Normalized defining polynomial
\( x^{27} - 9 x^{26} + 43 x^{25} - 96 x^{24} + 144 x^{23} - 267 x^{22} + 429 x^{21} + 20 x^{20} - 947 x^{19} + 1263 x^{18} + 1070 x^{17} - 4443 x^{16} + 3518 x^{15} + 4729 x^{14} + \cdots + 2075 \)
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: |
\(-20191329826970487018697554420683199956085496127\)
\(\medspace = -\,7^{13}\cdot 521^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(51.88\) | 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))
| |
Ramified primes: |
\(7\), \(521\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3647}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{35}a^{11}-\frac{13}{35}a^{10}-\frac{13}{35}a^{9}-\frac{13}{35}a^{8}-\frac{13}{35}a^{7}+\frac{1}{35}a^{6}+\frac{13}{35}a^{5}-\frac{1}{35}a^{4}+\frac{6}{35}a^{3}-\frac{3}{7}a^{2}+\frac{6}{35}a-\frac{3}{7}$, $\frac{1}{35}a^{12}-\frac{1}{5}a^{10}-\frac{1}{5}a^{9}-\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{9}{35}a^{6}-\frac{1}{5}a^{5}-\frac{1}{5}a^{4}-\frac{1}{5}a^{3}-\frac{2}{5}a^{2}-\frac{1}{5}a+\frac{3}{7}$, $\frac{1}{35}a^{13}+\frac{1}{5}a^{10}+\frac{1}{5}a^{9}-\frac{2}{5}a^{8}+\frac{1}{7}a^{7}+\frac{2}{5}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{13}{35}a$, $\frac{1}{35}a^{14}-\frac{1}{5}a^{10}+\frac{1}{5}a^{9}-\frac{9}{35}a^{8}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}-\frac{2}{5}a^{3}-\frac{13}{35}a^{2}-\frac{1}{5}a$, $\frac{1}{35}a^{15}-\frac{2}{5}a^{10}+\frac{1}{7}a^{9}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a^{4}-\frac{6}{35}a^{3}-\frac{1}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{35}a^{16}-\frac{2}{35}a^{10}-\frac{1}{5}a^{9}+\frac{2}{5}a^{8}-\frac{2}{5}a^{5}+\frac{3}{7}a^{4}+\frac{1}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a$, $\frac{1}{175}a^{17}-\frac{1}{175}a^{16}+\frac{1}{175}a^{15}+\frac{1}{175}a^{13}+\frac{1}{175}a^{12}-\frac{2}{175}a^{11}-\frac{54}{175}a^{10}+\frac{61}{175}a^{9}+\frac{1}{5}a^{8}-\frac{2}{175}a^{7}-\frac{16}{175}a^{6}+\frac{57}{175}a^{5}+\frac{4}{35}a^{4}-\frac{11}{35}a^{3}-\frac{3}{25}a^{2}-\frac{62}{175}a+\frac{2}{7}$, $\frac{1}{175}a^{18}+\frac{1}{175}a^{15}+\frac{1}{175}a^{14}+\frac{2}{175}a^{13}-\frac{1}{175}a^{12}-\frac{1}{175}a^{11}-\frac{8}{175}a^{10}+\frac{81}{175}a^{9}+\frac{18}{175}a^{8}-\frac{33}{175}a^{7}-\frac{79}{175}a^{6}-\frac{83}{175}a^{5}+\frac{17}{35}a^{4}+\frac{79}{175}a^{3}-\frac{33}{175}a^{2}-\frac{32}{175}a-\frac{3}{7}$, $\frac{1}{1225}a^{19}+\frac{1}{1225}a^{18}+\frac{3}{1225}a^{17}-\frac{1}{175}a^{16}-\frac{3}{245}a^{15}+\frac{8}{1225}a^{14}-\frac{6}{1225}a^{13}+\frac{16}{1225}a^{12}+\frac{3}{245}a^{11}-\frac{82}{175}a^{10}+\frac{37}{1225}a^{9}-\frac{41}{245}a^{8}+\frac{457}{1225}a^{7}+\frac{3}{35}a^{6}-\frac{32}{1225}a^{5}-\frac{18}{175}a^{4}+\frac{216}{1225}a^{3}-\frac{328}{1225}a^{2}-\frac{333}{1225}a+\frac{15}{49}$, $\frac{1}{1225}a^{20}+\frac{2}{1225}a^{18}-\frac{3}{1225}a^{17}-\frac{3}{245}a^{16}-\frac{1}{245}a^{15}-\frac{2}{175}a^{14}-\frac{6}{1225}a^{13}+\frac{6}{1225}a^{12}-\frac{8}{1225}a^{11}+\frac{93}{1225}a^{10}+\frac{121