LMFDB - Number field 22.8.15646172003756569249283257910156250000000000.4

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 19*x^20 - 395*x^18 + 615*x^16 + 20850*x^14 + 16842*x^12 - 223773*x^10 - 144525*x^8 + 284565*x^6 + 250165*x^4 + 38861*x^2 + 1)

gp: K = bnfinit(y^22 - 19*y^20 - 395*y^18 + 615*y^16 + 20850*y^14 + 16842*y^12 - 223773*y^10 - 144525*y^8 + 284565*y^6 + 250165*y^4 + 38861*y^2 + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 19*x^20 - 395*x^18 + 615*x^16 + 20850*x^14 + 16842*x^12 - 223773*x^10 - 144525*x^8 + 284565*x^6 + 250165*x^4 + 38861*x^2 + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 19*x^20 - 395*x^18 + 615*x^16 + 20850*x^14 + 16842*x^12 - 223773*x^10 - 144525*x^8 + 284565*x^6 + 250165*x^4 + 38861*x^2 + 1)

$ x^{22} - 19 x^{20} - 395 x^{18} + 615 x^{16} + 20850 x^{14} + 16842 x^{12} - 223773 x^{10} - 144525 x^{8} + \cdots + 1 $ x^22 - 19*x^20 - 395*x^18 + 615*x^16 + 20850*x^14 + 16842*x^12 - 223773*x^10 - 144525*x^8 + 284565*x^6 + 250165*x^4 + 38861*x^2 + 1

