Properties

Label 27.1.201...127.1
Degree $27$
Signature $[1, 13]$
Discriminant $-2.019\times 10^{46}$
Root discriminant \(51.88\)
Ramified primes $7,521$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{27}$ (as 27T8)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(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 - 7874*x^13 + 2497*x^12 + 13136*x^11 - 1996*x^10 - 8600*x^9 + 6002*x^8 + 17843*x^7 - 14482*x^6 + 6811*x^5 + 47488*x^4 + 900*x^3 - 46897*x^2 + 24633*x + 2075)
 
gp: K = bnfinit(y^27 - 9*y^26 + 43*y^25 - 96*y^24 + 144*y^23 - 267*y^22 + 429*y^21 + 20*y^20 - 947*y^19 + 1263*y^18 + 1070*y^17 - 4443*y^16 + 3518*y^15 + 4729*y^14 - 7874*y^13 + 2497*y^12 + 13136*y^11 - 1996*y^10 - 8600*y^9 + 6002*y^8 + 17843*y^7 - 14482*y^6 + 6811*y^5 + 47488*y^4 + 900*y^3 - 46897*y^2 + 24633*y + 2075, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(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 - 7874*x^13 + 2497*x^12 + 13136*x^11 - 1996*x^10 - 8600*x^9 + 6002*x^8 + 17843*x^7 - 14482*x^6 + 6811*x^5 + 47488*x^4 + 900*x^3 - 46897*x^2 + 24633*x + 2075);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(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 - 7874*x^13 + 2497*x^12 + 13136*x^11 - 1996*x^10 - 8600*x^9 + 6002*x^8 + 17843*x^7 - 14482*x^6 + 6811*x^5 + 47488*x^4 + 900*x^3 - 46897*x^2 + 24633*x + 2075)
 

\( x^{27} - 9 x^{26} + 43 x^{25} - 96 x^{24} + 144 x^{23} - 267 x^{22} + 429 x^{21} + 20 x^{20} - 947 x^{19} + 1263 x^{18} + 1070 x^{17} - 4443 x^{16} + 3518 x^{15} + 4729 x^{14} + \cdots + 2075 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

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:   \(-20191329826970487018697554420683199956085496127\) \(\medspace = -\,7^{13}\cdot 521^{13}\) Copy content Toggle raw display
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:   \(7\), \(521\) Copy content Toggle raw display
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{35}a^{11}-\frac{13}{35}a^{10}-\frac{13}{35}a^{9}-\frac{13}{35}a^{8}-\frac{13}{35}a^{7}+\frac{1}{35}a^{6}+\frac{13}{35}a^{5}-\frac{1}{35}a^{4}+\frac{6}{35}a^{3}-\frac{3}{7}a^{2}+\frac{6}{35}a-\frac{3}{7}$, $\frac{1}{35}a^{12}-\frac{1}{5}a^{10}-\frac{1}{5}a^{9}-\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{9}{35}a^{6}-\frac{1}{5}a^{5}-\frac{1}{5}a^{4}-\frac{1}{5}a^{3}-\frac{2}{5}a^{2}-\frac{1}{5}a+\frac{3}{7}$, $\frac{1}{35}a^{13}+\frac{1}{5}a^{10}+\frac{1}{5}a^{9}-\frac{2}{5}a^{8}+\frac{1}{7}a^{7}+\frac{2}{5}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{13}{35}a$, $\frac{1}{35}a^{14}-\frac{1}{5}a^{10}+\frac{1}{5}a^{9}-\frac{9}{35}a^{8}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}-\frac{2}{5}a^{3}-\frac{13}{35}a^{2}-\frac{1}{5}a$, $\frac{1}{35}a^{15}-\frac{2}{5}a^{10}+\frac{1}{7}a^{9}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a^{4}-\frac{6}{35}a^{3}-\frac{1}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{35}a^{16}-\frac{2}{35}a^{10}-\frac{1}{5}a^{9}+\frac{2}{5}a^{8}-\frac{2}{5}a^{5}+\frac{3}{7}a^{4}+\frac{1}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a$, $\frac{1}{175}a^{17}-\frac{1}{175}a^{16}+\frac{1}{175}a^{15}+\frac{1}{175}a^{13}+\frac{1}{175}a^{12}-\frac{2}{175}a^{11}-\frac{54}{175}a^{10}+\frac{61}{175}a^{9}+\frac{1}{5}a^{8}-\frac{2}{175}a^{7}-\frac{16}{175}a^{6}+\frac{57}{175}a^{5}+\frac{4}{35}a^{4}-\frac{11}{35}a^{3}-\frac{3}{25}a^{2}-\frac{62}{175}a+\frac{2}{7}$, $\frac{1}{175}a^{18}+\frac{1}{175}a^{15}+\frac{1}{175}a^{14}+\frac{2}{175}a^{13}-\frac{1}{175}a^{12}-\frac{1}{175}a^{11}-\frac{8}{175}a^{10}+\frac{81}{175}a^{9}+\frac{18}{175}a^{8}-\frac{33}{175}a^{7}-\frac{79}{175}a^{6}-\frac{83}{175}a^{5}+\frac{17}{35}a^{4}+\frac{79}{175}a^{3}-\frac{33}{175}a^{2}-\frac{32}{175}a-\frac{3}{7}$, $\frac{1}{1225}a^{19}+\frac{1}{1225}a^{18}+\frac{3}{1225}a^{17}-\frac{1}{175}a^{16}-\frac{3}{245}a^{15}+\frac{8}{1225}a^{14}-\frac{6}{1225}a^{13}+\frac{16}{1225}a^{12}+\frac{3}{245}a^{11}-\frac{82}{175}a^{10}+\frac{37}{1225}a^{9}-\frac{41}{245}a^{8}+\frac{457}{1225}a^{7}+\frac{3}{35}a^{6}-\frac{32}{1225}a^{5}-\frac{18}{175}a^{4}+\frac{216}{1225}a^{3}-\frac{328}{1225}a^{2}-\frac{333}{1225}a+\frac{15}{49}$, $\frac{1}{1225}a^{20}+\frac{2}{1225}a^{18}-\frac{3}{1225}a^{17}-\frac{3}{245}a^{16}-\frac{1}{245}a^{15}-\frac{2}{175}a^{14}-\frac{6}{1225}a^{13}+\frac{6}{1225}a^{12}-\frac{8}{1225}a^{11}+\frac{93}{1225}a^{10}+\frac{121}{245}a^{9}-\frac{213}{1225}a^{8}-\frac{436}{1225}a^{7}+\frac{101}{1225}a^{6}-\frac{107}{245}a^{5}-\frac{113}{1225}a^{4}-\frac{579}{1225}a^{3}-\frac{12}{1225}a^{2}+\frac{379}{1225}a-\frac{15}{49}$, $\frac{1}{1225}a^{21}+\frac{2}{1225}a^{18}-\frac{12}{1225}a^{16}+\frac{9}{1225}a^{15}-\frac{3}{245}a^{14}-\frac{17}{1225}a^{13}+\frac{9}{1225}a^{12}+\frac{2}{175}a^{11}+\frac{318}{1225}a^{10}-\frac{82}{175}a^{9}+\frac{69}{245}a^{8}+\frac{524}{1225}a^{7}+\frac{256}{1225}a^{6}-\frac{24}{175}a^{5}-\frac{292}{1225}a^{4}-\frac{591}{1225}a^{3}+\frac{608}{1225}a^{2}+\frac{82}{245}a+\frac{12}{49}$, $\frac{1}{1225}a^{22}-\frac{2}{1225}a^{18}+\frac{3}{1225}a^{17}+\frac{2}{1225}a^{16}+\frac{1}{1225}a^{15}+\frac{2}{1225}a^{14}+\frac{1}{175}a^{13}+\frac{3}{1225}a^{12}+\frac{1}{1225}a^{11}+\frac{1}{7}a^{10}-\frac{338}{1225}a^{9}+\frac{129}{1225}a^{8}-\frac{4}{35}a^{7}+\frac{38}{175}a^{6}+\frac{234}{1225}a^{5}+\frac{326}{1225}a^{4}-\frac{34}{1225}a^{3}-\frac{113}{245}a^{2}+\frac{87}{175}a+\frac{12}{49}$, $\frac{1}{44032625}a^{23}-\frac{16643}{44032625}a^{22}-\frac{1244}{8806525}a^{21}+\frac{1332}{44032625}a^{20}+\frac{916}{44032625}a^{19}+\frac{65157}{44032625}a^{18}-\frac{55838}{44032625}a^{17}-\frac{430902}{44032625}a^{16}+\frac{10729}{898625}a^{15}+\frac{143903}{44032625}a^{14}+\frac{116989}{8806525}a^{13}-\frac{512223}{44032625}a^{12}-\frac{415147}{44032625}a^{11}+\frac{16666339}{44032625}a^{10}-\frac{178393}{3387125}a^{9}+\frac{392046}{1258075}a^{8}+\frac{2794322}{8806525}a^{7}-\frac{2967036}{6290375}a^{6}+\frac{20112137}{44032625}a^{5}+\frac{19998639}{44032625}a^{4}+\frac{15494541}{44032625}a^{3}-\frac{5663993}{44032625}a^{2}-\frac{18964937}{44032625}a+\frac{146378}{1761305}$, $\frac{1}{1893402875}a^{24}+\frac{8}{1893402875}a^{23}-\frac{24336}{145646375}a^{22}-\frac{441683}{1893402875}a^{21}-\frac{321522}{1893402875}a^{20}-\frac{462382}{1893402875}a^{19}+\frac{2892924}{1893402875}a^{18}+\frac{178098}{75736115}a^{17}+\frac{7531299}{1893402875}a^{16}+\frac{17276354}{1893402875}a^{15}+\frac{18880158}{1893402875}a^{14}+\frac{6759047}{1893402875}a^{13}-\frac{3906692}{378680575}a^{12}-\frac{3374544}{270486125}a^{11}-\frac{3714890}{15147223}a^{10}-\frac{552542049}{1893402875}a^{9}+\frac{4312746}{75736115}a^{8}+\frac{701068378}{1893402875}a^{7}-\frac{223197}{677425}a^{6}-\frac{569685819}{1893402875}a^{5}-\frac{3650397}{54097225}a^{4}+\frac{789418808}{1893402875}a^{3}+\frac{7744369}{54097225}a^{2}+\frac{70928514}{270486125}a+\frac{29061023}{75736115}$, $\frac{1}{19310815922125}a^{25}-\frac{4408}{19310815922125}a^{24}-\frac{23323}{3862163184425}a^{23}+\frac{106054986}{2758687988875}a^{22}-\frac{726922324}{19310815922125}a^{21}+\frac{2721713062}{