}{245}a^{9}-\frac{213}{1225}a^{8}-\frac{436}{1225}a^{7}+\frac{101}{1225}a^{6}-\frac{107}{245}a^{5}-\frac{113}{1225}a^{4}-\frac{579}{1225}a^{3}-\frac{12}{1225}a^{2}+\frac{379}{1225}a-\frac{15}{49}$, $\frac{1}{1225}a^{21}+\frac{2}{1225}a^{18}-\frac{12}{1225}a^{16}+\frac{9}{1225}a^{15}-\frac{3}{245}a^{14}-\frac{17}{1225}a^{13}+\frac{9}{1225}a^{12}+\frac{2}{175}a^{11}+\frac{318}{1225}a^{10}-\frac{82}{175}a^{9}+\frac{69}{245}a^{8}+\frac{524}{1225}a^{7}+\frac{256}{1225}a^{6}-\frac{24}{175}a^{5}-\frac{292}{1225}a^{4}-\frac{591}{1225}a^{3}+\frac{608}{1225}a^{2}+\frac{82}{245}a+\frac{12}{49}$, $\frac{1}{1225}a^{22}-\frac{2}{1225}a^{18}+\frac{3}{1225}a^{17}+\frac{2}{1225}a^{16}+\frac{1}{1225}a^{15}+\frac{2}{1225}a^{14}+\frac{1}{175}a^{13}+\frac{3}{1225}a^{12}+\frac{1}{1225}a^{11}+\frac{1}{7}a^{10}-\frac{338}{1225}a^{9}+\frac{129}{1225}a^{8}-\frac{4}{35}a^{7}+\frac{38}{175}a^{6}+\frac{234}{1225}a^{5}+\frac{326}{1225}a^{4}-\frac{34}{1225}a^{3}-\frac{113}{245}a^{2}+\frac{87}{175}a+\frac{12}{49}$, $\frac{1}{44032625}a^{23}-\frac{16643}{44032625}a^{22}-\frac{1244}{8806525}a^{21}+\frac{1332}{44032625}a^{20}+\frac{916}{44032625}a^{19}+\frac{65157}{44032625}a^{18}-\frac{55838}{44032625}a^{17}-\frac{430902}{44032625}a^{16}+\frac{10729}{898625}a^{15}+\frac{143903}{44032625}a^{14}+\frac{116989}{8806525}a^{13}-\frac{512223}{44032625}a^{12}-\frac{415147}{44032625}a^{11}+\frac{16666339}{44032625}a^{10}-\frac{178393}{3387125}a^{9}+\frac{392046}{1258075}a^{8}+\frac{2794322}{8806525}a^{7}-\frac{2967036}{6290375}a^{6}+\frac{20112137}{44032625}a^{5}+\frac{19998639}{44032625}a^{4}+\frac{15494541}{44032625}a^{3}-\frac{5663993}{44032625}a^{2}-\frac{18964937}{44032625}a+\frac{146378}{1761305}$, $\frac{1}{1893402875}a^{24}+\frac{8}{1893402875}a^{23}-\frac{24336}{145646375}a^{22}-\frac{441683}{1893402875}a^{21}-\frac{321522}{1893402875}a^{20}-\frac{462382}{1893402875}a^{19}+\frac{2892924}{1893402875}a^{18}+\frac{178098}{75736115}a^{17}+\frac{7531299}{1893402875}a^{16}+\frac{17276354}{1893402875}a^{15}+\frac{18880158}{1893402875}a^{14}+\frac{6759047}{1893402875}a^{13}-\frac{3906692}{378680575}a^{12}-\frac{3374544}{270486125}a^{11}-\frac{3714890}{15147223}a^{10}-\frac{552542049}{1893402875}a^{9}+\frac{4312746}{75736115}a^{8}+\frac{701068378}{1893402875}a^{7}-\frac{223197}{677425}a^{6}-\frac{569685819}{1893402875}a^{5}-\frac{3650397}{54097225}a^{4}+\frac{789418808}{1893402875}a^{3}+\frac{7744369}{54097225}a^{2}+\frac{70928514}{270486125}a+\frac{29061023}{75736115}$, $\frac{1}{19310815922125}a^{25}-\frac{4408}{19310815922125}a^{24}-\frac{23323}{3862163184425}a^{23}+\frac{106054986}{2758687988875}a^{22}-\frac{726922324}{19310815922125}a^{21}+\frac{2721713062}{19310815922125}a^{20}-\frac{6880779358}{19310815922125}a^{19}+\frac{11722907778}{19310815922125}a^{18}-\frac{14857757319}{19310815922125}a^{17}-\frac{232017921272}{19310815922125}a^{16}-\frac{4892462241}{551737597775}a^{15}+\frac{1369907409}{1485447378625}a^{14}-\frac{240544771552}{19310815922125}a^{13}-\frac{66054933751}{19310815922125}a^{12}-\frac{128836764934}{19310815922125}a^{11}-\frac{133054197636}{297089475725}a^{10}+\frac{591948599141}{3862163184425}a^{9}-\frac{5240639872547}{19310815922125}a^{8}+\frac{1151008062952}{19310815922125}a^{7}+\frac{827061074447}{2758687988875}a^{6}-\frac{7599442030379}{19310815922125}a^{5}-\frac{1187173342723}{19310815922125}a^{4}+\frac{2134991397143}{19310815922125}a^{3}-\frac{201658403187}{551737597775}a^{2}+\frac{452859215253}{3862163184425}a+\frac{76389771697}{154486527377}$, $\frac{1}{64\!\cdots\!25}a^{26}+\frac{22836}{92\!\cdots\!75}a^{25}-\frac{1021108381}{64\!\cdots\!25}a^{24}+\frac{20333032619}{64\!\cdots\!25}a^{23}+\frac{397398971682214}{64\!\cdots\!25}a^{22}-\frac{322047617119439}{12\!\cdots\!25}a^{21}-\frac{325656430370671}{64\!\cdots\!25}a^{20}+\frac{26694702100179}{25\!\cdots\!05}a^{19}-\frac{12\!\cdots\!03}{64\!\cdots\!25}a^{18}-\frac{280075592992714}{49\!\cdots\!25}a^{17}+\frac{85\!\cdots\!61}{64\!\cdots\!25}a^{16}-\frac{273375427853587}{64\!\cdots\!25}a^{15}-\frac{80\!\cdots\!08}{12\!\cdots\!25}a^{14}-\frac{11\!\cdots\!97}{13\!\cdots\!25}a^{13}+\frac{58\!\cdots\!01}{64\!\cdots\!25}a^{12}+\frac{942503246043277}{13\!\cdots\!25}a^{11}-\frac{25\!\cdots\!22}{12\!\cdots\!25}a^{10}-\frac{36\!\cdots\!79}{92\!\cdots\!75}a^{9}-\frac{24\!\cdots\!13}{64\!\cdots\!25}a^{8}-\frac{17\!\cdots\!39}{64\!\cdots\!25}a^{7}+\frac{50\!\cdots\!06}{64\!\cdots\!25}a^{6}+\frac{23\!\cdots\!86}{64\!\cdots\!25}a^{5}+\frac{98\!\cdots\!33}{64\!\cdots\!25}a^{4}+\frac{69\!\cdots\!07}{64\!\cdots\!25}a^{3}+\frac{13\!\cdots\!99}{12\!\cdots\!25}a^{2}-\frac{16\!\cdots\!13}{64\!\cdots\!25}a+\frac{70\!\cdots\!67}{25\!\cdots\!05}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $5$, $7$ |
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: |
$\frac{11\!\cdots\!59}{64\!\cdots\!25}a^{26}-\frac{15\!\cdots\!53}{92\!\cdots\!75}a^{25}+\frac{50\!\cdots\!66}{64\!\cdots\!25}a^{24}-\frac{20\!\cdots\!52}{12\!\cdots\!25}a^{23}+\frac{12\!\cdots\!44}{64\!\cdots\!25}a^{22}-\frac{23\!\cdots\!18}{64\!\cdots\!25}a^{21}+\frac{42\!\cdots\!47}{64\!\cdots\!25}a^{20}+\frac{22\!\cdots\!61}{64\!\cdots\!25}a^{19}-\frac{13\!\cdots\!27}{64\!\cdots\!25}a^{18}+\frac{76\!\cdots\!23}{64\!\cdots\!25}a^{17}+\frac{29\!\cdots\!64}{64\!\cdots\!25}a^{16}-\frac{69\!\cdots\!96}{64\!\cdots\!25}a^{15}+\frac{21\!\cdots\!61}{64\!\cdots\!25}a^{14}+\frac{15\!\cdots\!96}{92\!\cdots\!75}a^{13}-\frac{12\!\cdots\!77}{64\!\cdots\!25}a^{12}-\frac{64\!\cdots\!73}{92\!\cdots\!75}a^{11}+\frac{82\!\cdots\!92}{25\!\cdots\!05}a^{10}-\frac{74\!\cdots\!32}{92\!\cdots\!75}a^{9}-\frac{22\!\cdots\!93}{49\!\cdots\!25}a^{8}-\frac{27\!\cdots\!52}{25\!\cdots\!05}a^{7}+\frac{18\!\cdots\!13}{64\!\cdots\!25}a^{6}-\frac{23\!\cdots\!52}{64\!\cdots\!25}a^{5}-\frac{69\!\cdots\!91}{20\!\cdots\!75}a^{4}+\frac{10\!\cdots\!93}{12\!\cdots\!25}a^{3}-\frac{34\!\cdots\!52}{12\!\cdots\!25}a^{2}-\frac{15\!\cdots\!23}{64\!\cdots\!25}a-\frac{10\!\cdots\!83}{25\!\cdots\!05}$, $\frac{59389260354803}{49\!\cdots\!25}a^{26}-\frac{199457870579008}{18\!\cdots\!75}a^{25}+\frac{33\!\cdots\!11}{64\!\cdots\!25}a^{24}-\frac{77\!