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$22$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[8, 7]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-15646172003756569249283257910156250000000000$ -15646172003756569249283257910156250000000000 $\medspace = -\,2^{10}\cdot 3^{20}\cdot 5^{20}\cdot 11^{16}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$91.91$	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:	$2^{31/16}3^{10/11}5^{10/11}11^{4/5}\approx 305.86771353088227$
Ramified primes:	$2$, $3$, $5$, $11$ 2, 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{-1}) $
$\card{ \Aut(K/\Q) }$:	$2$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}$, $\frac{1}{6}a^{11}-\frac{1}{6}a^{10}-\frac{1}{6}a^{9}-\frac{1}{6}a^{8}-\frac{1}{3}a^{7}+\frac{1}{6}a^{5}-\frac{1}{6}a^{4}-\frac{1}{6}a^{3}-\frac{1}{6}a^{2}+\frac{1}{6}a-\frac{1}{2}$, $\frac{1}{6}a^{12}-\frac{1}{6}a^{8}+\frac{1}{6}a^{6}+\frac{1}{3}a^{2}-\frac{1}{2}$, $\frac{1}{6}a^{13}-\frac{1}{6}a^{9}+\frac{1}{6}a^{7}+\frac{1}{3}a^{3}-\frac{1}{2}a$, $\frac{1}{6}a^{14}-\frac{1}{6}a^{10}-\frac{1}{6}a^{8}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{6}a^{2}-\frac{1}{3}$, $\frac{1}{6}a^{15}-\frac{1}{6}a^{10}-\frac{1}{6}a^{8}-\frac{1}{3}a^{7}-\frac{1}{2}a^{5}-\frac{1}{6}a^{4}+\frac{1}{3}a^{3}-\frac{1}{6}a^{2}+\frac{1}{6}a-\frac{1}{2}$, $\frac{1}{18}a^{16}-\frac{1}{18}a^{14}+\frac{1}{18}a^{12}+\frac{1}{9}a^{10}+\frac{1}{18}a^{8}-\frac{2}{9}a^{6}+\frac{1}{18}a^{4}-\frac{7}{18}a^{2}+\frac{2}{9}$, $\frac{1}{36}a^{17}-\frac{1}{36}a^{16}+\frac{1}{18}a^{15}+\frac{1}{36}a^{14}-\frac{1}{18}a^{13}+\frac{1}{18}a^{12}-\frac{1}{36}a^{11}-\frac{1}{18}a^{10}-\frac{5}{36}a^{9}+\frac{1}{18}a^{8}-\frac{1}{36}a^{7}+\frac{13}{36}a^{6}-\frac{11}{36}a^{5}-\frac{1}{36}a^{4}-\frac{4}{9}a^{3}+\frac{1}{36}a^{2}+\frac{1}{36}a-\frac{7}{36}$, $\frac{1}{36}a^{18}-\frac{1}{36}a^{16}-\frac{1}{12}a^{15}+\frac{1}{36}a^{14}-\frac{1}{36}a^{12}-\frac{1}{12}a^{11}-\frac{5}{36}a^{10}-\frac{1}{12}a^{9}+\frac{5}{36}a^{8}-\frac{1}{3}a^{7}+\frac{5}{18}a^{6}+\frac{1}{6}a^{5}-\frac{13}{36}a^{4}+\frac{1}{4}a^{3}-\frac{7}{18}a^{2}-\frac{1}{3}a+\frac{1}{12}$, $\frac{1}{36}a^{19}-\frac{1}{12}a^{15}-\frac{1}{12}a^{14}-\frac{1}{12}a^{13}-\frac{1}{12}a^{12}+\frac{1}{12}a^{10}-\frac{1}{6}a^{9}+\frac{1}{4}a^{7}-\frac{1}{12}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{5}{12}a^{2}+\frac{1}{9}a-\frac{1}{4}$, $\frac{1}{39\!\cdots\!88}a^{20}+\frac{58\!\cdots\!21}{66\!\cdots\!98}a^{18}+\frac{33\!\cdots\!11}{13\!\cdots\!96}a^{16}-\frac{1}{12}a^{15}-\frac{14\!\cdots\!47}{44\!\cdots\!32}a^{14}-\frac{1}{12}a^{13}-\frac{53\!\cdots\!26}{11\!\cdots\!83}a^{12}-\frac{1}{12}a^{11}-\frac{36\!\cdots\!67}{22\!\cdots\!66}a^{10}-\frac{1}{6}a^{9}-\frac{11\!\cdots\!31}{13\!\cdots\!96}a^{8}-\frac{1}{12}a^{7}+\frac{39\!\cdots\!97}{11\!\cdots\!83}a^{6}+\frac{1}{6}a^{5}+\frac{20\!\cdots\!61}{66\!\cdots\!98}a^{4}+\frac{5}{12}a^{3}+\frac{45\!\cdots\!95}{99\!\cdots\!47}a^{2}+\frac{1}{4}a+\frac{13\!\cdots\!85}{33\!\cdots\!49}$, $\frac{1}{39\!\cdots\!88}a^{21}+\frac{58\!\cdots\!21}{66\!\cdots\!98}a^{19}-\frac{48\!