Normalized defining polynomial
\( x^{27} - 6 x^{26} + 27 x^{25} - 119 x^{24} + 469 x^{23} - 1488 x^{22} + 4228 x^{21} - 11600 x^{20} + \cdots - 459999 \)
Invariants
Degree: | $27$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-55686266560375612156826962909758952173810975943\) \(\medspace = -\,3943^{13}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(53.87\) | 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: | $3943^{1/2}\approx 62.7933117457584$ | ||
Ramified primes: | \(3943\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3943}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{6}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{7}$, $\frac{1}{3}a^{16}-\frac{1}{3}a^{8}$, $\frac{1}{9}a^{17}+\frac{1}{9}a^{16}+\frac{1}{9}a^{15}+\frac{1}{9}a^{14}+\frac{1}{9}a^{13}+\frac{1}{9}a^{12}+\frac{1}{9}a^{11}+\frac{1}{9}a^{10}-\frac{1}{9}a^{9}-\frac{4}{9}a^{8}-\frac{4}{9}a^{7}-\frac{4}{9}a^{6}-\frac{4}{9}a^{5}-\frac{4}{9}a^{4}-\frac{4}{9}a^{3}-\frac{4}{9}a^{2}-\frac{1}{3}a$, $\frac{1}{45}a^{18}+\frac{1}{45}a^{17}-\frac{2}{45}a^{16}-\frac{1}{9}a^{15}+\frac{4}{45}a^{14}-\frac{2}{45}a^{13}-\frac{1}{9}a^{12}-\frac{2}{45}a^{11}+\frac{2}{45}a^{10}-\frac{4}{45}a^{9}-\frac{1}{45}a^{8}+\frac{2}{45}a^{7}-\frac{16}{45}a^{6}-\frac{2}{9}a^{5}-\frac{16}{45}a^{4}+\frac{8}{45}a^{3}+\frac{7}{15}a^{2}+\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{45}a^{19}+\frac{2}{45}a^{17}+\frac{2}{45}a^{16}-\frac{1}{45}a^{15}-\frac{1}{45}a^{14}+\frac{2}{45}a^{13}-\frac{7}{45}a^{12}-\frac{2}{15}a^{11}-\frac{1}{45}a^{10}-\frac{2}{45}a^{9}-\frac{17}{45}a^{8}+\frac{22}{45}a^{7}-\frac{14}{45}a^{6}+\frac{19}{45}a^{5}+\frac{19}{45}a^{4}+\frac{8}{45}a^{3}+\frac{13}{45}a^{2}-\frac{1}{3}a-\frac{1}{5}$, $\frac{1}{45}a^{20}+\frac{1}{15}a^{16}-\frac{2}{15}a^{15}-\frac{2}{15}a^{14}-\frac{1}{15}a^{13}+\frac{4}{45}a^{12}+\frac{1}{15}a^{11}-\frac{2}{15}a^{10}+\frac{2}{15}a^{9}-\frac{7}{15}a^{8}-\frac{1}{15}a^{7}+\frac{2}{15}a^{6}-\frac{2}{15}a^{5}-\frac{1}{9}a^{4}-\frac{1}{15}a^{3}-\frac{4}{15}a^{2}+\frac{1}{15}a-\frac{2}{5}$, $\frac{1}{405}a^{21}+\frac{2}{405}a^{20}+\frac{1}{135}a^{19}+\frac{1}{405}a^{18}+\frac{2}{81}a^{17}+\frac{49}{405}a^{16}-\frac{41}{405}a^{15}-\frac{29}{405}a^{14}-\frac{28}{405}a^{13}+\frac{8}{81}a^{11}-\frac{7}{405}a^{10}-\frac{4}{405}a^{9}+\frac{128}{405}a^{8}-\frac{187}{405}a^{7}+\frac{8}{405}a^{6}-\frac{8}{27}a^{5}-\frac{167}{405}a^{4}-\frac{46}{405}a^{3}+\frac{58}{135}a^{2}-\frac{22}{45}a-\frac{2}{15}$, $\frac{1}{1215}a^{22}-\frac{1}{1215}a^{21}-\frac{1}{405}a^{20}+\frac{2}{243}a^{19}-\frac{2}{1215}a^{18}-\frac{44}{1215}a^{17}-\frac{89}{1215}a^{16}+\frac{31}{1215}a^{15}-\frac{17}{243}a^{14}-\frac{29}{405}a^{13}+\frac{4}{1215}a^{12}-\frac{172}{1215}a^{11}+\frac{26}{1215}a^{10}-\frac{35}{243}a^{9}-\frac{238}{1215}a^{8}+\frac{92}{1215}a^{7}+\frac{19}{45}a^{