\cdots\!66}{64\!\cdots\!25}a^{23}+\frac{12\!\cdots\!87}{64\!\cdots\!25}a^{22}-\frac{22\!\cdots\!41}{64\!\cdots\!25}a^{21}+\frac{36\!\cdots\!33}{64\!\cdots\!25}a^{20}-\frac{23\!\cdots\!73}{64\!\cdots\!25}a^{19}-\frac{73\!\cdots\!02}{64\!\cdots\!25}a^{18}+\frac{11\!\cdots\!74}{64\!\cdots\!25}a^{17}+\frac{43\!\cdots\!29}{64\!\cdots\!25}a^{16}-\frac{46\!\cdots\!09}{10\!\cdots\!75}a^{15}+\frac{28\!\cdots\!47}{64\!\cdots\!25}a^{14}+\frac{74\!\cdots\!07}{18\!\cdots\!75}a^{13}-\frac{16\!\cdots\!83}{20\!\cdots\!75}a^{12}+\frac{38\!\cdots\!19}{13\!\cdots\!25}a^{11}+\frac{33\!\cdots\!36}{25\!\cdots\!05}a^{10}+\frac{16\!\cdots\!72}{92\!\cdots\!75}a^{9}-\frac{36\!\cdots\!26}{64\!\cdots\!25}a^{8}+\frac{49\!\cdots\!56}{64\!\cdots\!25}a^{7}+\frac{14\!\cdots\!22}{64\!\cdots\!25}a^{6}-\frac{13\!\cdots\!23}{64\!\cdots\!25}a^{5}+\frac{43\!\cdots\!87}{49\!\cdots\!25}a^{4}+\frac{28\!\cdots\!12}{64\!\cdots\!25}a^{3}-\frac{30\!\cdots\!44}{12\!\cdots\!25}a^{2}-\frac{27\!\cdots\!89}{64\!\cdots\!25}a+\frac{16\!\cdots\!56}{25\!\cdots\!05}$, $\frac{25\!\cdots\!79}{64\!\cdots\!25}a^{26}-\frac{35\!\cdots\!12}{92\!\cdots\!75}a^{25}+\frac{12\!\cdots\!57}{64\!\cdots\!25}a^{24}-\frac{34\!\cdots\!84}{64\!\cdots\!25}a^{23}+\frac{64\!\cdots\!87}{64\!\cdots\!25}a^{22}-\frac{12\!\cdots\!12}{64\!\cdots\!25}a^{21}+\frac{20\!\cdots\!32}{64\!\cdots\!25}a^{20}-\frac{33\!\cdots\!28}{15\!\cdots\!75}a^{19}-\frac{14\!\cdots\!38}{64\!\cdots\!25}a^{18}+\frac{79\!\cdots\!63}{99\!\cdots\!25}a^{17}-\frac{97\!\cdots\!12}{20\!\cdots\!75}a^{16}-\frac{69\!\cdots\!78}{64\!\cdots\!25}a^{15}+\frac{14\!\cdots\!31}{64\!\cdots\!25}a^{14}-\frac{26\!\cdots\!59}{92\!\cdots\!75}a^{13}-\frac{14\!\cdots\!44}{64\!\cdots\!25}a^{12}+\frac{16\!\cdots\!34}{59\!\cdots\!25}a^{11}+\frac{29\!\cdots\!06}{12\!\cdots\!25}a^{10}-\frac{12\!\cdots\!06}{92\!\cdots\!75}a^{9}-\frac{14\!\cdots\!28}{64\!\cdots\!25}a^{8}+\frac{18\!\cdots\!22}{51\!\cdots\!01}a^{7}+\frac{34\!\cdots\!91}{64\!\cdots\!25}a^{6}-\frac{65\!\cdots\!98}{64\!\cdots\!25}a^{5}+\frac{79\!\cdots\!03}{64\!\cdots\!25}a^{4}+\frac{14\!\cdots\!79}{16\!\cdots\!75}a^{3}-\frac{16\!\cdots\!89}{25\!\cdots\!05}a^{2}-\frac{52\!\cdots\!62}{64\!\cdots\!25}a+\frac{19\!\cdots\!06}{19\!\cdots\!85}$, $\frac{54298416131889}{49\!\cdots\!25}a^{26}-\frac{902897548145063}{92\!\cdots\!75}a^{25}+\frac{59\!\cdots\!16}{12\!\cdots\!25}a^{24}-\frac{64\!\cdots\!54}{64\!\cdots\!25}a^{23}+\frac{89\!\cdots\!06}{64\!\cdots\!25}a^{22}-\frac{17\!\cdots\!21}{64\!\cdots\!25}a^{21}+\frac{35\!\cdots\!72}{81\!\cdots\!75}a^{20}+\frac{74\!\cdots\!28}{64\!\cdots\!25}a^{19}-\frac{70\!\cdots\!04}{64\!\cdots\!25}a^{18}+\frac{34\!\cdots\!19}{30\!\cdots\!75}a^{17}+\frac{79\!\cdots\!96}{64\!\cdots\!25}a^{16}-\frac{30\!\cdots\!74}{64\!\cdots\!25}a^{15}+\frac{17\!\cdots\!42}{64\!\cdots\!25}a^{14}+\frac{58\!\cdots\!69}{92\!\cdots\!75}a^{13}-\frac{48\!\cdots\!49}{64\!\cdots\!25}a^{12}-\frac{48\!\cdots\!87}{92\!\cdots\!75}a^{11}+\frac{82\!\cdots\!13}{64\!\cdots\!25}a^{10}-\frac{38\!\cdots\!07}{21\!\cdots\!25}a^{9}-\frac{92\!\cdots\!61}{64\!