LMFDB - Number field 27.1.20191329826970487018697554420683199956085496127.1

Show commands: Magma / Oscar / PariGP / SageMath

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 $ 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

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:	$-20191329826970487018697554420683199956085496127$ -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:	$7$, $521$ 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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{35}a^{11}-\frac{13}{35}a^{10}-\frac{13}{35}a^{9}-\frac{13}{35}a^{8}-\frac{13}{35}a^{7}+\frac{1}{35}a^{6}+\frac{13}{35}a^{5}-\frac{1}{35}a^{4}+\frac{6}{35}a^{3}-\frac{3}{7}a^{2}+\frac{6}{35}a-\frac{3}{7}$, $\frac{1}{35}a^{12}-\frac{1}{5}a^{10}-\frac{1}{5}a^{9}-\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{9}{35}a^{6}-\frac{1}{5}a^{5}-\frac{1}{5}a^{4}-\frac{1}{5}a^{3}-\frac{2}{5}a^{2}-\frac{1}{5}a+\frac{3}{7}$, $\frac{1}{35}a^{13}+\frac{1}{5}a^{10}+\frac{1}{5}a^{9}-\frac{2}{5}a^{8}+\frac{1}{7}a^{7}+\frac{2}{5}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{13}{35}a$, $\frac{1}{35}a^{14}-\frac{1}{5}a^{10}+\frac{1}{5}a^{9}-\frac{9}{35}a^{8}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}-\frac{2}{5}a^{3}-\frac{13}{35}a^{2}-\frac{1}{5}a$, $\frac{1}{35}a^{15}-\frac{2}{5}a^{10}+\frac{1}{7}a^{9}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a^{4}-\frac{6}{35}a^{3}-\frac{1}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{35}a^{16}-\frac{2}{35}a^{10}-\frac{1}{5}a^{9}+\frac{2}{5}a^{8}-\frac{2}{5}a^{5}+\frac{3}{7}a^{4}+\frac{1}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a$, $\frac{1}{175}a^{17}-\frac{1}{175}a^{16}+\frac{1}{175}a^{15}+\frac{1}{175}a^{13}+\frac{1}{175}a^{12}-\frac{2}{175}a^{11}-\frac{54}{175}a^{10}+\frac{61}{175}a^{9}+\frac{1}{5}a^{8}-\frac{2}{175}a^{7}-\frac{16}{175}a^{6}+\frac{57}{175}a^{5}+\frac{4}{35}a^{4}-\frac{11}{35}a^{3}-\frac{3}{25}a^{2}-\frac{62}{175}a+\frac{2}{7}$, $\frac{1}{175}a^{18}+\frac{1}{175}a^{15}+\frac{1}{175}a^{14}+\frac{2}{175}a^{13}-\frac{1}{175}a^{12}-\frac{1}{175}a^{11}-\frac{8}{175}a^{10}+\frac{81}{175}a^{9}+\frac{18}{175}a^{8}-\frac{33}{175}a^{7}-\frac{79}{175}a^{6}-\frac{83}{175}a^{5}+\frac{17}{35}a^{4}+\frac{79}{175}a^{3}-\frac{33}{175}a^{2}-\frac{32}{175}a-\frac{3}{7}$, $\frac{1}{1225}a^{19}+\frac{1}{1225}a^{18}+\frac{3}{1225}a^{17}-\frac{1}{175}a^{16}-\frac{3}{245}a^{15}+\frac{8}{1225}a^{14}-\frac{6}{1225}a^{13}+\frac{16}{1225}a^{12}+\frac{3}{245}a^{11}-\frac{82}{175}a^{10}+\frac{37}{1225}a^{9}-\frac{41}{245}a^{8}+\frac{457}{1225}a^{7}+\frac{3}{35}a^{6}-\frac{32}{1225}a^{5}-\frac{18}{175}a^{4}+\frac{216}{1225}a^{3}-\frac{328}{1225}a^{2}-\frac{333}{1225}a+\frac{15}{49}$, $\frac{1}{1225}a^{20}+\frac{2}{1225}a^{18}-\frac{3}{1225}a^{17}-\frac{3}{245}a^{16}-\frac{1}{245}a^{15}-\frac{2}{175}a^{14}-\frac{6}{1225}a^{13}+\frac{6}{1225}a^{12}-\frac{8}{1225}a^{11}+\frac{93}{1225}a^{10}+\frac{121}{245}a^{9}-\frac{213}{1225}a^{8}-\frac{436}{1225}a^{7}+\frac{101}{1225}a^{6}-\frac{107}{245}a^{5}-\frac{113}{1225}a^{4}-\frac{579}{1225}a^{3}-\frac{12}{1225}a^{2}+\frac{379}{1225}a-\frac{15}{49}$, $\frac{1}{1225}a^{21}+\frac{2}{1225}a^{18}-\frac{12}{1225}a^{16}+\frac{9}{1225}a^{15}-\frac{3}{245}a^{14}-\frac{17}{1225}a^{13}+\frac{9}{1225}a^{12}+\frac{2}{175}a^{11}+\frac{318}{1225