6}-\frac{19}{243}a^{5}-\frac{49}{1215}a^{4}-\frac{202}{405}a^{3}-\frac{1}{27}a^{2}+\frac{17}{45}a-\frac{1}{15}$, $\frac{1}{1215}a^{23}-\frac{1}{1215}a^{21}+\frac{13}{1215}a^{20}-\frac{2}{243}a^{19}+\frac{11}{1215}a^{18}+\frac{32}{1215}a^{17}+\frac{62}{1215}a^{16}+\frac{8}{81}a^{15}+\frac{119}{1215}a^{14}-\frac{194}{1215}a^{13}-\frac{38}{405}a^{12}+\frac{163}{1215}a^{11}+\frac{20}{243}a^{10}+\frac{88}{1215}a^{9}+\frac{103}{1215}a^{8}-\frac{172}{1215}a^{7}-\frac{584}{1215}a^{6}+\frac{37}{135}a^{5}+\frac{572}{1215}a^{4}+\frac{34}{405}a^{3}-\frac{4}{135}a^{2}-\frac{4}{9}a+\frac{2}{5}$, $\frac{1}{8505}a^{24}-\frac{1}{2835}a^{23}+\frac{1}{2835}a^{22}+\frac{1}{2835}a^{21}-\frac{79}{8505}a^{20}-\frac{2}{315}a^{19}+\frac{1}{945}a^{18}-\frac{4}{405}a^{17}-\frac{1133}{8505}a^{16}-\frac{22}{189}a^{15}-\frac{46}{945}a^{14}-\frac{101}{2835}a^{13}-\frac{73}{8505}a^{12}+\frac{37}{405}a^{11}-\frac{122}{945}a^{10}+\frac{166}{2835}a^{9}+\frac{4084}{8505}a^{8}-\frac{74}{405}a^{7}+\frac{262}{567}a^{6}+\frac{262}{2835}a^{5}-\frac{1576}{8505}a^{4}-\frac{65}{567}a^{3}-\frac{40}{189}a^{2}+\frac{41}{315}a-\frac{1}{105}$, $\frac{1}{42525}a^{25}+\frac{1}{42525}a^{24}-\frac{1}{4725}a^{23}+\frac{1}{2835}a^{22}-\frac{4}{42525}a^{21}+\frac{134}{42525}a^{20}-\frac{23}{4725}a^{19}+\frac{68}{14175}a^{18}+\frac{1807}{42525}a^{17}-\frac{2057}{42525}a^{16}-\frac{479}{4725}a^{15}+\frac{2077}{14175}a^{14}+\frac{6401}{42525}a^{13}-\frac{1226}{8505}a^{12}-\frac{328}{2835}a^{11}+\frac{823}{14175}a^{10}-\frac{923}{6075}a^{9}-\frac{7583}{42525}a^{8}+\frac{662}{1575}a^{7}+\frac{8}{45}a^{6}+\frac{1277}{6075}a^{5}+\frac{1543}{8505}a^{4}-\frac{2299}{14175}a^{3}+\frac{41}{945}a^{2}+\frac{47}{225}a-\frac{193}{525}$, $\frac{1}{54\!\cdots\!25}a^{26}-\frac{29\!\cdots\!03}{18\!\cdots\!75}a^{25}-\frac{24\!\cdots\!11}{54\!\cdots\!75}a^{24}+\frac{17\!\cdots\!76}{10\!\cdots\!05}a^{23}-\frac{36\!\cdots\!49}{54\!\cdots\!25}a^{22}-\frac{53\!\cdots\!44}{60\!\cdots\!25}a^{21}-\frac{42\!\cdots\!17}{54\!\cdots\!25}a^{20}+\frac{68\!\cdots\!74}{54\!\cdots\!25}a^{19}+\frac{37\!\cdots\!57}{54\!\cdots\!25}a^{18}+\frac{16\!\cdots\!08}{54\!\cdots\!25}a^{17}+\frac{21\!\cdots\!89}{54\!\cdots\!25}a^{16}+\frac{79\!\cdots\!66}{54\!\cdots\!25}a^{15}-\frac{46\!\cdots\!62}{77\!\cdots\!75}a^{14}+\frac{78\!\cdots\!46}{98\!\cdots\!55}a^{13}-\frac{15\!\cdots\!43}{10\!\cdots\!05}a^{12}+\frac{11\!\cdots\!87}{77\!\cdots\!75}a^{11}-\frac{89\!\cdots\!26}{67\!\cdots\!25}a^{10}+\frac{44\!\cdots\!87}{54\!\cdots\!25}a^{9}-\frac{71\!\cdots\!23}{77\!\cdots\!75}a^{8}-\frac{14\!\cdots\!78}{36\!\cdots\!35}a^{7}+\frac{67\!\cdots\!63}{18\!\cdots\!75}a^{6}-\frac{31\!\cdots\!34}{10\!\cdots\!05}a^{5}+\frac{26\!\cdots\!67}{60\!\cdots\!25}a^{4}+\frac{30\!\cdots\!29}{12\!\cdots\!45}a^{3}+\frac{16\!\cdots\!49}{20\!\cdots\!75}a^{2}-\frac{27\!\cdots\!66}{22\!