19310815922125}a^{20}-\frac{6880779358}{19310815922125}a^{19}+\frac{11722907778}{19310815922125}a^{18}-\frac{14857757319}{19310815922125}a^{17}-\frac{232017921272}{19310815922125}a^{16}-\frac{4892462241}{551737597775}a^{15}+\frac{1369907409}{1485447378625}a^{14}-\frac{240544771552}{19310815922125}a^{13}-\frac{66054933751}{19310815922125}a^{12}-\frac{128836764934}{19310815922125}a^{11}-\frac{133054197636}{297089475725}a^{10}+\frac{591948599141}{3862163184425}a^{9}-\frac{5240639872547}{19310815922125}a^{8}+\frac{1151008062952}{19310815922125}a^{7}+\frac{827061074447}{2758687988875}a^{6}-\frac{7599442030379}{19310815922125}a^{5}-\frac{1187173342723}{19310815922125}a^{4}+\frac{2134991397143}{19310815922125}a^{3}-\frac{201658403187}{551737597775}a^{2}+\frac{452859215253}{3862163184425}a+\frac{76389771697}{154486527377}$, $\frac{1}{64\!\cdots\!25}a^{26}+\frac{22836}{92\!\cdots\!75}a^{25}-\frac{1021108381}{64\!\cdots\!25}a^{24}+\frac{20333032619}{64\!\cdots\!25}a^{23}+\frac{397398971682214}{64\!\cdots\!25}a^{22}-\frac{322047617119439}{12\!\cdots\!25}a^{21}-\frac{325656430370671}{64\!\cdots\!25}a^{20}+\frac{26694702100179}{25\!\cdots\!05}a^{19}-\frac{12\!\cdots\!03}{64\!\cdots\!25}a^{18}-\frac{280075592992714}{49\!\cdots\!25}a^{17}+\frac{85\!\cdots\!61}{64\!\cdots\!25}a^{16}-\frac{273375427853587}{64\!\cdots\!25}a^{15}-\frac{80\!\cdots\!08}{12\!\cdots\!25}a^{14}-\frac{11\!\cdots\!97}{13\!\cdots\!25}a^{13}+\frac{58\!\cdots\!01}{64\!\cdots\!25}a^{12}+\frac{942503246043277}{13\!\cdots\!25}a^{11}-\frac{25\!\cdots\!22}{12\!\cdots\!25}a^{10}-\frac{36\!\cdots\!79}{92\!\cdots\!75}a^{9}-\frac{24\!\cdots\!13}{64\!\cdots\!25}a^{8}-\frac{17\!\cdots\!39}{64\!\cdots\!25}a^{7}+\frac{50\!\cdots\!06}{64\!\cdots\!25}a^{6}+\frac{23\!\cdots\!86}{64\!\cdots\!25}a^{5}+\frac{98\!\cdots\!33}{64\!\cdots\!25}a^{4}+\frac{69\!\cdots\!07}{64\!\cdots\!25}a^{3}+\frac{13\!\cdots\!99}{12\!\cdots\!25}a^{2}-\frac{16\!\cdots\!13}{64\!\cdots\!25}a+\frac{70\!\cdots\!67}{25\!\cdots\!05}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $5$, $7$

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:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $\frac{11\!\cdots\!59}{64\!\cdots\!25}a^{26}-\frac{15\!\cdots\!53}{92\!\cdots\!75}a^{25}+\frac{50\!\cdots\!66}{64\!\cdots\!25}a^{24}-\frac{20\!\cdots\!52}{12\!\cdots\!25}a^{23}+\frac{12\!\cdots\!44}{64\!\cdots\!25}a^{22}-\frac{23\!\cdots\!18}{64\!\cdots\!25}a^{21}+\frac{42\!\cdots\!47}{64\!\cdots\!25}a^{20}+\frac{22\!\cdots\!61}{64\!\cdots\!25}a^{19}-\frac{13\!\cdots\!27}{64\!\cdots\!25}a^{18}+\frac{76\!\cdots\!23}{64\!\cdots\!25}a^{17}+\frac{29\!\cdots\!64}{64\!\cdots\!25}a^{16}-\frac{69\!\cdots\!96}{64\!\cdots\!25}a^{15}+\frac{21\!\cdots\!61}{64\!\cdots\!25}a^{14}+\frac{15\!\cdots\!96}{92\!\cdots\!75}a^{13}-\frac{12\!\cdots\!77}{64\!\cdots\!25}a^{12}-\frac{64\!\cdots\!73}{92\!\cdots\!75}a^{11}+\frac{82\!\cdots\!92}{25\!\cdots\!05}a^{10}-\frac{74\!\cdots\!32}{92\!\cdots\!75}a^{9}-\frac{22\!\cdots\!93}{49\!\cdots\!25}a^{8}-\frac{27\!\cdots\!52}{25\!\cdots\!05}a^{7}+\frac{18\!\cdots\!13}{64\!\cdots\!25}a^{6}-\frac{23\!\cdots\!52}{64\!\cdots\!25}a^{5}-\frac{69\!\cdots\!91}{20\!\cdots\!75}a^{4}+\frac{10\!\cdots\!93}{12\!\cdots\!25}a^{3}-\frac{34\!\cdots\!52}{12\!\cdots\!25}a^{2}-\frac{15\!\cdots\!23}{64\!\cdots\!25}a-\frac{10\!\cdots\!83}{25\!\cdots\!05}$, $\frac{59389260354803}{49\!\cdots\!25}a^{26}-\frac{199457870579008}{18\!\cdots\!75}a^{25}+\frac{33\!\cdots\!11}{64\!\cdots\!25}a^{24}-\frac{77\!\cdots\!66}{64\!\cdots\!25}a^{23}+\frac{12\!