\cdots\!25}a^{8}+\frac{50\!\cdots\!53}{64\!\cdots\!25}a^{7}+\frac{11\!\cdots\!38}{64\!\cdots\!25}a^{6}-\frac{10\!\cdots\!37}{64\!\cdots\!25}a^{5}-\frac{26\!\cdots\!16}{64\!\cdots\!25}a^{4}+\frac{25\!\cdots\!12}{64\!\cdots\!25}a^{3}-\frac{18\!\cdots\!11}{64\!\cdots\!25}a^{2}-\frac{45\!\cdots\!29}{64\!\cdots\!25}a-\frac{10\!\cdots\!39}{25\!\cdots\!05}$, $\frac{43700351657934}{92\!\cdots\!75}a^{26}-\frac{30103309449026}{70\!\cdots\!75}a^{25}+\frac{18\!\cdots\!56}{92\!\cdots\!75}a^{24}-\frac{155960005282377}{36\!\cdots\!15}a^{23}+\frac{50\!\cdots\!39}{92\!\cdots\!75}a^{22}-\frac{89\!\cdots\!22}{92\!\cdots\!75}a^{21}+\frac{13\!\cdots\!96}{92\!\cdots\!75}a^{20}+\frac{18\!\cdots\!06}{13\!\cdots\!25}a^{19}-\frac{59\!\cdots\!21}{92\!\cdots\!75}a^{18}+\frac{71\!\cdots\!97}{92\!\cdots\!75}a^{17}+\frac{30\!\cdots\!18}{92\!\cdots\!75}a^{16}-\frac{17\!\cdots\!82}{92\!\cdots\!75}a^{15}+\frac{10\!\cdots\!53}{92\!\cdots\!75}a^{14}+\frac{32\!\cdots\!29}{92\!\cdots\!75}a^{13}-\frac{47\!\cdots\!04}{92\!\cdots\!75}a^{12}+\frac{24\!\cdots\!59}{92\!\cdots\!75}a^{11}+\frac{50\!\cdots\!02}{92\!\cdots\!75}a^{10}+\frac{29\!\cdots\!43}{92\!\cdots\!75}a^{9}-\frac{16\!\cdots\!97}{26\!\cdots\!25}a^{8}+\frac{44\!\cdots\!33}{92\!\cdots\!75}a^{7}+\frac{38\!\cdots\!64}{92\!\cdots\!75}a^{6}-\frac{29\!\cdots\!54}{92\!\cdots\!75}a^{5}+\frac{19\!\cdots\!47}{92\!\cdots\!75}a^{4}+\frac{25\!\cdots\!99}{92\!\cdots\!75}a^{3}+\frac{17\!\cdots\!21}{92\!\cdots\!75}a^{2}-\frac{17\!\cdots\!67}{92\!\cdots\!75}a+\frac{408686444194319}{784906602233845}$, $\frac{135944007}{93760174328275}a^{26}-\frac{230580102}{9567364727375}a^{25}+\frac{83112830599}{468800871641375}a^{24}-\frac{350455873827}{468800871641375}a^{23}+\frac{36240216204}{18752034865655}a^{22}-\frac{1709470745288}{468800871641375}a^{21}+\frac{678130683444}{93760174328275}a^{20}-\frac{426325675136}{36061605510875}a^{19}+\frac{3819732365742}{468800871641375}a^{18}+\frac{233401371461}{36061605510875}a^{17}-\frac{1392268980078}{93760174328275}a^{16}+\frac{7434794265868}{468800871641375}a^{15}+\frac{5044772262808}{468800871641375}a^{14}-\frac{1950127818602}{66971553091625}a^{13}+\frac{18249488499754}{468800871641375}a^{12}+\frac{697847176928}{66971553091625}a^{11}+\frac{234540489}{1163277597125}a^{10}+\frac{4042785121189}{66971553091625}a^{9}-\frac{5659484479709}{468800871641375}a^{8}+\frac{7275729022973}{36061605510875}a^{7}+\frac{129468758084327}{468800871641375}a^{6}+\frac{174464049770663}{468800871641375}a^{5}+\frac{305232529877461}{468800871641375}a^{4}-\frac{172567854271581}{468800871641375}a^{3}+\frac{221635898507421}{468800871641375}a^{2}+\frac{398467816665794}{468800871641375}a+\frac{23685993561059}{18752034865655}$, $\frac{78378804390422}{25\!\cdots\!05}a^{26}-\frac{36121310865336}{14\!\cdots\!75}a^{25}+\frac{74\!\cdots\!94}{64\!\cdots\!25}a^{24}-\frac{28\!\cdots\!52}{12\!\cdots\!25}a^{23}+\frac{20\!\cdots\!19}{64\!\cdots\!25}a^{22}-\frac{42\!\cdots\!17}{64\!\cdots\!25}a^{21}+\frac{48\!\cdots\!71}{49\!\cdots\!25}a^{20}+\frac{29\!\cdots\!90}{51\!\cdots\!01}a^{19}-\frac{15\!