\cdots\!75}a+\frac{20\!\cdots\!34}{44\!\cdots\!35}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$, $5$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $13$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (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: | $\frac{98\!\cdots\!24}{16\!\cdots\!75}a^{26}-\frac{15\!\cdots\!42}{55\!\cdots\!25}a^{25}+\frac{61\!\cdots\!83}{50\!\cdots\!75}a^{24}-\frac{17\!\cdots\!41}{33\!\cdots\!15}a^{23}+\frac{34\!\cdots\!64}{16\!\cdots\!75}a^{22}-\frac{99\!\cdots\!04}{16\!\cdots\!75}a^{21}+\frac{27\!\cdots\!67}{16\!\cdots\!75}a^{20}-\frac{74\!\cdots\!34}{16\!\cdots\!75}a^{19}+\frac{50\!\cdots\!16}{55\!\cdots\!25}a^{18}-\frac{30\!\cdots\!33}{16\!\cdots\!75}a^{17}+\frac{83\!\cdots\!06}{16\!\cdots\!75}a^{16}-\frac{48\!\cdots\!67}{55\!\cdots\!25}a^{15}+\frac{40\!\cdots\!67}{23\!\cdots\!25}a^{14}-\frac{18\!\cdots\!59}{60\!\cdots\!93}a^{13}+\frac{51\!\cdots\!73}{11\!\cdots\!05}a^{12}-\frac{15\!\cdots\!72}{23\!\cdots\!25}a^{11}+\frac{16\!\cdots\!71}{16\!\cdots\!75}a^{10}-\frac{18\!\cdots\!02}{16\!\cdots\!75}a^{9}+\frac{30\!\cdots\!48}{23\!\cdots\!25}a^{8}-\frac{47\!\cdots\!99}{33\!\cdots\!15}a^{7}+\frac{71\!\cdots\!47}{55\!\cdots\!25}a^{6}-\frac{11\!\cdots\!81}{11\!\cdots\!05}a^{5}+\frac{16\!\cdots\!17}{16\!\cdots\!75}a^{4}-\frac{73\!\cdots\!44}{11\!\cdots\!05}a^{3}+\frac{72\!\cdots\!93}{18\!\cdots\!75}a^{2}-\frac{20\!\cdots\!56}{61\!\cdots\!25}a+\frac{29\!\cdots\!98}{13\!\cdots\!05}$, $\frac{34\!\cdots\!58}{54\!\cdots\!25}a^{26}-\frac{14\!\cdots\!34}{20\!\cdots\!75}a^{25}+\frac{41\!\cdots\!79}{16\!\cdots\!25}a^{24}-\frac{65\!\cdots\!81}{54\!\cdots\!25}a^{23}+\frac{26\!\cdots\!63}{54\!\cdots\!25}a^{22}-\frac{29\!\cdots\!03}{18\!\cdots\!75}a^{21}+\frac{47\!\cdots\!29}{10\!\cdots\!05}a^{20}-\frac{67\!\cdots\!01}{54\!\cdots\!25}a^{19}+\frac{16\!\cdots\!67}{54\!\cdots\!25}a^{18}-\frac{29\!\cdots\!38}{54\!\cdots\!25}a^{17}+\frac{75\!\cdots\!34}{54\!\cdots\!25}a^{16}-\frac{18\!\cdots\!76}{54\!\cdots\!25}a^{15}+\frac{32\!\cdots\!06}{77\!\cdots\!75}a^{14}-\frac{62\!\cdots\!81}{49\!\cdots\!75}a^{13}+\frac{17\!\cdots\!31}{10\!\cdots\!05}a^{12}-\frac{26\!\cdots\!67}{11\!\cdots\!25}a^{11}+\frac{79\!\cdots\!64}{20\!\cdots\!75}a^{10}-\frac{25\!\cdots\!53}{54\!\cdots\!25}a^{9}+\frac{62\!\cdots\!66}{15\!\cdots\!15}a^{8}-\frac{12\!\cdots\!63}{18\!\cdots\!75}a^{7}+\frac{76\!\cdots\!14}{18\!\cdots\!75}a^{6}-\frac{17\!\cdots\!54}{54\!\cdots\!25}a^{5}+\frac{85\!\cdots\!33}{18\!\cdots\!75}a^{4}-\frac{11\!\cdots\!47}{60\!\cdots\!25}a^{3}+\frac{13\!\cdots\!17}{20\!\cdots\!75}a^{2}-\frac{23\!\cdots\!52}{67\!\cdots\!25}a+\frac{14\!\cdots\!72}{74\!\cdots\!25}$, $\frac{19\!\cdots\!74}{18\!\cdots\!75}a^{26}-\frac{39\!\cdots\!34}{60\!\cdots\!25}a^{25}+\frac{45\!\cdots\!03}{16\!\cdots\!25}a^{24}-\frac{74\!\cdots\!72}{60\!\cdots\!25}a^{23}+\frac{87\!