\cdots\!75}{19\!\cdots\!94}a^{17}+\frac{30\!\cdots\!09}{39\!\cdots\!88}a^{15}+\frac{14\!\cdots\!15}{19\!\cdots\!94}a^{13}-\frac{1}{12}a^{12}+\frac{12\!\cdots\!75}{39\!\cdots\!88}a^{11}-\frac{1}{6}a^{10}-\frac{11\!\cdots\!19}{99\!\cdots\!47}a^{9}+\frac{1}{12}a^{8}+\frac{15\!\cdots\!71}{39\!\cdots\!88}a^{7}-\frac{5}{12}a^{6}+\frac{11\!\cdots\!83}{39\!\cdots\!88}a^{5}-\frac{1}{6}a^{4}+\frac{16\!\cdots\!01}{22\!\cdots\!66}a^{3}+\frac{1}{3}a^{2}-\frac{10\!\cdots\!55}{39\!\cdots\!88}a+\frac{5}{12}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, 1/3*a^8 + 1/3*a^6 + 1/3*a^2 + 1/3, 1/3*a^9 + 1/3*a^7 + 1/3*a^3 + 1/3*a, 1/3*a^10 - 1/3*a^6 + 1/3*a^4 - 1/3, 1/6*a^11 - 1/6*a^10 - 1/6*a^9 - 1/6*a^8 - 1/3*a^7 + 1/6*a^5 - 1/6*a^4 - 1/6*a^3 - 1/6*a^2 + 1/6*a - 1/2, 1/6*a^12 - 1/6*a^8 + 1/6*a^6 + 1/3*a^2 - 1/2, 1/6*a^13 - 1/6*a^9 + 1/6*a^7 + 1/3*a^3 - 1/2*a, 1/6*a^14 - 1/6*a^10 - 1/6*a^8 - 1/3*a^6 + 1/3*a^4 + 1/6*a^2 - 1/3, 1/6*a^15 - 1/6*a^10 - 1/6*a^8 - 1/3*a^7 - 1/2*a^5 - 1/6*a^4 + 1/3*a^3 - 1/6*a^2 + 1/6*a - 1/2, 1/18*a^16 - 1/18*a^14 + 1/18*a^12 + 1/9*a^10 + 1/18*a^8 - 2/9*a^6 + 1/18*a^4 - 7/18*a^2 + 2/9, 1/36*a^17 - 1/36*a^16 + 1/18*a^15 + 1/36*a^14 - 1/18*a^13 + 1/18*a^12 - 1/36*a^11 - 1/18*a^10 - 5/36*a^9 + 1/18*a^8 - 1/36*a^7 + 13/36*a^6 - 11/36*a^5 - 1/36*a^4 - 4/9*a^3 + 1/36*a^2 + 1/36*a - 7/36, 1/36*a^18 - 1/36*a^16 - 1/12*a^15 + 1/36*a^14 - 1/36*a^12 - 1/12*a^11 - 5/36*a^10 - 1/12*a^9 + 5/36*a^8 - 1/3*a^7 + 5/18*a^6 + 1/6*a^5 - 13/36*a^4 + 1/4*a^3 - 7/18*a^2 - 1/3*a + 1/12, 1/36*a^19 - 1/12*a^15 - 1/12*a^14 - 1/12*a^13 - 1/12*a^12 + 1/12*a^10 - 1/6*a^9 + 1/4*a^7 - 1/12*a^6 + 1/3*a^4 - 1/3*a^3 - 5/12*a^2 + 1/9*a - 1/4, 1/396625346636612525076016188*a^20 + 581153297873064590545021/66104224439435420846002698*a^18 + 3347938143559797428425211/132208448878870841692005396*a^16 - 1/12*a^15 - 1478303668483191633431847/44069482959623613897335132*a^14 - 1/12*a^13 - 533710856483595262696226/11017370739905903474333783*a^12 - 1/12*a^11 - 3601005789801083525861367/22034741479811806948667566*a^10 - 1/6*a^9 - 11208100826806731052172231/132208448878870841692005396*a^8 - 1/12*a^7 + 392336600510148328349897/11017370739905903474333783*a^6 + 1/6*a^5 + 20885859266492827408363961/66104224439435420846002698*a^4 + 5/12*a^3 + 45910709258694785135516395/99156336659153131269004047*a^2 + 1/4*a + 13954420542070994669944685/33052112219717710423001349, 1/396625346636612525076016188*a^21 + 581153297873064590545021/66104224439435420846002698*a^19 - 486778154613255594529075/198312673318306262538008094*a^17 + 30764749943274889196448509/396625346636612525076016188*a^15 + 1410575323201188745801715/198312673318306262538008094*a^13 - 1/12*a^12 + 12303490962921820854831875/396625346636612525076016188*a^11 - 1/6*a^10 - 11160418305081524157712619/99156336659153131269004047*a^9 + 1/12*a^8 + 157349937237142084986935471/396625346636612525076016188*a^7 - 5/12*a^6 + 114297784859051060975849983/396625346636612525076016188*a^5 - 1/6*a^4 + 1633313704227582883410701/22034741479811806948667566*a^3 + 1/3*a^2 - 107981221992795650819008355/396625346636612525076016188*a + 5/12