\cdots\!87}{64\!\cdots\!25}a^{22}-\frac{22\!\cdots\!41}{64\!\cdots\!25}a^{21}+\frac{36\!\cdots\!33}{64\!\cdots\!25}a^{20}-\frac{23\!\cdots\!73}{64\!\cdots\!25}a^{19}-\frac{73\!\cdots\!02}{64\!\cdots\!25}a^{18}+\frac{11\!\cdots\!74}{64\!\cdots\!25}a^{17}+\frac{43\!\cdots\!29}{64\!\cdots\!25}a^{16}-\frac{46\!\cdots\!09}{10\!\cdots\!75}a^{15}+\frac{28\!\cdots\!47}{64\!\cdots\!25}a^{14}+\frac{74\!\cdots\!07}{18\!\cdots\!75}a^{13}-\frac{16\!\cdots\!83}{20\!\cdots\!75}a^{12}+\frac{38\!\cdots\!19}{13\!\cdots\!25}a^{11}+\frac{33\!\cdots\!36}{25\!\cdots\!05}a^{10}+\frac{16\!\cdots\!72}{92\!\cdots\!75}a^{9}-\frac{36\!\cdots\!26}{64\!\cdots\!25}a^{8}+\frac{49\!\cdots\!56}{64\!\cdots\!25}a^{7}+\frac{14\!\cdots\!22}{64\!\cdots\!25}a^{6}-\frac{13\!\cdots\!23}{64\!\cdots\!25}a^{5}+\frac{43\!\cdots\!87}{49\!\cdots\!25}a^{4}+\frac{28\!\cdots\!12}{64\!\cdots\!25}a^{3}-\frac{30\!\cdots\!44}{12\!\cdots\!25}a^{2}-\frac{27\!\cdots\!89}{64\!\cdots\!25}a+\frac{16\!\cdots\!56}{25\!\cdots\!05}$, $\frac{25\!\cdots\!79}{64\!\cdots\!25}a^{26}-\frac{35\!\cdots\!12}{92\!\cdots\!75}a^{25}+\frac{12\!\cdots\!57}{64\!\cdots\!25}a^{24}-\frac{34\!\cdots\!84}{64\!\cdots\!25}a^{23}+\frac{64\!\cdots\!87}{64\!\cdots\!25}a^{22}-\frac{12\!\cdots\!12}{64\!\cdots\!25}a^{21}+\frac{20\!\cdots\!32}{64\!\cdots\!25}a^{20}-\frac{33\!\cdots\!28}{15\!\cdots\!75}a^{19}-\frac{14\!\cdots\!38}{64\!\cdots\!25}a^{18}+\frac{79\!\cdots\!63}{99\!\cdots\!25}a^{17}-\frac{97\!\cdots\!12}{20\!\cdots\!75}a^{16}-\frac{69\!\cdots\!78}{64\!\cdots\!25}a^{15}+\frac{14\!\cdots\!31}{64\!\cdots\!25}a^{14}-\frac{26\!\cdots\!59}{92\!\cdots\!75}a^{13}-\frac{14\!\cdots\!44}{64\!\cdots\!25}a^{12}+\frac{16\!\cdots\!34}{59\!\cdots\!25}a^{11}+\frac{29\!\cdots\!06}{12\!\cdots\!25}a^{10}-\frac{12\!\cdots\!06}{92\!\cdots\!75}a^{9}-\frac{14\!\cdots\!28}{64\!\cdots\!25}a^{8}+\frac{18\!\cdots\!22}{51\!\cdots\!01}a^{7}+\frac{34\!\cdots\!91}{64\!\cdots\!25}a^{6}-\frac{65\!\cdots\!98}{64\!\cdots\!25}a^{5}+\frac{79\!\cdots\!03}{64\!\cdots\!25}a^{4}+\frac{14\!\cdots\!79}{16\!\cdots\!75}a^{3}-\frac{16\!\cdots\!89}{25\!\cdots\!05}a^{2}-\frac{52\!\cdots\!62}{64\!\cdots\!25}a+\frac{19\!\cdots\!06}{19\!\cdots\!85}$, $\frac{54298416131889}{49\!\cdots\!25}a^{26}-\frac{902897548145063}{92\!\cdots\!75}a^{25}+\frac{59\!\cdots\!16}{12\!\cdots\!25}a^{24}-\frac{64\!\cdots\!54}{64\!\cdots\!25}a^{23}+\frac{89\!\cdots\!06}{64\!\cdots\!25}a^{22}-\frac{17\!\cdots\!21}{64\!\cdots\!25}a^{21}+\frac{35\!\cdots\!72}{81\!\cdots\!75}a^{20}+\frac{74\!\cdots\!28}{64\!\cdots\!25}a^{19}-\frac{70\!\cdots\!04}{64\!\cdots\!25}a^{18}+\frac{34\!\cdots\!19}{30\!\cdots\!75}a^{17}+\frac{79\!\cdots\!96}{64\!\cdots\!25}a^{16}-\frac{30\!\cdots\!74}{64\!\cdots\!25}a^{15}+\frac{17\!\cdots\!42}{64\!\cdots\!25}a^{14}+\frac{58\!\cdots\!69}{92\!\cdots\!75}a^{13}-\frac{48\!\cdots\!49}{64\!\cdots\!25}a^{12}-\frac{48\!\cdots\!87}{92\!\cdots\!75}a^{11}+\frac{82\!\cdots\!13}{64\!\cdots\!25}a^{10}-\frac{38\!\cdots\!07}{21\!\cdots\!25}a^{9}-\frac{92\!\cdots\!61}{64\!\cdots\!25}a^{8}+\frac{50\!\cdots\!53}{64\!\cdots\!25}a^{7}+\frac{11\!\cdots\!38}{64\!\cdots\!25}a^{6}-\frac{10\!\cdots\!37}{64\!\cdots\!25}a^{5}-\frac{26\!\cdots\!16}{64\!\cdots\!25}a^{4}+\frac{25\!\cdots\!12}{64\!\cdots\!25}a^{3}-\frac{18\!\cdots\!11}{64\!\cdots\!25}a^{2}-\frac{45\!\cdots\!29}{64\!\cdots\!25}a-\frac{10\!\cdots\!39}{25\!\cdots\!05}$, $\frac{43700351657934}{92\!\cdots\!75}a^{26}-\frac{30103309449026}{70\!\cdots\!75}a^{25}+\frac{18\!\cdots\!56}{92\!\cdots\!75}a^{24}-\frac{155960005282377}{36\!