\cdots\!34}{18\!\cdots\!75}a^{22}-\frac{54\!\cdots\!11}{36\!\cdots\!35}a^{21}+\frac{75\!\cdots\!28}{18\!\cdots\!75}a^{20}-\frac{20\!\cdots\!57}{18\!\cdots\!75}a^{19}+\frac{15\!\cdots\!08}{60\!\cdots\!25}a^{18}-\frac{18\!\cdots\!11}{36\!\cdots\!35}a^{17}+\frac{23\!\cdots\!53}{18\!\cdots\!75}a^{16}-\frac{95\!\cdots\!49}{36\!\cdots\!35}a^{15}+\frac{40\!\cdots\!93}{86\!\cdots\!75}a^{14}-\frac{56\!\cdots\!23}{60\!\cdots\!75}a^{13}+\frac{34\!\cdots\!79}{24\!\cdots\!69}a^{12}-\frac{53\!\cdots\!22}{25\!\cdots\!25}a^{11}+\frac{55\!\cdots\!32}{18\!\cdots\!75}a^{10}-\frac{70\!\cdots\!21}{18\!\cdots\!75}a^{9}+\frac{35\!\cdots\!74}{86\!\cdots\!75}a^{8}-\frac{89\!\cdots\!49}{18\!\cdots\!75}a^{7}+\frac{83\!\cdots\!66}{18\!\cdots\!75}a^{6}-\frac{66\!\cdots\!44}{18\!\cdots\!75}a^{5}+\frac{59\!\cdots\!52}{18\!\cdots\!75}a^{4}-\frac{19\!\cdots\!09}{67\!\cdots\!25}a^{3}+\frac{23\!\cdots\!17}{22\!\cdots\!75}a^{2}-\frac{12\!\cdots\!23}{89\!\cdots\!47}a+\frac{16\!\cdots\!33}{22\!\cdots\!75}$, $\frac{29\!\cdots\!79}{54\!\cdots\!25}a^{26}-\frac{11\!\cdots\!29}{36\!\cdots\!35}a^{25}+\frac{48\!\cdots\!07}{32\!\cdots\!85}a^{24}-\frac{34\!\cdots\!54}{54\!\cdots\!25}a^{23}+\frac{13\!\cdots\!59}{54\!\cdots\!25}a^{22}-\frac{47\!\cdots\!17}{60\!\cdots\!25}a^{21}+\frac{12\!\cdots\!61}{54\!\cdots\!25}a^{20}-\frac{33\!\cdots\!41}{54\!\cdots\!25}a^{19}+\frac{74\!\cdots\!77}{54\!\cdots\!25}a^{18}-\frac{15\!\cdots\!16}{54\!\cdots\!25}a^{17}+\frac{37\!\cdots\!89}{54\!\cdots\!25}a^{16}-\frac{76\!\cdots\!22}{54\!\cdots\!25}a^{15}+\frac{59\!\cdots\!93}{22\!\cdots\!45}a^{14}-\frac{24\!\cdots\!39}{49\!\cdots\!75}a^{13}+\frac{88\!\cdots\!97}{10\!\cdots\!05}a^{12}-\frac{93\!\cdots\!27}{77\!\cdots\!75}a^{11}+\frac{62\!\cdots\!68}{36\!\cdots\!35}a^{10}-\frac{12\!\cdots\!33}{54\!\cdots\!25}a^{9}+\frac{28\!\cdots\!62}{11\!\cdots\!25}a^{8}-\frac{50\!\cdots\!82}{18\!\cdots\!75}a^{7}+\frac{49\!\cdots\!42}{18\!\cdots\!75}a^{6}-\frac{12\!\cdots\!66}{54\!\cdots\!25}a^{5}+\frac{10\!\cdots\!33}{60\!\cdots\!25}a^{4}-\frac{27\!\cdots\!71}{20\!\cdots\!75}a^{3}+\frac{47\!\cdots\!77}{67\!\cdots\!25}a^{2}-\frac{20\!\cdots\!54}{67\!\cdots\!25}a+\frac{14\!\cdots\!23}{74\!\cdots\!25}$, $\frac{94\!\cdots\!62}{77\!\cdots\!75}a^{26}-\frac{10\!\cdots\!78}{12\!\cdots\!25}a^{25}+\frac{32\!\cdots\!09}{78\!\cdots\!25}a^{24}-\frac{14\!\cdots\!71}{77\!\cdots\!75}a^{23}+\frac{80\!\cdots\!66}{11\!\cdots\!25}a^{22}-\frac{62\!\cdots\!66}{25\!\cdots\!25}a^{21}+\frac{54\!\cdots\!27}{77\!\cdots\!75}a^{20}-\frac{30\!\cdots\!53}{15\!\cdots\!15}a^{19}+\frac{72\!\cdots\!42}{15\!\cdots\!15}a^{18}-\frac{10\!\cdots\!53}{11\!\cdots\!25}a^{17}+\frac{34\!\cdots\!91}{15\!\cdots\!15}a^{16}-\frac{37\!\cdots\!27}{77\!\cdots\!75}a^{15}+\frac{72\!\cdots\!56}{77\!\cdots\!75}a^{14}-\frac{12\!\cdots\!36}{70\!\cdots\!25}a^{13}+\frac{46\!\cdots\!52}{15\!