\cdots\!03}{64\!\cdots\!25}a^{18}+\frac{15\!\cdots\!21}{64\!\cdots\!25}a^{17}+\frac{51\!\cdots\!38}{12\!\cdots\!25}a^{16}-\frac{61\!\cdots\!91}{64\!\cdots\!25}a^{15}+\frac{23\!\cdots\!81}{64\!\cdots\!25}a^{14}+\frac{13\!\cdots\!14}{92\!\cdots\!75}a^{13}-\frac{57\!\cdots\!04}{64\!\cdots\!25}a^{12}-\frac{23\!\cdots\!09}{92\!\cdots\!75}a^{11}+\frac{20\!\cdots\!17}{64\!\cdots\!25}a^{10}+\frac{38\!\cdots\!93}{13\!\cdots\!25}a^{9}-\frac{48\!\cdots\!06}{25\!\cdots\!05}a^{8}-\frac{55\!\cdots\!38}{64\!\cdots\!25}a^{7}+\frac{45\!\cdots\!09}{64\!\cdots\!25}a^{6}-\frac{15\!\cdots\!24}{12\!\cdots\!25}a^{5}-\frac{31\!\cdots\!48}{64\!\cdots\!25}a^{4}+\frac{21\!\cdots\!12}{12\!\cdots\!25}a^{3}+\frac{60\!\cdots\!36}{64\!\cdots\!25}a^{2}-\frac{65\!\cdots\!64}{64\!\cdots\!25}a-\frac{21\!\cdots\!84}{25\!\cdots\!05}$, $\frac{140967034366906}{64\!\cdots\!25}a^{26}-\frac{28506386777677}{13\!\cdots\!25}a^{25}+\frac{58178315852217}{51\!\cdots\!01}a^{24}-\frac{14\!\cdots\!77}{49\!\cdots\!25}a^{23}+\frac{204615166494356}{41\!\cdots\!75}a^{22}-\frac{46\!\cdots\!03}{64\!\cdots\!25}a^{21}+\frac{74\!\cdots\!36}{64\!\cdots\!25}a^{20}-\frac{10\!\cdots\!09}{25\!\cdots\!05}a^{19}-\frac{85\!\cdots\!63}{25\!\cdots\!05}a^{18}+\frac{44\!\cdots\!52}{64\!\cdots\!25}a^{17}+\frac{28\!\cdots\!21}{64\!\cdots\!25}a^{16}-\frac{91\!\cdots\!86}{64\!\cdots\!25}a^{15}+\frac{12\!\cdots\!34}{64\!\cdots\!25}a^{14}-\frac{60217690714042}{11\!\cdots\!25}a^{13}-\frac{46\!\cdots\!82}{12\!\cdots\!25}a^{12}+\frac{29\!\cdots\!78}{92\!\cdots\!75}a^{11}+\frac{32\!\cdots\!88}{64\!\cdots\!25}a^{10}-\frac{18\!\cdots\!51}{73\!\cdots\!43}a^{9}-\frac{91\!\cdots\!48}{64\!\cdots\!25}a^{8}+\frac{10\!\cdots\!19}{64\!\cdots\!25}a^{7}+\frac{14\!\cdots\!62}{64\!\cdots\!25}a^{6}-\frac{54\!\cdots\!94}{64\!\cdots\!25}a^{5}+\frac{68\!\cdots\!01}{64\!\cdots\!25}a^{4}+\frac{18\!\cdots\!78}{81\!\cdots\!75}a^{3}-\frac{75\!\cdots\!71}{64\!\cdots\!25}a^{2}-\frac{97\!\cdots\!59}{64\!\cdots\!25}a+\frac{29\!\cdots\!71}{25\!\cdots\!05}$, $\frac{30\!\cdots\!88}{64\!\cdots\!25}a^{26}-\frac{116924489840968}{29\!\cdots\!25}a^{25}+\frac{11\!\cdots\!99}{64\!\cdots\!25}a^{24}-\frac{19\!\cdots\!04}{64\!\cdots\!25}a^{23}+\frac{51\!\cdots\!99}{15\!\cdots\!75}a^{22}-\frac{73\!\cdots\!06}{99\!\cdots\!25}a^{21}+\frac{67\!\cdots\!13}{64\!\cdots\!25}a^{20}+\frac{21\!\cdots\!23}{12\!\cdots\!25}a^{19}-\frac{29\!\cdots\!94}{64\!\cdots\!25}a^{18}+\frac{17\!\cdots\!14}{64\!\cdots\!25}a^{17}+\frac{64\!\cdots\!49}{64\!\cdots\!25}a^{16}-\frac{12\!\cdots\!46}{64\!\cdots\!25}a^{15}+\frac{27\!\cdots\!02}{64\!\cdots\!25}a^{14}+\frac{99\!\cdots\!56}{29\!\cdots\!25}a^{13}-\frac{15\!\cdots\!53}{64\!\cdots\!25}a^{12}-\frac{78\!\cdots\!19}{92\!\cdots\!75}a^{11}+\frac{35\!\cdots\!13}{49\!\cdots\!25}a^{10}+\frac{23\!\cdots\!74}{92\!\cdots\!75}a^{9}-\frac{98\!\cdots\!33}{25\!\cdots\!05}a^{8}+\frac{15\!\cdots\!82}{64\!\cdots\!25}a^{7}+\frac{58\!\cdots\!78}{64\!\cdots\!25}a^{6}-\frac{41\!\cdots\!71}{64\!\cdots\!25}a^{5}-\frac{51\!\cdots\!76}{64\!\cdots\!25}a^{4}+\frac{16\!\cdots\!54}{64\!\cdots\!25}a^{3}+\frac{97\!