\cdots\!15}a^{23}+\frac{50\!\cdots\!39}{92\!\cdots\!75}a^{22}-\frac{89\!\cdots\!22}{92\!\cdots\!75}a^{21}+\frac{13\!\cdots\!96}{92\!\cdots\!75}a^{20}+\frac{18\!\cdots\!06}{13\!\cdots\!25}a^{19}-\frac{59\!\cdots\!21}{92\!\cdots\!75}a^{18}+\frac{71\!\cdots\!97}{92\!\cdots\!75}a^{17}+\frac{30\!\cdots\!18}{92\!\cdots\!75}a^{16}-\frac{17\!\cdots\!82}{92\!\cdots\!75}a^{15}+\frac{10\!\cdots\!53}{92\!\cdots\!75}a^{14}+\frac{32\!\cdots\!29}{92\!\cdots\!75}a^{13}-\frac{47\!\cdots\!04}{92\!\cdots\!75}a^{12}+\frac{24\!\cdots\!59}{92\!\cdots\!75}a^{11}+\frac{50\!\cdots\!02}{92\!\cdots\!75}a^{10}+\frac{29\!\cdots\!43}{92\!\cdots\!75}a^{9}-\frac{16\!\cdots\!97}{26\!\cdots\!25}a^{8}+\frac{44\!\cdots\!33}{92\!\cdots\!75}a^{7}+\frac{38\!\cdots\!64}{92\!\cdots\!75}a^{6}-\frac{29\!\cdots\!54}{92\!\cdots\!75}a^{5}+\frac{19\!\cdots\!47}{92\!\cdots\!75}a^{4}+\frac{25\!\cdots\!99}{92\!\cdots\!75}a^{3}+\frac{17\!\cdots\!21}{92\!\cdots\!75}a^{2}-\frac{17\!\cdots\!67}{92\!\cdots\!75}a+\frac{408686444194319}{784906602233845}$, $\frac{135944007}{93760174328275}a^{26}-\frac{230580102}{9567364727375}a^{25}+\frac{83112830599}{468800871641375}a^{24}-\frac{350455873827}{468800871641375}a^{23}+\frac{36240216204}{18752034865655}a^{22}-\frac{1709470745288}{468800871641375}a^{21}+\frac{678130683444}{93760174328275}a^{20}-\frac{426325675136}{36061605510875}a^{19}+\frac{3819732365742}{468800871641375}a^{18}+\frac{233401371461}{36061605510875}a^{17}-\frac{1392268980078}{93760174328275}a^{16}+\frac{7434794265868}{468800871641375}a^{15}+\frac{5044772262808}{468800871641375}a^{14}-\frac{1950127818602}{66971553091625}a^{13}+\frac{18249488499754}{468800871641375}a^{12}+\frac{697847176928}{66971553091625}a^{11}+\frac{234540489}{1163277597125}a^{10}+\frac{4042785121189}{66971553091625}a^{9}-\frac{5659484479709}{468800871641375}a^{8}+\frac{7275729022973}{36061605510875}a^{7}+\frac{129468758084327}{468800871641375}a^{6}+\frac{174464049770663}{468800871641375}a^{5}+\frac{305232529877461}{468800871641375}a^{4}-\frac{172567854271581}{468800871641375}a^{3}+\frac{221635898507421}{468800871641375}a^{2}+\frac{398467816665794}{468800871641375}a+\frac{23685993561059}{18752034865655}$, $\frac{78378804390422}{25\!\cdots\!05}a^{26}-\frac{36121310865336}{14\!\cdots\!75}a^{25}+\frac{74\!\cdots\!94}{64\!\cdots\!25}a^{24}-\frac{28\!\cdots\!52}{12\!\cdots\!25}a^{23}+\frac{20\!\cdots\!19}{64\!\cdots\!25}a^{22}-\frac{42\!\cdots\!17}{64\!\cdots\!25}a^{21}+\frac{48\!\cdots\!71}{49\!\cdots\!25}a^{20}+\frac{29\!\cdots\!90}{51\!\cdots\!01}a^{19}-\frac{15\!\cdots\!03}{64\!\cdots\!25}a^{18}+\frac{15\!\cdots\!21}{64\!\cdots\!25}a^{17}+\frac{51\!\cdots\!38}{12\!\cdots\!25}a^{16}-\frac{61\!\cdots\!91}{64\!\cdots\!25}a^{15}+\frac{23\!\cdots\!81}{64\!\cdots\!25}a^{14}+\frac{13\!\cdots\!14}{92\!\cdots\!75}a^{13}-\frac{57\!\cdots\!04}{64\!\cdots\!25}a^{12}-\frac{23\!\cdots\!09}{92\!\cdots\!75}a^{11}+\frac{20\!\cdots\!17}{64\!\cdots\!25}a^{10}+\frac{38\!\cdots\!93}{13\!\cdots\!25}a^{9}-\frac{48\!\cdots\!06}{25\!\cdots\!05}a^{8}-\frac{55\!\cdots\!38}{64\!\cdots\!25}a^{7}+\frac{45\!\cdots\!09}{64\!\cdots\!25}a^{6}-\frac{15\!\cdots\!24}{12\!\cdots\!25}a^{5}-\frac{31\!\cdots\!48}{64\!\cdots\!25}a^{4}+\frac{21\!\cdots\!12}{12\!\cdots\!25}a^{3}+\frac{60\!\cdots\!36}{64\!\cdots\!25}a^{2}-\frac{65\!\cdots\!64}{64\!\cdots\!25}a-\frac{21\!\cdots\!84}{25\!\cdots\!05}$, $\frac{140967034366906}{64\!\cdots\!25}a^{26}-\frac{28506386777677}{13\!\cdots\!25}a^{25}+\frac{58178315852217}{51\!\cdots\!01}a^{24}-\frac{14\!\cdots\!77}{49\!