\cdots\!15}a^{12}-\frac{34\!\cdots\!47}{77\!\cdots\!75}a^{11}+\frac{54\!\cdots\!81}{86\!\cdots\!75}a^{10}-\frac{12\!\cdots\!03}{15\!\cdots\!15}a^{9}+\frac{71\!\cdots\!01}{77\!\cdots\!75}a^{8}-\frac{24\!\cdots\!13}{25\!\cdots\!25}a^{7}+\frac{80\!\cdots\!18}{95\!\cdots\!75}a^{6}-\frac{45\!\cdots\!94}{77\!\cdots\!75}a^{5}+\frac{10\!\cdots\!21}{36\!\cdots\!75}a^{4}-\frac{72\!\cdots\!87}{86\!\cdots\!75}a^{3}-\frac{18\!\cdots\!47}{28\!\cdots\!25}a^{2}+\frac{98\!\cdots\!04}{10\!\cdots\!75}a-\frac{32\!\cdots\!54}{15\!\cdots\!25}$, $\frac{18\!\cdots\!39}{54\!\cdots\!25}a^{26}-\frac{39\!\cdots\!72}{18\!\cdots\!75}a^{25}+\frac{15\!\cdots\!73}{16\!\cdots\!25}a^{24}-\frac{43\!\cdots\!22}{10\!\cdots\!05}a^{23}+\frac{84\!\cdots\!79}{54\!\cdots\!25}a^{22}-\frac{87\!\cdots\!13}{18\!\cdots\!75}a^{21}+\frac{70\!\cdots\!27}{54\!\cdots\!25}a^{20}-\frac{18\!\cdots\!04}{54\!\cdots\!25}a^{19}+\frac{41\!\cdots\!38}{54\!\cdots\!25}a^{18}-\frac{78\!\cdots\!43}{54\!\cdots\!25}a^{17}+\frac{19\!\cdots\!31}{54\!\cdots\!25}a^{16}-\frac{40\!\cdots\!36}{54\!\cdots\!25}a^{15}+\frac{10\!\cdots\!07}{77\!\cdots\!75}a^{14}-\frac{22\!\cdots\!07}{98\!\cdots\!55}a^{13}+\frac{39\!\cdots\!44}{10\!\cdots\!05}a^{12}-\frac{52\!\cdots\!31}{11\!\cdots\!25}a^{11}+\frac{12\!\cdots\!43}{20\!\cdots\!75}a^{10}-\frac{40\!\cdots\!82}{54\!\cdots\!25}a^{9}+\frac{54\!\cdots\!03}{77\!\cdots\!75}a^{8}-\frac{27\!\cdots\!07}{44\!\cdots\!35}a^{7}+\frac{10\!\cdots\!77}{18\!\cdots\!75}a^{6}-\frac{33\!\cdots\!31}{10\!\cdots\!05}a^{5}+\frac{30\!\cdots\!21}{20\!\cdots\!75}a^{4}-\frac{33\!\cdots\!54}{12\!\cdots\!45}a^{3}+\frac{34\!\cdots\!51}{20\!\cdots\!75}a^{2}+\frac{23\!\cdots\!86}{22\!\cdots\!75}a-\frac{79\!\cdots\!58}{89\!\cdots\!47}$, $\frac{76\!\cdots\!21}{49\!\cdots\!75}a^{26}-\frac{35\!\cdots\!49}{10\!\cdots\!95}a^{25}-\frac{21\!\cdots\!95}{21\!\cdots\!79}a^{24}+\frac{34\!\cdots\!59}{49\!\cdots\!75}a^{23}-\frac{20\!\cdots\!94}{49\!\cdots\!75}a^{22}+\frac{39\!\cdots\!06}{16\!\cdots\!25}a^{21}-\frac{47\!\cdots\!46}{49\!\cdots\!75}a^{20}+\frac{15\!\cdots\!91}{49\!\cdots\!75}a^{19}-\frac{47\!\cdots\!57}{49\!\cdots\!75}a^{18}+\frac{12\!\cdots\!91}{49\!\cdots\!75}a^{17}-\frac{26\!\cdots\!29}{49\!\cdots\!75}a^{16}+\frac{61\!\cdots\!07}{49\!\cdots\!75}a^{15}-\frac{41\!\cdots\!91}{14\!\cdots\!65}a^{14}+\frac{28\!\cdots\!69}{49\!\cdots\!75}a^{13}-\frac{11\!\cdots\!47}{98\!\cdots\!55}a^{12}+\frac{15\!\cdots\!37}{70\!\cdots\!25}a^{11}-\frac{22\!\cdots\!66}{65\!\cdots\!37}a^{10}+\frac{25\!\cdots\!28}{49\!\cdots\!75}a^{9}-\frac{51\!\cdots\!09}{70\!\cdots\!25}a^{8}+\frac{14\!\cdots\!27}{16\!\cdots\!25}a^{7}-\frac{16\!\cdots\!12}{16\!\cdots\!25}a^{6}+\frac{51\!\cdots\!11}{49\!\cdots\!75}a^{5}-\frac{14\!\cdots\!99}{16\!\cdots\!25}a^{4}+\frac{36\!\cdots\!38}{54\!\cdots\!75}a^{3}-\frac{81\!\cdots\!66}{18\!\cdots\!25}a^{2}+\frac{11\!\cdots\!39}{60\!