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

Rank:	$14$	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:	$a$, $\frac{20\!\cdots\!17}{19\!\cdots\!94}a^{21}-\frac{12\!\cdots\!79}{66\!\cdots\!98}a^{19}-\frac{80\!\cdots\!55}{19\!\cdots\!94}a^{17}+\frac{66\!\cdots\!68}{99\!\cdots\!47}a^{15}+\frac{42\!\cdots\!29}{19\!\cdots\!94}a^{13}+\frac{15\!\cdots\!49}{99\!\cdots\!47}a^{11}-\frac{45\!\cdots\!77}{19\!\cdots\!94}a^{9}-\frac{24\!\cdots\!59}{19\!\cdots\!94}a^{7}+\frac{29\!\cdots\!24}{99\!\cdots\!47}a^{5}+\frac{22\!\cdots\!89}{99\!\cdots\!47}a^{3}+\frac{24\!\cdots\!00}{99\!\cdots\!47}a$, $\frac{19\!\cdots\!85}{99\!\cdots\!47}a^{21}-\frac{37\!\cdots\!70}{99\!\cdots\!47}a^{19}-\frac{77\!\cdots\!65}{99\!\cdots\!47}a^{17}+\frac{24\!\cdots\!21}{19\!\cdots\!94}a^{15}+\frac{41\!\cdots\!75}{99\!\cdots\!47}a^{13}+\frac{66\!\cdots\!43}{19\!\cdots\!94}a^{11}-\frac{88\!\cdots\!23}{19\!\cdots\!94}a^{9}-\frac{28\!\cdots\!69}{99\!\cdots\!47}a^{7}+\frac{55\!\cdots\!88}{99\!\cdots\!47}a^{5}+\frac{98\!\cdots\!03}{19\!\cdots\!94}a^{3}+\frac{84\!\cdots\!55}{11\!\cdots\!83}a$, $\frac{13\!\cdots\!75}{19\!\cdots\!94}a^{21}-\frac{44\!\cdots\!11}{33\!\cdots\!49}a^{19}-\frac{55\!\cdots\!11}{19\!\cdots\!94}a^{17}+\frac{41\!\cdots\!25}{99\!\cdots\!47}a^{15}+\frac{14\!\cdots\!15}{99\!\cdots\!47}a^{13}+\frac{24\!\cdots\!91}{19\!\cdots\!94}a^{11}-\frac{15\!\cdots\!39}{99\!\cdots\!47}a^{9}-\frac{10\!\cdots\!92}{99\!\cdots\!47}a^{7}+\frac{19\!\cdots\!40}{99\!\cdots\!47}a^{5}+\frac{40\!\cdots\!25}{22\!\cdots\!66}a^{3}+\frac{32\!\cdots\!03}{99\!\cdots\!47}a$, $\frac{26\!\cdots\!13}{19\!\cdots\!94}a^{20}-\frac{48\!\cdots\!19}{19\!\cdots\!94}a^{18}-\frac{10\!\cdots\!57}{19\!\cdots\!94}a^{16}+\frac{10\!\cdots\!65}{19\!\cdots\!94}a^{14}+\frac{55\!\cdots\!89}{19\!\cdots\!94}a^{12}+\frac{74\!\cdots\!55}{19\!\cdots\!94}a^{10}-\frac{29\!\cdots\!27}{99\!\cdots\!47}a^{8}-\frac{71\!\cdots\!43}{19\!\cdots\!94}a^{6}+\frac{41\!\cdots\!55}{99\!\cdots\!47}a^{4}+\frac{11\!\cdots\!71}{22\!\cdots\!66}a^{2}+\frac{82\!\cdots\!80}{99\!\cdots\!47}$, $\frac{16\!\cdots\!61}{99\!\cdots\!47}a^{21}-\frac{63\!\cdots\!07}{19\!\cdots\!94}a^{19}-\frac{65\!\cdots\!10}{99\!\cdots\!47}a^{17}+\frac{20\!\cdots\!11}{19\!\cdots\!94}a^{15}+\frac{34\!\cdots\!22}{99\!\cdots\!47}a^{13}+\frac{27\!\cdots\!26}{99\!\cdots\!47}a^{11}-\frac{74\!\cdots\!97}{19\!\cdots\!94}a^{9}-\frac{23\!\cdots\!99}{99\!\cdots\!47}a^{7}+\frac{47\!\cdots\!12}{99\!\cdots\!47}a^{5}+\frac{40\!\cdots\!75}{99\!\cdots\!47}a^{3}+\frac{12\!\cdots\!61}{19\!\cdots\!94}a$, $\frac{99\!\cdots\!73}{19\!\cdots\!94}a^{20}-\frac{18\!\cdots\!57}{19\!\cdots\!94}a^{18}-\frac{39\!\cdots\!03}{19\!\cdots\!94}a^{16}+\frac{56\!\cdots\!11}{19\!\cdots\!94}a^{14}+\frac{10\!\cdots\!96}{99\!\cdots\!47}a^{12}+\frac{19\!\cdots\!39}{19\!\cdots\!94}a^{10}-\frac{21\!\cdots\!45}{19\!\cdots\!94}a^{8}-\frac{85\!\cdots\!38}{99\!\cdots\!47}a^{6}+\frac{10\!\cdots\!60}{99\!\cdots\!47}a^{4}+\frac{30\!\cdots\!87}{19\!\cdots\!94}a^{2}+\frac{16\!\cdots\!47}{66\!\cdots\!98}$, $\frac{21\!\cdots\!95}{19\!\cdots\!94}a^{20}-\frac{44\!\cdots\!89}{22\!\cdots\!66}a^{18}-\frac{42\!\cdots\!77}{99\!\cdots\!47}a^{16}+\frac{11\!\cdots\!07}{19\!\cdots\!94}a^{14}+\frac{44\!\cdots\!75}{19\!\cdots\!94}a^{12}+\frac{21\!\cdots\!70}{99\!\cdots\!47}a^{10}-\frac{47\!\cdots\!85}{19\!\cdots\!94}a^{8}-\frac{19\!\cdots\!57}{99\!\cdots\!47}a^{6}+\frac{59\!\cdots\!33}{19\!\cdots\!94}a^{4}+\frac{22\!\cdots\!85}{66\!\cdots\!98}a^{2}+\frac{15\!\cdots\!39}{19\!\cdots\!94}$, $\frac{65\!\cdots\!23}{33\!\cdots\!49}a^{20}-\frac{38\!\cdots\!14}{99\!\cdots\!47}a^{18}-\frac{15\!\cdots\!61}{19\!