\cdots\!72}{64\!\cdots\!25}a^{2}-\frac{13\!\cdots\!88}{64\!\cdots\!25}a-\frac{41\!\cdots\!28}{25\!\cdots\!05}$, $\frac{18\!\cdots\!89}{64\!\cdots\!25}a^{26}-\frac{382502559002826}{13\!\cdots\!25}a^{25}+\frac{99\!\cdots\!74}{64\!\cdots\!25}a^{24}-\frac{28\!\cdots\!51}{64\!\cdots\!25}a^{23}+\frac{22\!\cdots\!61}{27\!\cdots\!75}a^{22}-\frac{89\!\cdots\!63}{64\!\cdots\!25}a^{21}+\frac{13\!\cdots\!62}{64\!\cdots\!25}a^{20}-\frac{73\!\cdots\!01}{64\!\cdots\!25}a^{19}-\frac{23\!\cdots\!31}{64\!\cdots\!25}a^{18}+\frac{15\!\cdots\!03}{15\!\cdots\!75}a^{17}-\frac{43\!\cdots\!56}{49\!\cdots\!25}a^{16}-\frac{43\!\cdots\!43}{64\!\cdots\!25}a^{15}+\frac{15\!\cdots\!63}{64\!\cdots\!25}a^{14}-\frac{13\!\cdots\!77}{92\!\cdots\!75}a^{13}-\frac{19\!\cdots\!72}{12\!\cdots\!25}a^{12}+\frac{65\!\cdots\!53}{18\!\cdots\!75}a^{11}-\frac{28\!\cdots\!69}{49\!\cdots\!25}a^{10}-\frac{13\!\cdots\!54}{92\!\cdots\!75}a^{9}+\frac{87\!\cdots\!27}{64\!\cdots\!25}a^{8}+\frac{86\!\cdots\!01}{64\!\cdots\!25}a^{7}+\frac{25\!\cdots\!22}{64\!\cdots\!25}a^{6}-\frac{61\!\cdots\!67}{64\!\cdots\!25}a^{5}+\frac{79\!\cdots\!11}{64\!\cdots\!25}a^{4}+\frac{55\!\cdots\!52}{12\!\cdots\!25}a^{3}-\frac{98\!\cdots\!46}{64\!\cdots\!25}a^{2}+\frac{11\!\cdots\!06}{12\!\cdots\!25}a-\frac{47\!\cdots\!93}{51\!\cdots\!01}$, $\frac{6695897567}{468800871641375}a^{26}-\frac{9863796}{76538917819}a^{25}+\frac{280571090011}{468800871641375}a^{24}-\frac{565231937353}{468800871641375}a^{23}+\frac{550681081744}{468800871641375}a^{22}-\frac{700043322702}{468800871641375}a^{21}+\frac{182154983326}{93760174328275}a^{20}+\frac{3993167192876}{468800871641375}a^{19}-\frac{13513183851582}{468800871641375}a^{18}+\frac{13784719176629}{468800871641375}a^{17}+\frac{10977972577042}{468800871641375}a^{16}-\frac{38802159632914}{468800871641375}a^{15}+\frac{20179394499148}{468800871641375}a^{14}+\frac{7324795530112}{66971553091625}a^{13}-\frac{78682074424322}{468800871641375}a^{12}+\frac{2583525605666}{66971553091625}a^{11}+\frac{116758789216679}{468800871641375}a^{10}-\frac{909268618509}{5151657930125}a^{9}-\frac{49883700126303}{468800871641375}a^{8}+\frac{59603271271527}{468800871641375}a^{7}+\frac{146202608706659}{468800871641375}a^{6}-\frac{280605849127894}{468800871641375}a^{5}+\frac{15094996466619}{36061605510875}a^{4}+\frac{157790698152551}{468800871641375}a^{3}-\frac{90872287756098}{468800871641375}a^{2}-\frac{2305658274558}{3750406973131}a+\frac{4786241078783}{3750406973131}$, $\frac{5076242311062}{49\!\cdots\!25}a^{26}-\frac{159918308489664}{92\!\cdots\!75}a^{25}+\frac{293449616015516}{25\!\cdots\!05}a^{24}-\frac{26\!\cdots\!74}{64\!\cdots\!25}a^{23}+\frac{47\!\cdots\!09}{64\!\cdots\!25}a^{22}-\frac{63\!\cdots\!23}{64\!\cdots\!25}a^{21}+\frac{27\!\cdots\!04}{12\!\cdots\!25}a^{20}-\frac{20\!\cdots\!08}{64\!\cdots\!25}a^{19}-\frac{81\!\cdots\!87}{49\!\cdots\!25}a^{18}+\frac{51\!\cdots\!91}{64\!\cdots\!25}a^{17}+\frac{41\!\cdots\!87}{64\!\cdots\!25}a^{16}-\frac{38\!\cdots\!57}{13\!\cdots\!75}a^{15}+\frac{10\!\cdots\!68}{41\!\cdots\!75}a^{14}+\frac{22\!\cdots\!04}{70\!