\cdots\!75}a-\frac{36\!\cdots\!34}{20\!\cdots\!25}$, $\frac{67\!\cdots\!02}{54\!\cdots\!25}a^{26}-\frac{34\!\cdots\!13}{18\!\cdots\!75}a^{25}+\frac{18\!\cdots\!57}{16\!\cdots\!25}a^{24}-\frac{27\!\cdots\!16}{54\!\cdots\!25}a^{23}+\frac{81\!\cdots\!22}{54\!\cdots\!25}a^{22}-\frac{45\!\cdots\!16}{18\!\cdots\!75}a^{21}+\frac{43\!\cdots\!72}{54\!\cdots\!25}a^{20}-\frac{22\!\cdots\!76}{10\!\cdots\!05}a^{19}-\frac{63\!\cdots\!62}{10\!\cdots\!05}a^{18}-\frac{24\!\cdots\!66}{54\!\cdots\!25}a^{17}+\frac{34\!\cdots\!89}{10\!\cdots\!05}a^{16}+\frac{27\!\cdots\!73}{54\!\cdots\!25}a^{15}+\frac{99\!\cdots\!34}{11\!\cdots\!25}a^{14}+\frac{33\!\cdots\!09}{49\!\cdots\!75}a^{13}-\frac{13\!\cdots\!56}{10\!\cdots\!05}a^{12}+\frac{89\!\cdots\!34}{77\!\cdots\!75}a^{11}-\frac{42\!\cdots\!42}{18\!\cdots\!75}a^{10}+\frac{95\!\cdots\!84}{10\!\cdots\!05}a^{9}-\frac{24\!\cdots\!87}{77\!\cdots\!75}a^{8}+\frac{18\!\cdots\!97}{18\!\cdots\!75}a^{7}-\frac{22\!\cdots\!74}{18\!\cdots\!75}a^{6}+\frac{32\!\cdots\!26}{54\!\cdots\!25}a^{5}-\frac{13\!\cdots\!63}{18\!\cdots\!75}a^{4}+\frac{23\!\cdots\!11}{20\!\cdots\!75}a^{3}+\frac{49\!\cdots\!27}{22\!\cdots\!75}a^{2}+\frac{34\!\cdots\!41}{67\!\cdots\!25}a-\frac{83\!\cdots\!29}{22\!\cdots\!75}$, $\frac{15\!\cdots\!94}{54\!\cdots\!25}a^{26}-\frac{36\!\cdots\!99}{18\!\cdots\!75}a^{25}+\frac{13\!\cdots\!81}{16\!\cdots\!25}a^{24}-\frac{20\!\cdots\!11}{54\!\cdots\!25}a^{23}+\frac{80\!\cdots\!74}{54\!\cdots\!25}a^{22}-\frac{17\!\cdots\!57}{36\!\cdots\!35}a^{21}+\frac{70\!\cdots\!13}{54\!\cdots\!25}a^{20}-\frac{19\!\cdots\!12}{54\!\cdots\!25}a^{19}+\frac{44\!\cdots\!74}{54\!\cdots\!25}a^{18}-\frac{17\!\cdots\!01}{10\!\cdots\!05}a^{17}+\frac{21\!\cdots\!43}{54\!\cdots\!25}a^{16}-\frac{94\!\cdots\!97}{10\!\cdots\!05}a^{15}+\frac{11\!\cdots\!04}{77\!\cdots\!75}a^{14}-\frac{14\!\cdots\!96}{49\!\cdots\!75}a^{13}+\frac{50\!\cdots\!54}{10\!\cdots\!05}a^{12}-\frac{51\!\cdots\!02}{77\!\cdots\!75}a^{11}+\frac{18\!\cdots\!94}{18\!\cdots\!75}a^{10}-\frac{68\!\cdots\!51}{54\!\cdots\!25}a^{9}+\frac{10\!\cdots\!62}{77\!\cdots\!75}a^{8}-\frac{29\!\cdots\!38}{18\!\cdots\!75}a^{7}+\frac{26\!\cdots\!02}{18\!\cdots\!75}a^{6}-\frac{60\!\cdots\!89}{54\!\cdots\!25}a^{5}+\frac{21\!\cdots\!99}{18\!\cdots\!75}a^{4}-\frac{16\!\cdots\!99}{20\!\cdots\!75}a^{3}+\frac{23\!\cdots\!07}{67\!\cdots\!25}a^{2}-\frac{68\!\cdots\!77}{13\!\cdots\!05}a+\frac{46\!\cdots\!11}{22\!\cdots\!75}$, $\frac{13\!\cdots\!48}{31\!\cdots\!25}a^{26}-\frac{94\!\cdots\!48}{51\!\cdots\!05}a^{25}+\frac{37\!\cdots\!76}{47\!\cdots\!55}a^{24}-\frac{92\!\cdots\!08}{25\!\cdots\!25}a^{23}+\frac{11\!\cdots\!11}{86\!\cdots\!75}a^{22}-\frac{10\!\cdots\!34}{28\!\cdots\!25}a^{21}+\frac{30\!\cdots\!88}{28\!\cdots\!25}a^{20}-\frac{74\!\cdots\!72}{25\!\cdots\!25}a^{19}+\frac{14\!\cdots\!59}{25\!\cdots\!25}a^{18}-\frac{30\!