\cdots\!94}a^{16}+\frac{17\!\cdots\!01}{99\!\cdots\!47}a^{14}+\frac{39\!\cdots\!77}{99\!\cdots\!47}a^{12}+\frac{13\!\cdots\!17}{19\!\cdots\!94}a^{10}-\frac{88\!\cdots\!91}{19\!\cdots\!94}a^{8}+\frac{24\!\cdots\!91}{19\!\cdots\!94}a^{6}+\frac{10\!\cdots\!17}{19\!\cdots\!94}a^{4}+\frac{13\!\cdots\!38}{99\!\cdots\!47}a^{2}-\frac{33\!\cdots\!95}{66\!\cdots\!98}$, $\frac{16\!\cdots\!39}{66\!\cdots\!98}a^{20}-\frac{97\!\cdots\!91}{19\!\cdots\!94}a^{18}-\frac{61\!\cdots\!75}{66\!\cdots\!98}a^{16}+\frac{52\!\cdots\!19}{22\!\cdots\!66}a^{14}+\frac{32\!\cdots\!73}{66\!\cdots\!98}a^{12}-\frac{34\!\cdots\!73}{66\!\cdots\!98}a^{10}-\frac{60\!\cdots\!43}{11\!\cdots\!83}a^{8}+\frac{10\!\cdots\!33}{66\!\cdots\!98}a^{6}+\frac{19\!\cdots\!62}{33\!\cdots\!49}a^{4}+\frac{10\!\cdots\!37}{66\!\cdots\!98}a^{2}+\frac{70\!\cdots\!03}{99\!\cdots\!47}$, $\frac{18\!\cdots\!56}{99\!\cdots\!47}a^{21}-\frac{33\!\cdots\!11}{39\!\cdots\!88}a^{20}-\frac{46\!\cdots\!53}{13\!\cdots\!96}a^{19}+\frac{18\!\cdots\!15}{11\!\cdots\!83}a^{18}-\frac{80\!\cdots\!52}{11\!\cdots\!83}a^{17}+\frac{12\!\cdots\!29}{39\!\cdots\!88}a^{16}+\frac{12\!\cdots\!49}{11\!\cdots\!83}a^{15}-\frac{69\!\cdots\!83}{99\!\cdots\!47}a^{14}+\frac{12\!\cdots\!82}{33\!\cdots\!49}a^{13}-\frac{68\!\cdots\!21}{39\!\cdots\!88}a^{12}+\frac{40\!\cdots\!83}{13\!\cdots\!96}a^{11}-\frac{18\!\cdots\!05}{39\!\cdots\!88}a^{10}-\frac{27\!\cdots\!99}{66\!\cdots\!98}a^{9}+\frac{77\!\cdots\!09}{39\!\cdots\!88}a^{8}-\frac{86\!\cdots\!68}{33\!\cdots\!49}a^{7}+\frac{12\!\cdots\!19}{39\!\cdots\!88}a^{6}+\frac{17\!\cdots\!54}{33\!\cdots\!49}a^{5}-\frac{24\!\cdots\!49}{99\!\cdots\!47}a^{4}+\frac{18\!\cdots\!23}{39\!\cdots\!88}a^{3}-\frac{24\!\cdots\!95}{13\!\cdots\!96}a^{2}+\frac{88\!\cdots\!27}{13\!\cdots\!96}a-\frac{17\!\cdots\!27}{39\!\cdots\!88}$, $\frac{11\!\cdots\!55}{19\!\cdots\!94}a^{21}-\frac{86\!\cdots\!04}{99\!\cdots\!47}a^{20}-\frac{21\!\cdots\!61}{19\!\cdots\!94}a^{19}+\frac{10\!\cdots\!83}{66\!\cdots\!98}a^{18}-\frac{28\!\cdots\!59}{13\!\cdots\!96}a^{17}+\frac{13\!\cdots\!61}{39\!\cdots\!88}a^{16}+\frac{81\!\cdots\!01}{22\!\cdots\!66}a^{15}-\frac{15\!\cdots\!07}{39\!\cdots\!88}a^{14}+\frac{12\!\cdots\!38}{11\!\cdots\!83}a^{13}-\frac{35\!\cdots\!15}{19\!\cdots\!94}a^{12}+\frac{10\!\cdots\!85}{13\!\cdots\!96}a^{11}-\frac{43\!\cdots\!33}{19\!\cdots\!94}a^{10}-\frac{16\!\cdots\!41}{13\!\cdots\!96}a^{9}+\frac{16\!\cdots\!59}{99\!\cdots\!47}a^{8}-\frac{90\!\cdots\!25}{13\!\cdots\!96}a^{7}+\frac{64\!\cdots\!63}{39\!\cdots\!88}a^{6}+\frac{23\!\cdots\!95}{13\!\cdots\!96}a^{5}-\frac{46\!\cdots\!31}{39\!\cdots\!88}a^{4}+\frac{14\!\cdots\!30}{99\!\cdots\!47}a^{3}-\frac{29\!\cdots\!53}{39\!\cdots\!88}a^{2}+\frac{89\!\cdots\!73}{39\!\cdots\!88}a+\frac{43\!\cdots\!41}{39\!\cdots\!88}$, $\frac{55\!\cdots\!41}{39\!\cdots\!88}a^{21}-\frac{18\!\cdots\!00}{33\!\cdots\!49}a^{20}-\frac{26\!\cdots\!35}{99\!\cdots\!47}a^{19}+\frac{42\!\cdots\!51}{39\!\cdots\!88}a^{18}-\frac{24\!\cdots\!61}{44\!\cdots\!32}a^{17}+\frac{22\!\cdots\!64}{99\!\cdots\!47}a^{16}+\frac{28\!\cdots\!51}{33\!\cdots\!49}a^{15}-\frac{36\!\cdots\!53}{99\!\cdots\!47}a^{14}+\frac{32\!\cdots\!65}{11\!\cdots\!83}a^{13}-\frac{23\!\cdots\!67}{19\!\cdots\!94}a^{12}+\frac{31\!\cdots\!63}{13\!\cdots\!96}a^{11}-\frac{34\!\cdots\!19}{39\!\cdots\!88}a^{10}-\frac{10\!\cdots\!38}{33\!\cdots\!49}a^{9}+\frac{25\!\cdots\!73}{19\!\cdots\!94}a^{8}-\frac{13\!\cdots\!07}{66\!\cdots\!98}a^{7}+\frac{15\!\cdots\!81}{19\!\cdots\!94}a^{6}+\frac{26\!\cdots\!41}{66\!\cdots\!98}a^{5}-\frac{34\!\cdots\!27}{19\!\cdots\!94}a^{4}+\frac{13\!\cdots\!61}{39\!\cdots\!88}a^{3}-\frac{60\!