\cdots\!25}a^{23}+\frac{204615166494356}{41\!\cdots\!75}a^{22}-\frac{46\!\cdots\!03}{64\!\cdots\!25}a^{21}+\frac{74\!\cdots\!36}{64\!\cdots\!25}a^{20}-\frac{10\!\cdots\!09}{25\!\cdots\!05}a^{19}-\frac{85\!\cdots\!63}{25\!\cdots\!05}a^{18}+\frac{44\!\cdots\!52}{64\!\cdots\!25}a^{17}+\frac{28\!\cdots\!21}{64\!\cdots\!25}a^{16}-\frac{91\!\cdots\!86}{64\!\cdots\!25}a^{15}+\frac{12\!\cdots\!34}{64\!\cdots\!25}a^{14}-\frac{60217690714042}{11\!\cdots\!25}a^{13}-\frac{46\!\cdots\!82}{12\!\cdots\!25}a^{12}+\frac{29\!\cdots\!78}{92\!\cdots\!75}a^{11}+\frac{32\!\cdots\!88}{64\!\cdots\!25}a^{10}-\frac{18\!\cdots\!51}{73\!\cdots\!43}a^{9}-\frac{91\!\cdots\!48}{64\!\cdots\!25}a^{8}+\frac{10\!\cdots\!19}{64\!\cdots\!25}a^{7}+\frac{14\!\cdots\!62}{64\!\cdots\!25}a^{6}-\frac{54\!\cdots\!94}{64\!\cdots\!25}a^{5}+\frac{68\!\cdots\!01}{64\!\cdots\!25}a^{4}+\frac{18\!\cdots\!78}{81\!\cdots\!75}a^{3}-\frac{75\!\cdots\!71}{64\!\cdots\!25}a^{2}-\frac{97\!\cdots\!59}{64\!\cdots\!25}a+\frac{29\!\cdots\!71}{25\!\cdots\!05}$, $\frac{30\!\cdots\!88}{64\!\cdots\!25}a^{26}-\frac{116924489840968}{29\!\cdots\!25}a^{25}+\frac{11\!\cdots\!99}{64\!\cdots\!25}a^{24}-\frac{19\!\cdots\!04}{64\!\cdots\!25}a^{23}+\frac{51\!\cdots\!99}{15\!\cdots\!75}a^{22}-\frac{73\!\cdots\!06}{99\!\cdots\!25}a^{21}+\frac{67\!\cdots\!13}{64\!\cdots\!25}a^{20}+\frac{21\!\cdots\!23}{12\!\cdots\!25}a^{19}-\frac{29\!\cdots\!94}{64\!\cdots\!25}a^{18}+\frac{17\!\cdots\!14}{64\!\cdots\!25}a^{17}+\frac{64\!\cdots\!49}{64\!\cdots\!25}a^{16}-\frac{12\!\cdots\!46}{64\!\cdots\!25}a^{15}+\frac{27\!\cdots\!02}{64\!\cdots\!25}a^{14}+\frac{99\!\cdots\!56}{29\!\cdots\!25}a^{13}-\frac{15\!\cdots\!53}{64\!\cdots\!25}a^{12}-\frac{78\!\cdots\!19}{92\!\cdots\!75}a^{11}+\frac{35\!\cdots\!13}{49\!\cdots\!25}a^{10}+\frac{23\!\cdots\!74}{92\!\cdots\!75}a^{9}-\frac{98\!\cdots\!33}{25\!\cdots\!05}a^{8}+\frac{15\!\cdots\!82}{64\!\cdots\!25}a^{7}+\frac{58\!\cdots\!78}{64\!\cdots\!25}a^{6}-\frac{41\!\cdots\!71}{64\!\cdots\!25}a^{5}-\frac{51\!\cdots\!76}{64\!\cdots\!25}a^{4}+\frac{16\!\cdots\!54}{64\!\cdots\!25}a^{3}+\frac{97\!\cdots\!72}{64\!\cdots\!25}a^{2}-\frac{13\!\cdots\!88}{64\!\cdots\!25}a-\frac{41\!\cdots\!28}{25\!\cdots\!05}$, $\frac{18\!\cdots\!89}{64\!\cdots\!25}a^{26}-\frac{382502559002826}{13\!\cdots\!25}a^{25}+\frac{99\!\cdots\!74}{64\!\cdots\!25}a^{24}-\frac{28\!\cdots\!51}{64\!\cdots\!25}a^{23}+\frac{22\!\cdots\!61}{27\!\cdots\!75}a^{22}-\frac{89\!\cdots\!63}{64\!\cdots\!25}a^{21}+\frac{13\!\cdots\!62}{64\!\cdots\!25}a^{20}-\frac{73\!\cdots\!01}{64\!\cdots\!25}a^{19}-\frac{23\!\cdots\!31}{64\!\cdots\!25}a^{18}+\frac{15\!\cdots\!03}{15\!\cdots\!75}a^{17}-\frac{43\!\cdots\!56}{49\!\cdots\!25}a^{16}-\frac{43\!\cdots\!43}{64\!\cdots\!25}a^{15}+\frac{15\!\cdots\!63}{64\!\cdots\!25}a^{14}-\frac{13\!\cdots\!77}{92\!\cdots\!75}a^{13}-\frac{19\!\cdots\!72}{12\!\cdots\!25}a^{12}+\frac{65\!\cdots\!53}{18\!\cdots\!75}a^{11}-\frac{28\!\cdots\!69}{49\!\cdots\!25}a^{10}-\frac{13\!\cdots\!54}{92\!\cdots\!75}a^{9}+\frac{87\!\cdots\!27}{64\!\cdots\!25}a^{8}+\frac{86\!\cdots\!01}{64\!\cdots\!25}a^{7}+\frac{25\!\cdots\!22}{64\!\cdots\!25}a^{6}-\frac{61\!\cdots\!67}{64\!\cdots\!25}a^{5}+\frac{79\!\cdots\!11}{64\!\cdots\!25}a^{4}+\frac{55\!\cdots\!52}{12\!\cdots\!25}a^{3}-\frac{98\!\cdots\!46}{64\!\cdots\!25}a^{2}+\frac{11\!\cdots\!06}{12\!\cdots\!25}a-\frac{47\!\cdots\!93}{51\!\cdots\!01}$, $\frac{6695897567}{468800871641375}a^{26}-\frac{9863796}{76538917819}a^{25}+\frac{280571090011}{468800871641375}a^{24}-\frac{565231937353}{468800871641375}a^{23}+\frac{550681081744}{468800871641375}a^{22}-\frac{700043322702}{468800871641375}a^{21}+\frac{182154983326}{93760174328275}a^{20}+\frac{3993167192876}{468800871641375}a^{19}-\frac{13513183851582}{468800871641375}a^{18}+\frac{13784719176629}{468800871641375}a^{17}+\frac{10977972577042}{468800871641375}a^{16}-\frac{38802159632914}{468800871641375}a^{15}+\frac{20179394499148}{468800871641375}a^{14}+\frac{7324795530112}{66971553091625}a^{13}-\frac{78682074424322}{468800871641375}a^{12}+\frac{2583525605666}{66971553091625}a^{11}+\frac{116758789216679}{468800871641375}a^{10}-\frac{909268618509}{5151657930125}a^{9}-\frac{49883700126303}{468800871641375}a^{8}+\frac{59603271271527}{468800871641375}a^{7}+\frac{146202608706659}{468800871641375}a^{6}-\frac{280605849127894}{468800871641375}a^{5}+\frac{15094996466619}{36061605510875}a^{4}+\frac{157790698152551}{468800871641375}a^{3}-\frac{90872287756098}{468800871641375}a^{2}-\frac{2305658274558}{3750406973131}a+\frac{4786241078783}{3750406973131}$, $\frac{5076242311062}{49\!\cdots\!25}a^{26}-\frac{159918308489664}{92\!\cdots\!75}a^{25}+\frac{293449616015516}{25\!\cdots\!05}a^{24}-\frac{26\!\cdots\!74}{64\!\cdots\!25}a^{23}+\frac{47\!\cdots\!09}{64\!\cdots\!25}a^{22}-\frac{63\!\cdots\!23}{64\!\cdots\!25}a^{21}+\frac{27\!\cdots\!04}{12\!\cdots\!25}a^{20}-\frac{20\!\cdots\!08}{64\!\cdots\!25}a^{19}-\frac{81\!\cdots\!87}{49\!\cdots\!25}a^{18}+\frac{51\!\cdots\!91}{64\!\cdots\!25}a^{17}+\frac{41\!\cdots\!87}{64\!\cdots\!25}a^{16}-\frac{38\!\cdots\!57}{13\!\cdots\!75}a^{15}+\frac{10\!\cdots\!68}{41\!\cdots\!75}a^{14}+\frac{22\!\cdots\!04}{70\!\cdots\!75}a^{13}-\frac{48\!\cdots\!21}{64\!\cdots\!25}a^{12}-\frac{43\!\cdots\!44}{92\!\cdots\!75}a^{11}+\frac{55\!\cdots\!86}{64\!\cdots\!25}a^{10}-\frac{16\!\cdots\!43}{16\!\cdots\!75}a^{9}-\frac{45\!\cdots\!36}{49\!\cdots\!25}a^{8}+\frac{61\!\cdots\!59}{64\!\cdots\!25}a^{7}+\frac{10\!\cdots\!16}{49\!\cdots\!25}a^{6}-\frac{10\!\cdots\!60}{51\!\cdots\!01}a^{5}-\frac{20\!\cdots\!16}{64\!\cdots\!25}a^{4}+\frac{70\!\cdots\!99}{64\!\cdots\!25}a^{3}-\frac{41\!\cdots\!02}{64\!\cdots\!25}a^{2}-\frac{24\!\cdots\!13}{64\!\cdots\!25}a+\frac{36\!\cdots\!72}{25\!\cdots\!05}$, $\frac{62098536544473}{49\!\cdots\!25}a^{26}-\frac{10\!\cdots\!54}{92\!\cdots\!75}a^{25}+\frac{33\!\cdots\!74}{64\!\cdots\!25}a^{24}-\frac{71\!\cdots\!08}{64\!\cdots\!25}a^{23}+\frac{10\!\cdots\!76}{64\!\cdots\!25}a^{22}-\frac{14\!\cdots\!07}{49\!\cdots\!25}a^{21}+\frac{27\!\cdots\!66}{64\!\cdots\!25}a^{20}+\frac{13\!\cdots\!99}{64\!\cdots\!25}a^{19}-\frac{65\!\cdots\!61}{49\!\cdots\!25}a^{18}+\frac{11\!\cdots\!43}{64\!\cdots\!25}a^{17}+\frac{51\!\cdots\!48}{49\!\cdots\!25}a^{16}-\frac{34\!\cdots\!47}{64\!\cdots\!25}a^{15}+\frac{32\!\cdots\!36}{64\!\cdots\!25}a^{14}+\frac{18\!\cdots\!08}{39\!\cdots\!25}a^{13}-\frac{59\!\cdots\!18}{64\!\cdots\!25}a^{12}+\frac{52\!\cdots\!42}{92\!\cdots\!75}a^{11}+\frac{95\!\cdots\!14}{64\!\cdots\!25}a^{10}-\frac{98\!\cdots\!83}{92\!\cdots\!75}a^{9}-\frac{67\!\cdots\!46}{12\!\cdots\!25}a^{8}+\frac{33\!\cdots\!04}{49\!\cdots\!25}a^{7}+\frac{13\!\cdots\!33}{64\!\cdots\!25}a^{6}-\frac{11\!\cdots\!63}{60\!\cdots\!35}a^{5}+\frac{40\!\cdots\!19}{64\!\cdots\!25}a^{4}+\frac{47\!\cdots\!41}{64\!\cdots\!25}a^{3}+\frac{45\!\cdots\!14}{49\!\cdots\!25}a^{2}-\frac{64\!\cdots\!92}{13\!\cdots\!75}a+\frac{58\!\cdots\!16}{25\!\cdots\!05}$ Copy content Toggle raw display (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:  \( 16038112241729.484 \) (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)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(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 - 7874*x^13 + 2497*x^12 + 13136*x^11 - 1996*x^10 - 8600*x^9 + 6002*x^8 + 17843*x^7 - 14482*x^6 + 6811*x^5 + 47488*x^4 + 900*x^3 - 46897*x^2 + 24633*x + 2075)
 