\cdots\!67}{25\!\cdots\!25}a^{17}+\frac{96\!\cdots\!47}{28\!\cdots\!25}a^{16}-\frac{21\!\cdots\!38}{41\!\cdots\!75}a^{15}+\frac{58\!\cdots\!43}{51\!\cdots\!05}a^{14}-\frac{47\!\cdots\!68}{23\!\cdots\!75}a^{13}+\frac{14\!\cdots\!18}{51\!\cdots\!05}a^{12}-\frac{11\!\cdots\!58}{25\!\cdots\!25}a^{11}+\frac{32\!\cdots\!78}{51\!\cdots\!05}a^{10}-\frac{59\!\cdots\!92}{86\!\cdots\!75}a^{9}+\frac{21\!\cdots\!16}{25\!\cdots\!25}a^{8}-\frac{23\!\cdots\!02}{25\!\cdots\!25}a^{7}+\frac{18\!\cdots\!47}{25\!\cdots\!25}a^{6}-\frac{24\!\cdots\!76}{36\!\cdots\!75}a^{5}+\frac{16\!\cdots\!69}{25\!\cdots\!25}a^{4}-\frac{29\!\cdots\!13}{86\!\cdots\!75}a^{3}+\frac{73\!\cdots\!86}{28\!\cdots\!25}a^{2}-\frac{23\!\cdots\!64}{95\!\cdots\!75}a+\frac{10\!\cdots\!29}{15\!\cdots\!25}$, $\frac{27\!\cdots\!03}{77\!\cdots\!75}a^{26}-\frac{89\!\cdots\!47}{25\!\cdots\!25}a^{25}+\frac{13\!\cdots\!66}{78\!\cdots\!25}a^{24}-\frac{58\!\cdots\!04}{77\!\cdots\!75}a^{23}+\frac{23\!\cdots\!78}{77\!\cdots\!75}a^{22}-\frac{28\!\cdots\!79}{25\!\cdots\!25}a^{21}+\frac{24\!\cdots\!53}{77\!\cdots\!75}a^{20}-\frac{38\!\cdots\!97}{44\!\cdots\!29}a^{19}+\frac{96\!\cdots\!41}{44\!\cdots\!29}a^{18}-\frac{35\!\cdots\!89}{77\!\cdots\!75}a^{17}+\frac{15\!\cdots\!89}{15\!\cdots\!15}a^{16}-\frac{17\!\cdots\!98}{77\!\cdots\!75}a^{15}+\frac{49\!\cdots\!47}{11\!\cdots\!25}a^{14}-\frac{58\!\cdots\!34}{70\!\cdots\!25}a^{13}+\frac{21\!\cdots\!59}{15\!\cdots\!15}a^{12}-\frac{17\!\cdots\!28}{77\!\cdots\!75}a^{11}+\frac{87\!\cdots\!93}{28\!\cdots\!25}a^{10}-\frac{61\!\cdots\!09}{15\!\cdots\!15}a^{9}+\frac{37\!\cdots\!54}{77\!\cdots\!75}a^{8}-\frac{21\!\cdots\!84}{41\!\cdots\!75}a^{7}+\frac{12\!\cdots\!59}{25\!\cdots\!25}a^{6}-\frac{49\!\cdots\!63}{11\!\cdots\!25}a^{5}+\frac{98\!\cdots\!53}{25\!\cdots\!25}a^{4}-\frac{21\!\cdots\!38}{86\!\cdots\!75}a^{3}+\frac{47\!\cdots\!32}{28\!\cdots\!25}a^{2}-\frac{51\!\cdots\!42}{31\!\cdots\!25}a+\frac{23\!\cdots\!19}{31\!\cdots\!25}$, $\frac{22\!\cdots\!67}{54\!\cdots\!25}a^{26}-\frac{12\!\cdots\!14}{18\!\cdots\!75}a^{25}+\frac{43\!\cdots\!46}{16\!\cdots\!25}a^{24}-\frac{64\!\cdots\!39}{54\!\cdots\!25}a^{23}+\frac{27\!\cdots\!87}{54\!\cdots\!25}a^{22}-\frac{31\!\cdots\!42}{18\!\cdots\!75}a^{21}+\frac{51\!\cdots\!06}{10\!\cdots\!05}a^{20}-\frac{71\!\cdots\!14}{54\!\cdots\!25}a^{19}+\frac{18\!\cdots\!73}{54\!\cdots\!25}a^{18}-\frac{34\!\cdots\!37}{54\!\cdots\!25}a^{17}+\frac{76\!\cdots\!61}{54\!\cdots\!25}a^{16}-\frac{20\!\cdots\!39}{54\!\cdots\!25}a^{15}+\frac{42\!\cdots\!59}{77\!\cdots\!75}a^{14}-\frac{61\!\cdots\!89}{49\!\cdots\!75}a^{13}+\frac{22\!\cdots\!01}{10\!\cdots\!05}a^{12}-\frac{21\!\cdots\!96}{77\!\cdots\!75}a^{11}+\frac{85\!\cdots\!21}{20\!\cdots\!75}a^{10}-\frac{31\!\cdots\!02}{54\!\cdots\!25}a^{9}+\frac{86\!\cdots\!38}{15\!\cdots\!15}a^{8}-\frac{13\!\cdots\!97}{18\!\cdots\!75}a^{7}+\frac{53\!\cdots\!07}{74\!\cdots\!25}a^{6}-\frac{26\!\cdots\!76}{54\!\cdots\!25}a^{5}+\frac{97\!\cdots\!77}{18\!\cdots\!75}a^{4}-\frac{29\!\cdots\!42}{67\!\cdots\!25}a^{3}+\frac{95\!\cdots\!86}{67\!\cdots\!25}a^{2}-\frac{16\!\cdots\!78}{67\!\cdots\!25}a+\frac{31\!\cdots\!34}{22\!\cdots\!75}$, $\frac{27\!\cdots\!57}{20\!\cdots\!75}a^{26}-\frac{81\!\cdots\!77}{18\!\cdots\!75}a^{25}+\frac{32\!\cdots\!53}{16\!\cdots\!25}a^{24}-\frac{32\!\cdots\!38}{36\!\cdots\!35}a^{23}+\frac{56\!\cdots\!63}{18\!\cdots\!75}a^{22}-\frac{14\!\cdots\!83}{18\!\cdots\!75}a^{21}+\frac{39\!\cdots\!99}{18\!\cdots\!75}a^{20}-\frac{35\!\cdots\!51}{60\!\cdots\!25}a^{19}+\frac{18\!\cdots\!39}{20\!\cdots\!75}a^{18}-\frac{35\!\cdots\!91}{18\!\cdots\!75}a^{17}+\frac{12\!\cdots\!32}{18\!\cdots\!75}a^{16}-\frac{10\!\cdots\!47}{18\!\cdots\!75}a^{15}+\frac{68\!\cdots\!02}{36\!\cdots\!75}a^{14}-\frac{96\!\cdots\!43}{32\!\cdots\!85}a^{13}+\frac{15\!\cdots\!20}{72\!\cdots\!07}a^{12}-\frac{58\!\cdots\!94}{12\!\cdots\!25}a^{11}+\frac{10\!\cdots\!07}{22\!\cdots\!75}a^{10}+\frac{44\!\cdots\!86}{18\!\cdots\!75}a^{9}+\frac{51\!\cdots\!81}{25\!\cdots\!25}a^{8}+\frac{18\!\cdots\!06}{36\!\cdots\!35}a^{7}-\frac{32\!\cdots\!53}{18\!\cdots\!75}a^{6}+\frac{23\!\cdots\!11}{36\!\cdots\!35}a^{5}-\frac{20\!\cdots\!76}{18\!\cdots\!75}a^{4}+\frac{85\!\cdots\!19}{40\!\cdots\!15}a^{3}+\frac{38\!\cdots\!88}{74\!\cdots\!25}a^{2}+\frac{43\!\cdots\!46}{22\!\cdots\!75}a-\frac{56\!\cdots\!47}{89\!\cdots\!47}$ (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: | \( 29482123669434.47 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 29482123669434.47 \cdot 1}{2\cdot\sqrt{55686266560375612156826962909758952173810975943}}\cr\approx \mathstrut & 2.97182342252825 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 54 |
The 15 conjugacy class representatives for $D_{27}$ |
Character table for $D_{27}$ |
Intermediate fields
3.1.3943.1, 9.1.241716951468001.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $27$ | ${\href{/padicField/3.2.0.1}{2} }^{13}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.2.0.1}{2} }^{13}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.2.0.1}{2} }^{13}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.2.0.1}{2} }^{13}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $27$ | $27$ | ${\href{/padicField/19.3.0.1}{3} }^{9}$ | $27$ | ${\href{/padicField/29.9.0.1}{9} }^{3}$ | $27$ | $27$ | ${\href{/padicField/41.2.0.1}{2} }^{13}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.9.0.1}{9} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{13}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{13}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{13}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3943\) | $\Q_{3943}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |