Normalized defining polynomial

\( x^{19} - 38 x^{17} + 703 x^{15} - 228 x^{14} - 7239 x^{13} + 6384 x^{12} + 39159 x^{11} - 64068 x^{10} + \cdots - 110592 \) x^19 - 38*x^17 + 703*x^15 - 228*x^14 - 7239*x^13 + 6384*x^12 + 39159*x^11 - 64068*x^10 - 78147*x^9 + 263112*x^8 - 144818*x^7 - 236892*x^6 + 532855*x^5 - 695400*x^4 + 696844*x^3 - 445968*x^2 + 229824*x - 110592

Invariants

Degree:		$19$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[1, 9]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(-33600614943460448322716069311260139\) -33600614943460448322716069311260139 \(\medspace = -\,19^{27}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(65.64\)	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:		$19^{55/38}\approx 70.9296113719033$
Ramified primes:		\(19\) 19	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q(\sqrt{-19}) \)
$\card{ \Aut(K/\Q) }$:		$1$	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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{6}a^{7}-\frac{1}{6}a^{5}-\frac{1}{3}a^{3}-\frac{1}{2}a^{2}-\frac{1}{6}a$, $\frac{1}{12}a^{8}+\frac{1}{6}a^{6}-\frac{1}{6}a^{4}-\frac{1}{2}a^{3}-\frac{1}{12}a^{2}-\frac{1}{2}a$, $\frac{1}{12}a^{9}+\frac{1}{4}a^{3}-\frac{1}{3}a$, $\frac{1}{24}a^{10}-\frac{1}{24}a^{9}-\frac{1}{24}a^{8}-\frac{1}{12}a^{7}+\frac{1}{6}a^{6}-\frac{1}{6}a^{5}-\frac{1}{24}a^{4}+\frac{1}{24}a^{3}+\frac{3}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{144}a^{11}-\frac{1}{48}a^{10}-\frac{1}{144}a^{9}+\frac{1}{18}a^{7}-\frac{5}{144}a^{5}+\frac{1}{48}a^{4}+\frac{61}{144}a^{3}+\frac{1}{3}a^{2}+\frac{17}{36}a+\frac{1}{3}$, $\frac{1}{144}a^{12}+\frac{1}{72}a^{10}-\frac{1}{48}a^{9}-\frac{1}{36}a^{8}-\frac{29}{144}a^{6}+\frac{1}{12}a^{5}-\frac{7}{72}a^{4}+\frac{7}{16}a^{3}-\frac{5}{18}a^{2}+\frac{5}{12}a$, $\frac{1}{288}a^{13}-\frac{1}{288}a^{11}-\frac{7}{288}a^{9}+\frac{1}{48}a^{8}+\frac{7}{288}a^{7}-\frac{1}{24}a^{6}+\frac{49}{288}a^{5}-\frac{1}{24}a^{4}+\frac{71}{288}a^{3}-\frac{11}{48}a^{2}+\frac{5}{12}a$, $\frac{1}{288}a^{14}-\frac{1}{288}a^{12}+\frac{5}{288}a^{10}-\frac{1}{48}a^{9}-\frac{5}{288}a^{8}+\frac{1}{24}a^{7}-\frac{47}{288}a^{6}+\frac{1}{8}a^{5}+\frac{59}{288}a^{4}-\frac{1}{48}a^{3}-\frac{5}{24}a^{2}+\frac{1}{12}a$, $\frac{1}{288}a^{15}-\frac{1}{48}a^{10}+\frac{1}{72}a^{9}+\frac{1}{48}a^{8}-\frac{1}{4}a^{6}-\frac{1}{18}a^{5}+\frac{5}{48}a^{4}+\frac{91}{288}a^{3}-\frac{5}{48}a^{2}