\cdots\!75}a^{13}-\frac{48\!\cdots\!21}{64\!\cdots\!25}a^{12}-\frac{43\!\cdots\!44}{92\!\cdots\!75}a^{11}+\frac{55\!\cdots\!86}{64\!\cdots\!25}a^{10}-\frac{16\!\cdots\!43}{16\!\cdots\!75}a^{9}-\frac{45\!\cdots\!36}{49\!\cdots\!25}a^{8}+\frac{61\!\cdots\!59}{64\!\cdots\!25}a^{7}+\frac{10\!\cdots\!16}{49\!\cdots\!25}a^{6}-\frac{10\!\cdots\!60}{51\!\cdots\!01}a^{5}-\frac{20\!\cdots\!16}{64\!\cdots\!25}a^{4}+\frac{70\!\cdots\!99}{64\!\cdots\!25}a^{3}-\frac{41\!\cdots\!02}{64\!\cdots\!25}a^{2}-\frac{24\!\cdots\!13}{64\!\cdots\!25}a+\frac{36\!\cdots\!72}{25\!\cdots\!05}$, $\frac{62098536544473}{49\!\cdots\!25}a^{26}-\frac{10\!\cdots\!54}{92\!\cdots\!75}a^{25}+\frac{33\!\cdots\!74}{64\!\cdots\!25}a^{24}-\frac{71\!\cdots\!08}{64\!\cdots\!25}a^{23}+\frac{10\!\cdots\!76}{64\!\cdots\!25}a^{22}-\frac{14\!\cdots\!07}{49\!\cdots\!25}a^{21}+\frac{27\!\cdots\!66}{64\!\cdots\!25}a^{20}+\frac{13\!\cdots\!99}{64\!\cdots\!25}a^{19}-\frac{65\!\cdots\!61}{49\!\cdots\!25}a^{18}+\frac{11\!\cdots\!43}{64\!\cdots\!25}a^{17}+\frac{51\!\cdots\!48}{49\!\cdots\!25}a^{16}-\frac{34\!\cdots\!47}{64\!\cdots\!25}a^{15}+\frac{32\!\cdots\!36}{64\!\cdots\!25}a^{14}+\frac{18\!\cdots\!08}{39\!\cdots\!25}a^{13}-\frac{59\!\cdots\!18}{64\!\cdots\!25}a^{12}+\frac{52\!\cdots\!42}{92\!\cdots\!75}a^{11}+\frac{95\!\cdots\!14}{64\!\cdots\!25}a^{10}-\frac{98\!\cdots\!83}{92\!\cdots\!75}a^{9}-\frac{67\!\cdots\!46}{12\!\cdots\!25}a^{8}+\frac{33\!\cdots\!04}{49\!\cdots\!25}a^{7}+\frac{13\!\cdots\!33}{64\!\cdots\!25}a^{6}-\frac{11\!\cdots\!63}{60\!\cdots\!35}a^{5}+\frac{40\!\cdots\!19}{64\!\cdots\!25}a^{4}+\frac{47\!\cdots\!41}{64\!\cdots\!25}a^{3}+\frac{45\!\cdots\!14}{49\!\cdots\!25}a^{2}-\frac{64\!\cdots\!92}{13\!\cdots\!75}a+\frac{58\!\cdots\!16}{25\!\cdots\!05}$
| 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: | \( 16038112241729.484 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 16038112241729.484 \cdot 1}{2\cdot\sqrt{20191329826970487018697554420683199956085496127}}\cr\approx \mathstrut & 2.68478229796354 \end{aligned}\] (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
3.1.3647.1, 9.1.176906199770881.1 |
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 | $27$ | $27$ | ${\href{/padicField/5.2.0.1}{2} }^{13}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | ${\href{/padicField/11.9.0.1}{9} }^{3}$ | ${\href{/padicField/13.2.0.1}{2} }^{13}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $27$ | ${\href{/padicField/19.9.0.1}{9} }^{3}$ | ${\href{/padicField/23.2.0.1}{2} }^{13}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.3.0.1}{3} }^{9}$ | ${\href{/padicField/31.2.0.1}{2} }^{13}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $27$ | $27$ | ${\href{/padicField/43.2.0.1}{2} }^{13}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }^{13}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $27$ | ${\href{/padicField/59.2.0.1}{2} }^{13}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(521\)
| $\Q_{521}$ | $x$ | $1$ | $1$ | $0$ | 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |