\frac{19167927313}{6561}a^{8}-\frac{95133454859}{6561}a^{7}+\frac{154740321767}{2187}a^{6}-\frac{3227070785}{9}a^{5}+\frac{293663187961}{243}a^{4}-\frac{190818906721}{81}a^{3}+\frac{38120180860}{9}a^{2}-\frac{184976750099}{9}a+\frac{208216946765}{3}$ 1/19683*a^10 + 2/19683*a^8 + 1/729*a^7 - 11/2187*a^6 + 2/729*a^5 + 1/27*a^4 - 11/81*a^3 + 2/27*a^2 + a - 14/3, 166161325075/6561*a^10 - 98773576097/2187*a^9 - 125606170516/6561*a^8 + 494500694048/2187*a^7 + 64188571019/729*a^6 - 728579225062/243*a^5 + 16501580095/81*a^4 + 36991264597*a^3 - 218858353025/9*a^2 - 1381210944052/3*a + 533567788456, 1879276056552508/98415*a^10 - 59954792757868/1215*a^9 + 31707223438692491/98415*a^8 - 2732395124652803/3645*a^7 + 5266592988586673/3645*a^6 - 6373988185696933/3645*a^5 - 896871395613838/405*a^4 + 390327917864194/15*a^3 - 16525140584309896/135*a^2 + 6751694514813749/15*a - 7199327588318333/5, 10311512857948/6561*a^10 - 8252865544889/2187*a^9 + 163799809525355/6561*a^8 - 114797452906183/2187*a^7 + 178881696877805/2187*a^6 - 4371161822185/243*a^5 - 45516154448965/81*a^4 + 257049106890149/81*a^3 - 113193971169722/9*a^2 + 126665492882459/3*a - 378727237275808/3, 189880673/6561*a^10 - 18637019983/6561*a^9 + 19167927313/6561*a^8 - 95133454859/6561*a^7 + 154740321767/2187*a^6 - 3227070785/9*a^5 + 293663187961/243*a^4 - 190818906721/81*a^3 + 38120180860/9*a^2 - 184976750099/9*a + 208216946765/3 (assuming GRH)	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:		\( 678643197.709 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{5}\cdot 678643197.709 \cdot 3}{2\cdot\sqrt{162512240934979374477176655}}\cr\approx \mathstrut & 1.56393629046 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

 

x = polygen(QQ); K.<a> = NumberField(x^11 + 11*x^9 - 216513)

 

DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()

 

hK = K.class_number(); wK = K.unit_group().torsion_generator().order();

 

2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

 

# self-contained Pari/GP code snippet to compute the analytic class number formula

 

K = bnfinit(x^11 + 11*x^9 - 216513, 1);

 

[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

 

/* self-contained Magma code snippet to compute the analytic class number formula */

 

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^11 + 11*x^9 - 216513);

 

OK := Integers(K); DK := Discriminant(OK);

 

UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);

 

r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);

 

hK := #clK; wK := #TorsionSubgroup(UK);

 

2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

 

# self-contained Oscar code snippet to compute the analytic class number formula

 

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^11 + 11*x^9 - 216513);

 

OK = ring_of_integers(K); DK = discriminant(OK);

 

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

 

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

 

hK = order(clK); wK = torsion_units_order(K);

 

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

 

Galois group

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

A non-solvable group of order 39916800

The 56 conjugacy class representatives for $S_{11}$

Character table for $S_{11}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.11.0.1}{11} }$	R	R	${\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$	R	${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$	${\href{/padicField/17.11.0.1}{11} }$	${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.1.0.1}{1} }$	${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.2.0.1}{2} }$	${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$	${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$	${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.4.0.1}{4} }$	${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$	${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$	${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$	${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.3.0.1}{3} }$	${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(3\) 3	$\Q_{3}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{3}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
	3.3.3.1	$x^{3} + 6 x + 3$	$3$	$1$	$3$	$S_3$	$[3/2]_{2}$
	3.6.9.16	$x^{6} + 3 x^{5} + 3 x^{4} + 12$	$6$	$1$	$9$	$S_3^2$	$[3/2, 2]_{2}^{2}$
\(5\) 5	5.2.1.1	$x^{2} + 5$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	5.2.0.1	$x^{2} + 4 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	5.3.0.1	$x^{3} + 3 x + 3$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	5.4.0.1	$x^{4} + 4 x^{2} + 4 x + 2$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
\(11\) 11	11.11.19.9	$x^{11} + 99 x^{9} + 11$	$11$	$1$	$19$	$F_{11}$	$[19/10]_{10}$

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