-\frac{4}{9}a+\frac{1}{3}$, $\frac{1}{1728}a^{16}+\frac{1}{1728}a^{15}+\frac{1}{1728}a^{13}-\frac{1}{432}a^{12}-\frac{1}{576}a^{11}-\frac{5}{864}a^{10}+\frac{47}{1728}a^{9}-\frac{1}{54}a^{8}-\frac{1}{192}a^{7}-\frac{1}{54}a^{6}+\frac{331}{1728}a^{5}-\frac{1}{64}a^{4}-\frac{85}{216}a^{3}-\frac{145}{432}a^{2}-\frac{13}{36}a-\frac{1}{3}$, $\frac{1}{44928}a^{17}-\frac{7}{44928}a^{16}+\frac{1}{864}a^{15}-\frac{53}{44928}a^{14}+\frac{5}{3744}a^{13}-\frac{1}{3456}a^{12}+\frac{61}{22464}a^{11}-\frac{515}{44928}a^{10}+\frac{25}{936}a^{9}+\frac{1837}{44928}a^{8}+\frac{23}{5616}a^{7}-\frac{10219}{44928}a^{6}-\frac{227}{3456}a^{5}-\frac{2071}{22464}a^{4}-\frac{7}{936}a^{3}+\frac{851}{2808}a^{2}-\frac{7}{36}a+\frac{3}{13}$, $\frac{1}{43\!\cdots\!56}a^{18}+\frac{37\!\cdots\!95}{35\!\cdots\!88}a^{17}-\frac{81\!\cdots\!73}{43\!\cdots\!56}a^{16}-\frac{80\!\cdots\!53}{53\!\cdots\!76}a^{15}+\frac{40\!\cdots\!65}{33\!\cdots\!12}a^{14}+\frac{11\!\cdots\!09}{14\!\cdots\!52}a^{13}-\frac{13\!\cdots\!69}{47\!\cdots\!84}a^{12}-\frac{27\!\cdots\!17}{47\!\cdots\!84}a^{11}-\frac{73\!\cdots\!81}{47\!\cdots\!84}a^{10}+\frac{10\!\cdots\!89}{47\!\cdots\!84}a^{9}-\frac{26\!\cdots\!55}{14\!\cdots\!52}a^{8}+\frac{13\!\cdots\!93}{47\!\cdots\!84}a^{7}-\frac{24\!\cdots\!09}{10\!\cdots\!64}a^{6}-\frac{22\!\cdots\!57}{14\!\cdots\!52}a^{5}-\frac{21\!\cdots\!79}{21\!\cdots\!28}a^{4}+\frac{13\!\cdots\!35}{99\!\cdots\!08}a^{3}+\frac{28\!\cdots\!09}{13\!\cdots\!08}a^{2}-\frac{86\!\cdots\!33}{44\!\cdots\!36}a+\frac{17\!\cdots\!05}{37\!\cdots\!03}$ 1, a, a^2, a^3, 1/2*a^4 - 1/2*a, 1/2*a^5 - 1/2*a^2, 1/2*a^6 - 1/2*a^3, 1/6*a^7 - 1/6*a^5 - 1/3*a^3 - 1/2*a^2 - 1/6*a, 1/12*a^8 + 1/6*a^6 - 1/6*a^4 - 1/2*a^3 - 1/12*a^2 - 1/2*a, 1/12*a^9 + 1/4*a^3 - 1/3*a, 1/24*a^10 - 1/24*a^9 - 1/24*a^8 - 1/12*a^7 + 1/6*a^6 - 1/6*a^5 - 1/24*a^4 + 1/24*a^3 + 3/8*a^2 + 1/4*a, 1/144*a^11 - 1/48*a^10 - 1/144*a^9 + 1/18*a^7 - 5/144*a^5 + 1/48*a^4 + 61/144*a^3 + 1/3*a^2 + 17/36*a + 1/3, 1/144*a^12 + 1/72*a^10 - 1/48*a^9 - 1/36*a^8 - 29/144*a^6 + 1/12*a^5 - 7/72*a^4 + 7/16*a^3 - 5/18*a^2 + 5/12*a, 1/288*a^13 - 1/288*a^11 - 7/288*a^9 + 1/48*a^8 + 7/288*a^7 - 1/24*a^6 + 49/288*a^5 - 1/24*a^4 + 71/288*a^3 - 11/48*a^2 + 5/12*a, 1/288*a^14 - 1/288*a^12 + 5/288*a^10 - 1/48*a^9 - 5/288*a^8 + 1/24*a^7 - 47/288*a^6 + 1/8*a^5 + 59/288*a^4 - 1/48*a^3 - 5/24*a^2 + 1/12*a, 1/288*a^15 - 1/48*a^10 + 1/72*a^9 + 1/48*a^8 - 1/4*a^6 - 