\cdots\!33}{39\!\cdots\!88}a^{2}+\frac{54\!\cdots\!37}{99\!\cdots\!47}a-\frac{98\!\cdots\!09}{39\!\cdots\!88}$, $\frac{39\!\cdots\!45}{99\!\cdots\!47}a^{21}-\frac{49\!\cdots\!11}{39\!\cdots\!88}a^{20}-\frac{75\!\cdots\!23}{99\!\cdots\!47}a^{19}+\frac{99\!\cdots\!01}{39\!\cdots\!88}a^{18}-\frac{15\!\cdots\!81}{99\!\cdots\!47}a^{17}+\frac{93\!\cdots\!03}{19\!\cdots\!94}a^{16}+\frac{51\!\cdots\!01}{19\!\cdots\!94}a^{15}-\frac{13\!\cdots\!34}{99\!\cdots\!47}a^{14}+\frac{32\!\cdots\!81}{39\!\cdots\!88}a^{13}-\frac{10\!\cdots\!75}{39\!\cdots\!88}a^{12}+\frac{11\!\cdots\!25}{19\!\cdots\!94}a^{11}+\frac{24\!\cdots\!87}{39\!\cdots\!88}a^{10}-\frac{35\!\cdots\!21}{39\!\cdots\!88}a^{9}+\frac{71\!\cdots\!29}{19\!\cdots\!94}a^{8}-\frac{19\!\cdots\!75}{39\!\cdots\!88}a^{7}+\frac{19\!\cdots\!87}{19\!\cdots\!94}a^{6}+\frac{11\!\cdots\!02}{99\!\cdots\!47}a^{5}-\frac{21\!\cdots\!33}{39\!\cdots\!88}a^{4}+\frac{89\!\cdots\!59}{99\!\cdots\!47}a^{3}-\frac{20\!\cdots\!45}{66\!\cdots\!98}a^{2}+\frac{38\!\cdots\!61}{39\!\cdots\!88}a+\frac{28\!\cdots\!81}{39\!\cdots\!88}$ a, 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1962700048016361818783379662/33052112219717710423001349a^4 + 1035197181621000279435135437/66104224439435420846002698a^2 + 70091389991119323005628403/99156336659153131269004047, 1836018963327896082986356/99156336659153131269004047a^21 - 33393770172099941386211/396625346636612525076016188a^20 - 46544233310721365981720353/132208448878870841692005396a^19 + 18157326766616491067215/11017370739905903474333783a^18 - 80529756227399111531218852/11017370739905903474333783a^17 + 12821858101278850961802829/396625346636612525076016188a^16 + 126491585357322173029499349/11017370739905903474333783a^15 - 6998109709861242094372183/99156336659153131269004047a^14 + 12754485918860578676176065182/33052112219717710423001349a^13 - 683523475382267575884018521/396625346636612525076016188a^12 + 40571927565228192433040125883/132208448878870841692005396a^11 - 182298859731879631618419505/396625346636612525076016188a^10 - 274068838199488161502610703499/66104224439435420846002698a^9 + 7704999608013992953768964509/396625346636612525076016188a^8 - 86614087190337807408501031868/33052112219717710423001349a^7 + 1225603716504667040702281919/396625346636612525076016188a^6 + 174832518525933252222980757854/33052112219717710423001349a^5 - 2468209160208446553705486749/99156336659153131269004047a^4 + 1805413551667804864050727908823/396625346636612525076016188a^3 - 2439233628270546872186981995/132208448878870841692005396a^2 + 88794396130918437124233184627/132208448878870841692005396a - 1797361074562076923719972227/396625346636612525076016188, 1104351508354183004655755/198312673318306262538008094a^21 - 86992978448161435274204/99156336659153131269004047a^20 - 21116540360248334835005161/198312673318306262538008094a^19 + 1079599757556069699672383/66104224439435420846002698a^18 - 289202734345294646834462059/132208448878870841692005396a^17 + 139564464036646799200388261/396625346636612525076016188a^16 + 81587099405554593401107101/22034741479811806948667566a^15 - 153394651390427384410963007/396625346636612525076016188a^14 + 1277664893532722704472250338/11017370739905903474333783a^13 - 3580803923924759715816942215/198312673318306262538008094a^12 + 10545031708057629285632886985/132208448878870841692005396a^11 - 4372178570343928679879672833/198312673318306262538008094a^10 - 168052768135557052386762925741/132208448878870841692005396a^9 + 16684645952489821802790189959/99156336659153131269004047a^8 - 90810796261217212572614739325/132208448878870841692005396a^7 + 64710811537166675525091530563/396625346636612525076016188a^6 + 235612855670921943326395207595/132208448878870841692005396a^5 - 4678918776826864515282518131/396625346636612525076016188a^4 + 145933832724288602137211103130/99156336659153131269004047a^3 - 2909729830092739995843815953/396625346636612525076016188a^2 + 89315417599919592660957410473/396625346636612525076016188a + 43008025672917187088979041/396625346636612525076016188, 55944992388737054480878541/396625346636612525076016188a^21 - 18780915710484360970300/33052112219717710423001349a^20 - 265737370589670818789193835/99156336659153131269004047a^19 + 4296578147964882440699551/396625346636612525076016188a^18 - 2455371550813721684583160361/44069482959623613897335132a^17 + 22191023902502135119811464/99156336659153131269004047a^16 + 2866958177506910059694314451/33052112219717710423001349a^15 - 36168062498711872807162153/99156336659153131269004047a^14 + 32401226118528137199730337065/11017370739905903474333783a^13 - 2348926250721421457017158067/198312673318306262538008094a^12 + 314115316239357041250589455863/132208448878870841692005396a^11 - 3490704222395809631855489419/396625346636612525076016188a^10 - 1043189598997259646910296654838/33052112219717710423001349a^9 + 25540248347986000120619179373/198312673318306262538008094a^8 - 1347612115035727494577082965607/66104224439435420846002698a^7 + 15083951597747053896498837781/198312673318306262538008094a^6 + 2652354737778890688101998537741/66104224439435420846002698a^5 - 34861767590543704745111254327/198312673318306262538008094a^4 + 13990204221990241963181777999561/396625346636612525076016188a^3 - 60534286001027481241375515133/396625346636612525076016188a^2 + 543252852286270780519902757937/99156336659153131269004047a - 9874448238155192236208493109/396625346636612525076016188, 395873944724227786308145/99156336659153131269004047a^21 - 49502785143542613851611/396625346636612525076016188a^20 - 7559605216770331998835423/99156336659153131269004047a^19 + 997034292135543403003601/396625346636612525076016188a^18 - 155655842911500240078670081/99156336659153131269004047a^17 + 9303886216236361046882203/198312673318306262538008094a^16 + 517122512991607767133950401/198312673318306262538008094a^15 - 13696186257669892974232634/99156336659153131269004047a^14 + 32938106977193881714572126281/396625346636612525076016188a^13 - 1053844977480732224430634375/396625346636612525076016188a^12 + 11790443173079471016926764025/198312673318306262538008094a^11 + 249363441057052809599851687/396625346636612525076016188a^10 - 357384792735351621182082468121/396625346636612525076016188a^9 + 7191001778373566227238340529/198312673318306262538008094a^8 - 199530576596051135839215257975/396625346636612525076016188a^7 + 1923624338252008410042662987/198312673318306262538008094a^6 + 115862293674257353332196993502/99156336659153131269004047a^5 - 21770494301036859039098233333/396625346636612525076016188a^4 + 89416648058065614400721814359/99156336659153131269004047a^3 - 2031175564242091285047642845/66104224439435420846002698a^2 + 38269652135264009109577354861/396625346636612525076016188a + 285726339619857111705073081/396625346636612525076016188 (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:	$ 70053485057500 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{7}\cdot 70053485057500 \cdot 1}{2\cdot\sqrt{15646172003756569249283257910156250000000000}}\cr\approx \mathstrut & 0.876384612575901 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^22 - 19*x^20 - 395*x^18 + 615*x^16 + 20850*x^14 + 16842*x^12 - 223773*x^10 - 144525*x^8 + 284565*x^6 + 250165*x^4 + 38861*x^2 + 1)

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^22 - 19*x^20 - 395*x^18 + 615*x^16 + 20850*x^14 + 16842*x^12 - 223773*x^10 - 144525*x^8 + 284565*x^6 + 250165*x^4 + 38861*x^2 + 1, 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^22 - 19*x^20 - 395*x^18 + 615*x^16 + 20850*x^14 + 16842*x^12 - 223773*x^10 - 144525*x^8 + 284565*x^6 + 250165*x^4 + 38861*x^2 + 1);

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^22 - 19*x^20 - 395*x^18 + 615*x^16 + 20850*x^14 + 16842*x^12 - 223773*x^10 - 144525*x^8 + 284565*x^6 + 250165*x^4 + 38861*x^2 + 1);

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

$C_2^{10}.C_{11}:C_{10}$ (as 22T36):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

A solvable group of order 112640

The 80 conjugacy class representatives for $C_2^{10}.C_{11}:C_{10}$

Character table for $C_2^{10}.C_{11}:C_{10}$

Intermediate fields

11.11.123610132462587890625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]

gp: L = nfsubfields(K); L[2..length(b)]

magma: L := Subfields(K); L[2..#L];

oscar: subfields(K)[2:end-1]

Sibling fields

Degree 22 siblings:	data not computed
Degree 44 siblings:	data not computed
Minimal sibling:	This field is its own minimal sibling

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	R	${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$	R	${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$	${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$	${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$	$22$	${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$	${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$	${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$	${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$	$22$	${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$	${\href{/padicField/53.5.0.1}{5} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$	${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2	2.2.0.1	$x^{2} + x + 1$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	2.10.10.8	$x^{10} + 4 x^{9} + 14 x^{8} + 240 x^{7} + 928 x^{6} + 4400 x^{5} + 6368 x^{4} + 13888 x^{3} - 336 x^{2} + 2432 x - 17632$	$2$	$5$	$10$	$C_2 \times (C_2^4 : C_5)$	$[2, 2, 2, 2, 2]^{5}$
	2.10.0.1	$x^{10} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1$	$1$	$10$	$0$	$C_{10}$	$[\ ]^{10}$
$3$ 3	3.22.20.1	$x^{22} + 22 x^{21} + 242 x^{20} + 1760 x^{19} + 9460 x^{18} + 39864 x^{17} + 136488 x^{16} + 388608 x^{15} + 934560 x^{14} + 1918400 x^{13} + 3384128 x^{12} + 5149702 x^{11} + 6768322 x^{10} + 7673600 x^{9} + 7474500 x^{8} + 6209808 x^{7} + 4356528 x^{6} + 2551296 x^{5} + 1226720 x^{4} + 466400 x^{3} + 129184 x^{2} + 22528 x + 1865$	$11$	$2$	$20$	22T5	$[\ ]_{11}^{10}$
$5$ 5	5.11.10.1	$x^{11} + 5$	$11$	$1$	$10$	$C_{11}:C_5$	$[\ ]_{11}^{5}$
$5$ 5	5.11.10.1	$x^{11} + 5$	$11$	$1$	$10$	$C_{11}:C_5$	$[\ ]_{11}^{5}$
$11$ 11	11.2.0.1	$x^{2} + 7 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	11.5.4.4	$x^{5} + 11$	$5$	$1$	$4$	$C_5$	$[\ ]_{5}$
	11.5.4.4	$x^{5} + 11$	$5$	$1$	$4$	$C_5$	$[\ ]_{5}$
	11.5.4.4	$x^{5} + 11$	$5$	$1$	$4$	$C_5$	$[\ ]_{5}$
	11.5.4.4	$x^{5} + 11$	$5$	$1$	$4$	$C_5$	$[\ ]_{5}$

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