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^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 - 7874*x^13 + 2497*x^12 + 13136*x^11 - 1996*x^10 - 8600*x^9 + 6002*x^8 + 17843*x^7 - 14482*x^6 + 6811*x^5 + 47488*x^4 + 900*x^3 - 46897*x^2 + 24633*x + 2075, 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^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 - 7874*x^13 + 2497*x^12 + 13136*x^11 - 1996*x^10 - 8600*x^9 + 6002*x^8 + 17843*x^7 - 14482*x^6 + 6811*x^5 + 47488*x^4 + 900*x^3 - 46897*x^2 + 24633*x + 2075);
 
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^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 - 7874*x^13 + 2497*x^12 + 13136*x^11 - 1996*x^10 - 8600*x^9 + 6002*x^8 + 17843*x^7 - 14482*x^6 + 6811*x^5 + 47488*x^4 + 900*x^3 - 46897*x^2 + 24633*x + 2075);
 
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

$D_{27}$ (as 27T8):

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 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.

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]
 

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.

# 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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(7\) Copy content Toggle raw display $\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}$
\(521\) Copy content Toggle raw display $\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}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3647.2t1.a.a$1$ $ 7 \cdot 521 $ \(\Q(\sqrt{-3647}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.3647.3t2.a.a$2$ $ 7 \cdot 521 $ 3.1.3647.1 $S_3$ (as 3T2) $1$ $0$
* 2.3647.9t3.a.c$2$ $ 7 \cdot 521 $ 9.1.176906199770881.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.3647.9t3.a.a$2$ $ 7 \cdot 521 $ 9.1.176906199770881.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.3647.9t3.a.b$2$ $ 7 \cdot 521 $ 9.1.176906199770881.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.3647.27t8.a.e$2$ $ 7 \cdot 521 $ 27.1.20191329826970487018697554420683199956085496127.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3647.27t8.a.f$2$ $ 7 \cdot 521 $ 27.1.20191329826970487018697554420683199956085496127.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3647.27t8.a.a$2$ $ 7 \cdot 521 $ 27.1.20191329826970487018697554420683199956085496127.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3647.27t8.a.h$2$ $ 7 \cdot 521 $ 27.1.20191329826970487018697554420683199956085496127.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3647.27t8.a.c$2$ $ 7 \cdot 521 $ 27.1.20191329826970487018697554420683199956085496127.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3647.27t8.a.i$2$ $ 7 \cdot 521 $ 27.1.20191329826970487018697554420683199956085496127.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3647.27t8.a.g$2$ $ 7 \cdot 521 $ 27.1.20191329826970487018697554420683199956085496127.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3647.27t8.a.d$2$ $ 7 \cdot 521 $ 27.1.20191329826970487018697554420683199956085496127.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3647.27t8.a.b$2$ $ 7 \cdot 521 $ 27.1.20191329826970487018697554420683199956085496127.1 $D_{27}$ (as 27T8) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.