1/18*a^5 + 5/48*a^4 + 91/288*a^3 - 5/48*a^2 - 4/9*a + 1/3, 1/1728*a^16 + 1/1728*a^15 + 1/1728*a^13 - 1/432*a^12 - 1/576*a^11 - 5/864*a^10 + 47/1728*a^9 - 1/54*a^8 - 1/192*a^7 - 1/54*a^6 + 331/1728*a^5 - 1/64*a^4 - 85/216*a^3 - 145/432*a^2 - 13/36*a - 1/3, 1/44928*a^17 - 7/44928*a^16 + 1/864*a^15 - 53/44928*a^14 + 5/3744*a^13 - 1/3456*a^12 + 61/22464*a^11 - 515/44928*a^10 + 25/936*a^9 + 1837/44928*a^8 + 23/5616*a^7 - 10219/44928*a^6 - 227/3456*a^5 - 2071/22464*a^4 - 7/936*a^3 + 851/2808*a^2 - 7/36*a + 3/13, 1/431059976754143919901056*a^18 + 373866811795322195/35921664729511993325088*a^17 - 81016404862559107673/431059976754143919901056*a^16 - 8031016140055928353/5321728108075850862976*a^15 + 40510207904925414565/33158459750318763069312*a^14 + 119186151865216585309/143686658918047973300352*a^13 - 135061596995156953769/47895552972682657766784*a^12 - 27797133245399049917/47895552972682657766784*a^11 - 734181150814994075581/47895552972682657766784*a^10 + 1034003639877070564589/47895552972682657766784*a^9 - 2699142237496533274855/143686658918047973300352*a^8 + 1359106788617060853493/47895552972682657766784*a^7 - 24613953369902649414209/107764994188535979975264*a^6 - 22876120399774580903257/143686658918047973300352*a^5 - 21865467284824721147179/215529988377071959950528*a^4 + 135904126339845164035/997824020264222036808*a^3 + 2857439909628598537709/13470624273566997496908*a^2 - 860680176964427119933/4490208091188999165636*a + 171942530252109252305/374184007599083263803

Class group and class number

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

Unit group

Rank:		$9$	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:		$\frac{12\!\cdots\!17}{43\!\cdots\!56}a^{18}+\frac{34\!\cdots\!61}{14\!\cdots\!52}a^{17}-\frac{23\!\cdots\!41}{21\!\cdots\!28}a^{16}-\frac{14\!\cdots\!27}{14\!\cdots\!52}a^{15}+\frac{33\!\cdots\!23}{16\!\cdots\!56}a^{14}+\frac{61\!\cdots\!79}{47\!\cdots\!84}a^{13}-\frac{76\!\cdots\!47}{35\!\cdots\!88}a^{12}-\frac{44\!\cdots\!09}{14\!\cdots\!52}a^{11}+\frac{10\!\cdots\!01}{79\!\cdots\!64}a^{10}-\frac{75\!\cdots\!33}{14\!\cdots\!52}a^{9}-\frac{28\!\cdots\!31}{79\!\cdots\!64}a^{8}+\frac{54\!\cdots\!79}{14\!\cdots\!52}a^{7}+\frac{79\!\cdots\!39}{43\!\cdots\!56}a^{6}-\frac{72\!\cdots\!79}{13\!\cdots\!44}a^{5}+\frac{45\!\cdots\!99}{53\!\cdots\!32}a^{4}-\frac{22\!\cdots\!77}{22\!\cdots\!18}a^{3}+\frac{21\!\cdots\!05}{33\!\cdots\!27}a^{2}-\frac{16\!\cdots\!35}{44\!\cdots\!36}a+\frac{72\!\cdots\!89}{37\!\cdots\!03}$, $\frac{71\!\cdots\!71}{43\!\cdots\!56}a^{18}-\frac{73\!\cdots\!93}{14\!\cdots\!52}a^{17}-\frac{10\!\cdots\!07}{21\!\cdots\!28}a^{16}+\frac{21\!\cdots\!87}{14\!\cdots\!52}a^{15}+\frac{16\!\cdots\!99}{21\!\cdots\!28}a^{14}-\frac{12\!\cdots\!79}{47\!\cdots\!84}a^{13}-\frac{40\!\cdots\!89}{89\!\cdots\!72}a^{12}+\frac{11\!\cdots\!63}{47\!\cdots\!84}a^{11}-\frac{49\!\cdots\!63}{71\!\cdots\!76}a^{10}-\frac{13\!\cdots\!71}{14\!\cdots\!52}a^{9}+\frac{11\!\cdots\!21}{71\!\cdots\!76}a^{8}-\frac{37\!\cdots\!29}{47\!\cdots\!84}a^{7}-\frac{13\!\cdots\!31}{43\!\cdots\!56}a^{6}+\frac{52\!\cdots\!03}{89\!\cdots\!72}a^{5}-\frac{77\!\cdots\!73}{10\!\cdots\!64}a^{4}+\frac{22\!\cdots\!33}{35\!\cdots\!88}a^{3}-\frac{20\!\cdots\!75}{53\!\cdots\!32}a^{2}+\frac{19\!\cdots\!96}{11\!\cdots\!09}a-\frac{19\!\cdots\!83}{37\!\cdots\!03}$, $\frac{71\!\cdots\!65}{14\!\cdots\!52}a^{18}+\frac{59\!\cdots\!51}{23\!\cdots\!92}a^{17}-\frac{28\!\cdots\!97}{14\!\cdots\!52}a^{16}-\frac{52\!\cdots\!17}{47\!\cdots\!84}a^{15}+\frac{52\!\cdots\!99}{14\!\cdots\!52}a^{14}+\frac{19\!\cdots\!29}{15\!\cdots\!28}a^{13}-\frac{62\!\cdots\!23}{15\!\cdots\!28}a^{12}+\frac{21\!\cdots\!51}{47\!\cdots\!84}a^{11}+\frac{37\!\cdots\!41}{15\!\cdots\!28}a^{10}-\frac{70\!\cdots\!59}{47\!\cdots\!84}a^{9}-\frac{32\!\cdots\!49}{47\!\cdots\!84}a^{8}+\frac{39\!\cdots\!65}{47\!\cdots\!84}a^{7}+\frac{21\!\cdots\!51}{71\!\cdots\!76}a^{6}-\frac{17\!\cdots\!03}{15\!\cdots\!28}a^{5}+\frac{29\!\cdots\!55}{17\!\cdots\!44}a^{4}-\frac{24\!\cdots\!01}{11\!\cdots\!96}a^{3}+\frac{15\!\cdots\!11}{11\!\cdots\!09}a^{2}-\frac{10\!\cdots\!19}{14\!\cdots\!12}a+\frac{17\!\cdots\!81}{41\!\cdots\!67}$, $\frac{17\!\cdots\!37}{43\!\cdots\!56}a^{18}-\frac{11\!\cdots\!69}{14\!\cdots\!52}a^{17}-\frac{42\!\cdots\!69}{21\!\cdots\!28}a^{16}+\frac{26\!\cdots\!95}{14\!\cdots\!52}a^{15}+\frac{45\!\cdots\!65}{10\!\cdots\!64}a^{14}-\frac{26\!\cdots\!49}{14\!\cdots\!52}a^{13}-\frac{32\!\cdots\!69}{71\!\cdots\!76}a^{12}+\frac{31\!\cdots\!81}{14\!\cdots\!52}a^{11}+\frac{88\!\cdots\!95}{35\!\cdots\!88}a^{10}-\frac{33\!\cdots\!31}{15\!\cdots\!28}a^{9}-\frac{76\!\cdots\!45}{13\!\cdots\!88}a^{8}+\frac{12\!\cdots\!93}{14\!\cdots\!52}a^{7}-\frac{96\!\cdots\!63}{43\!\cdots\!56}a^{6}-\frac{74\!\cdots\!67}{17\!\cdots\!44}a^{5}+\frac{33\!\cdots\!09}{21\!\cdots\!28}a^{4}-\frac{51\!\cdots\!61}{13\!\cdots\!44}a^{3}+\frac{11\!\cdots\!51}{26\!\cdots\!16}a^{2}-\frac{52\!\cdots\!55}{22\!\cdots\!18}a+\frac{71\!\cdots\!13}{37\!\cdots\!03}$, $\frac{13\!\cdots\!23}{16\!\cdots\!56}a^{18}-\frac{21\!\cdots\!79}{15\!\cdots\!28}a^{17}-\frac{15\!\cdots\!87}{43\!\cdots\!56}a^{16}+\frac{24\!\cdots\!21}{61\!\cdots\!28}a^{15}+\frac{31\!\cdots\!43}{43\!\cdots\!56}a^{14}-\frac{17\!\cdots\!37}{23\!\cdots\!92}a^{13}-\frac{34\!\cdots\!49}{40\!\cdots\!52}a^{12}+\frac{82\!\cdots\!45}{89\!\cdots\!72}a^{11}+\frac{72\!\cdots\!79}{14\!\cdots\!52}a^{10}-\frac{51\!\cdots\!03}{71\!\cdots\!76}a^{9}-\frac{18\!\cdots\!25}{14\!\cdots\!52}a^{8}+\frac{20\!\cdots\!81}{71\!\cdots\!76}a^{7}-\frac{73\!\cdots\!65}{43\!\cdots\!56}a^{6}-\frac{40\!\cdots\!65}{11\!\cdots\!04}a^{5}+\frac{98\!\cdots\!65}{21\!\cdots\!28}a^{4}-\frac{94\!\cdots\!59}{17\!\cdots\!44}a^{3}+\frac{63\!\cdots\!31}{13\!\cdots\!08}a^{2}-\frac{48\!\cdots\!13}{34\!\cdots\!72}a+\frac{58\!\cdots\!61}{37\!\cdots\!03}$, $\frac{57\!\cdots\!63}{10\!\cdots\!64}a^{18}-\frac{12\!\cdots\!97}{71\!\cdots\!76}a^{17}-\frac{16\!\cdots\!19}{67\!\cdots\!54}a^{16}+\frac{42\!\cdots\!03}{71\!\cdots\!76}a^{15}+\frac{90\!\cdots\!33}{16\!\cdots\!56}a^{14}-\frac{86\!\cdots\!79}{79\!\cdots\!64}a^{13}-\frac{47\!\cdots\!53}{71\!\cdots\!76}a^{12}+\frac{29\!\cdots\!25}{23\!\cdots\!92}a^{11}+\frac{30\!\cdots\!47}{71\!\cdots\!76}a^{10}-\frac{58\!\cdots\!49}{71\!\cdots\!76}a^{9}-\frac{86\!\cdots\!59}{71\!\cdots\!76}a^{8}+\frac{67\!\cdots\!89}{23\!\cdots\!92}a^{7}+\frac{33\!\cdots\!33}{21\!\cdots\!28}a^{6}-\frac{95\!\cdots\!79}{35\!\cdots\!88}a^{5}+\frac{77\!\cdots\!87}{21\!\cdots\!28}a^{4}-\frac{20\!\cdots\!21}{35\!\cdots\!88}a^{3}+\frac{54\!\cdots\!75}{13\!\cdots\!08}a^{2}-\frac{42\!\cdots\!37}{22\!\cdots\!18}a+\frac{56\!\cdots\!09}{37\!\cdots\!03}$, $\frac{24\!\cdots\!19}{43\!\cdots\!56}a^{18}-\frac{24\!\cdots\!41}{14\!\cdots\!52}a^{17}-\frac{39\!\cdots\!21}{21\!\cdots\!28}a^{16}+\frac{85\!\cdots\!15}{14\!\cdots\!52}a^{15}+\frac{62\!\cdots\!99}{21\!\cdots\!28}a^{14}-\frac{15\!\cdots\!57}{14\!\cdots\!52}a^{13}-\frac{21\!\cdots\!32}{11\!\cdots\!09}a^{12}+\frac{54\!\cdots\!63}{47\!\cdots\!84}a^{11}-\frac{20\!\cdots\!59}{71\!\cdots\!76}a^{10}-\frac{72\!\cdots\!47}{14\!\cdots\!52}a^{9}+\frac{19\!\cdots\!19}{23\!\cdots\!92}a^{8}+\frac{42\!\cdots\!19}{53\!\cdots\!76}a^{7}-\frac{79\!\cdots\!59}{43\!\cdots\!56}a^{6}+\frac{36\!\cdots\!05}{11\!\cdots\!96}a^{5}-\frac{46\!\cdots\!71}{13\!\cdots\!08}a^{4}+\frac{11\!\cdots\!11}{35\!\cdots\!88}a^{3}-\frac{10\!\cdots\!23}{53\!\cdots\!32}a^{2}+\frac{10\!\cdots\!25}{11\!\cdots\!09}a-\frac{15\!\cdots\!13}{37\!\cdots\!03}$, $\frac{41\!\cdots\!97}{23\!\cdots\!92}a^{18}-\frac{99\!\cdots\!93}{14\!\cdots\!52}a^{17}-\frac{74\!\cdots\!11}{14\!\cdots\!52}a^{16}+\frac{17\!\cdots\!83}{79\!\cdots\!64}a^{15}+\frac{87\!\cdots\!83}{11\!\cdots\!04}a^{14}-\frac{27\!\cdots\!79}{71\!\cdots\!76}a^{13}-\frac{20\!\cdots\!75}{47\!\cdots\!84}a^{12}+\frac{12\!\cdots\!29}{35\!\cdots\!88}a^{11}-\frac{23\!\cdots\!79}{14\!\cdots\!52}a^{10}-\frac{94\!\cdots\!91}{71\!\cdots\!76}a^{9}+\frac{10\!\cdots\!63}{47\!\cdots\!84}a^{8}-\frac{25\!\cdots\!07}{71\!\cdots\!76}a^{7}-\frac{63\!\cdots\!09}{14\!\cdots\!52}a^{6}+\frac{38\!\cdots\!03}{47\!\cdots\!84}a^{5}-\frac{75\!\cdots\!75}{71\!\cdots\!76}a^{4}+\frac{36\!\cdots\!41}{35\!\cdots\!88}a^{3}-\frac{37\!\cdots\!15}{59\!\cdots\!48}a^{2}+\frac{25\!\cdots\!71}{74\!\cdots\!06}a-\frac{21\!\cdots\!77}{12\!\cdots\!01}$, $\frac{78\!\cdots\!83}{21\!\cdots\!28}a^{18}+\frac{17\!\cdots\!47}{15\!\cdots\!28}a^{17}-\frac{51\!\cdots\!55}{43\!\cdots\!56}a^{16}-\frac{33\!\cdots\!79}{89\!\cdots\!72}a^{15}+\frac{62\!\cdots\!05}{33\!\cdots\!12}a^{14}+\frac{95\!\cdots\!61}{17\!\cdots\!44}a^{13}-\frac{25\!\cdots\!73}{14\!\cdots\!52}a^{12}-\frac{25\!\cdots\!55}{71\!\cdots\!76}a^{11}+\frac{50\!\cdots\!31}{47\!\cdots\!84}a^{10}+\frac{34\!\cdots\!49}{35\!\cdots\!88}a^{9}-\frac{55\!\cdots\!67}{14\!\cdots\!52}a^{8}+\frac{13\!\cdots\!07}{35\!\cdots\!88}a^{7}+\frac{28\!\cdots\!69}{43\!\cdots\!56}a^{6}-\frac{78\!\cdots\!05}{14\!\cdots\!52}a^{5}-\frac{55\!\cdots\!05}{10\!\cdots\!64}a^{4}+\frac{12\!\cdots\!81}{89\!\cdots\!72}a^{3}-\frac{12\!\cdots\!85}{53\!\cdots\!32}a^{2}+\frac{69\!\cdots\!57}{22\!\cdots\!18}a-\frac{38\!\cdots\!39}{37\!\cdots\!03}$ 1228302611227758388417/431059976754143919901056*a^18 + 346471637894485440061/143686658918047973300352*a^17 - 23496007293603157314841/215529988377071959950528*a^16 - 14066113123839848667827/143686658918047973300352*a^15 + 33491645557943616013223/16579229875159381534656*a^14 + 61052068484860780958879/47895552972682657766784*a^13 - 764009726072459727707347/35921664729511993325088*a^12 - 449505183334399239506009/143686658918047973300352*a^11 + 1004258245158604715411701/7982592162113776294464*a^10 - 7541964651443415993502733/143686658918047973300352*a^9 - 2891625298865416752318131/7982592162113776294464*a^8 + 54277590896832893440729379/143686658918047973300352*a^7 + 79548035225776415775230839/431059976754143919901056*a^6 - 727429121726374687074579/1330432027018962715744*a^5 + 45088997029171408839223199/53882497094267989987632*a^4 - 2259117630769670581391977/2245104045594499582818*a^3 + 2155148397259209508203305/3367656068391749374227*a^2 - 1601096639680193142044035/4490208091188999165636*a + 72848259057858927865289/374184007599083263803, 71412969429815481571/431059976754143919901056*a^18 - 73484966319147460093/143686658918047973300352*a^17 - 1059546931870648732507/215529988377071959950528*a^16 + 2185486454528367508787/143686658918047973300352*a^15 + 16202222347911658864199/215529988377071959950528*a^14 - 12937075559794206348379/47895552972682657766784*a^13 - 4000148111853775679189/8980416182377998331272*a^12 + 118760668209305457972863/47895552972682657766784*a^11 - 49275285593082819131863/71843329459023986650176*a^10 - 1338018686364800443975271/143686658918047973300352*a^9 + 1123674463371608566755521/71843329459023986650176*a^8 - 37802636294360199315029/47895552972682657766784*a^7 - 13241659846968585578889431/431059976754143919901056*a^6 + 526656438101797070667403/8980416182377998331272*a^5 - 7726825395054470161820273/107764994188535979975264*a^4 + 2233681512752647959405233/35921664729511993325088*a^3 - 2023679574052608817805275/53882497094267989987632*a^2 + 19727843871026799062996/1122552022797249791409*a - 1976689960704743034583/374184007599083263803, 717864687667893401765/143686658918047973300352*a^18 + 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55862518317722782424105/107764994188535979975264*a^4 + 12595126605690364495381/8980416182377998331272*a^3 - 125465869750746896409085/53882497094267989987632*a^2 + 6956183581342455708757/2245104045594499582818*a - 381770783128515572239/374184007599083263803 (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:		\( 215015238772 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 215015238772 \cdot 1}{2\cdot\sqrt{33600614943460448322716069311260139}}\cr\approx \mathstrut & 17.9025313582229 \end{aligned}\] (assuming GRH)

Galois group

$D_{19}$ (as 19T2):

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
\(19\) 19	19.19.27.10	$x^{19} + 323 x^{9} + 19$	$19$	$1$	$27$	$D_{19}$	$[3/2]_{2}$