}a^{10}-\frac{82}{175}a^{9}+\frac{69}{245}a^{8}+\frac{524}{1225}a^{7}+\frac{256}{1225}a^{6}-\frac{24}{175}a^{5}-\frac{292}{1225}a^{4}-\frac{591}{1225}a^{3}+\frac{608}{1225}a^{2}+\frac{82}{245}a+\frac{12}{49}$, $\frac{1}{1225}a^{22}-\frac{2}{1225}a^{18}+\frac{3}{1225}a^{17}+\frac{2}{1225}a^{16}+\frac{1}{1225}a^{15}+\frac{2}{1225}a^{14}+\frac{1}{175}a^{13}+\frac{3}{1225}a^{12}+\frac{1}{1225}a^{11}+\frac{1}{7}a^{10}-\frac{338}{1225}a^{9}+\frac{129}{1225}a^{8}-\frac{4}{35}a^{7}+\frac{38}{175}a^{6}+\frac{234}{1225}a^{5}+\frac{326}{1225}a^{4}-\frac{34}{1225}a^{3}-\frac{113}{245}a^{2}+\frac{87}{175}a+\frac{12}{49}$, $\frac{1}{44032625}a^{23}-\frac{16643}{44032625}a^{22}-\frac{1244}{8806525}a^{21}+\frac{1332}{44032625}a^{20}+\frac{916}{44032625}a^{19}+\frac{65157}{44032625}a^{18}-\frac{55838}{44032625}a^{17}-\frac{430902}{44032625}a^{16}+\frac{10729}{898625}a^{15}+\frac{143903}{44032625}a^{14}+\frac{116989}{8806525}a^{13}-\frac{512223}{44032625}a^{12}-\frac{415147}{44032625}a^{11}+\frac{16666339}{44032625}a^{10}-\frac{178393}{3387125}a^{9}+\frac{392046}{1258075}a^{8}+\frac{2794322}{8806525}a^{7}-\frac{2967036}{6290375}a^{6}+\frac{20112137}{44032625}a^{5}+\frac{19998639}{44032625}a^{4}+\frac{15494541}{44032625}a^{3}-\frac{5663993}{44032625}a^{2}-\frac{18964937}{44032625}a+\frac{146378}{1761305}$, $\frac{1}{1893402875}a^{24}+\frac{8}{1893402875}a^{23}-\frac{24336}{145646375}a^{22}-\frac{441683}{1893402875}a^{21}-\frac{321522}{1893402875}a^{20}-\frac{462382}{1893402875}a^{19}+\frac{2892924}{1893402875}a^{18}+\frac{178098}{75736115}a^{17}+\frac{7531299}{1893402875}a^{16}+\frac{17276354}{1893402875}a^{15}+\frac{18880158}{1893402875}a^{14}+\frac{6759047}{1893402875}a^{13}-\frac{3906692}{378680575}a^{12}-\frac{3374544}{270486125}a^{11}-\frac{3714890}{15147223}a^{10}-\frac{552542049}{1893402875}a^{9}+\frac{4312746}{75736115}a^{8}+\frac{701068378}{1893402875}a^{7}-\frac{223197}{677425}a^{6}-\frac{569685819}{1893402875}a^{5}-\frac{3650397}{54097225}a^{4}+\frac{789418808}{1893402875}a^{3}+\frac{7744369}{54097225}a^{2}+\frac{70928514}{270486125}a+\frac{29061023}{75736115}$, $\frac{1}{19310815922125}a^{25}-\frac{4408}{19310815922125}a^{24}-\frac{23323}{3862163184425}a^{23}+\frac{106054986}{2758687988875}a^{22}-\frac{726922324}{19310815922125}a^{21}+\frac{2721713062}{19310815922125}a^{20}-\frac{6880779358}{19310815922125}a^{19}+\frac{11722907778}{19310815922125}a^{18}-\frac{14857757319}{19310815922125}a^{17}-\frac{232017921272}{19310815922125}a^{16}-\frac{4892462241}{551737597775}a^{15}+\frac{1369907409}{1485447378625}a^{14}-\frac{240544771552}{19310815922125}a^{13}-\frac{66054933751}{19310815922125}a^{12}-\frac{128836764934}{19310815922125}a^{11}-\frac{133054197636}{297089475725}a^{10}+\frac{591948599141}{3862163184425}a^{9}-\frac{5240639872547}{19310815922125}a^{8}+\frac{1151008062952}{19310815922125}a^{7}+\frac{827061074447}{2758687988875}a^{6}-\frac{7599442030379}{19310815922125}a^{5}-\frac{1187173342723}{19310815922125}a^{4}+\frac{2134991397143}{19310815922125}a^{3}-\frac{201658403187}{551737597775}a^{2}+\frac{452859215253}{3862163184425}a+\frac{76389771697}{154486527377}$, $\frac{1}{64\!\cdots\!25}a^{26}+\frac{22836}{92\!\cdots\!75}a^{25}-\frac{1021108381}{64\!\cdots\!25}a^{24}+\frac{20333032619}{64\!\cdots\!25}a^{23}+\frac{397398971682214}{64\!\cdots\!25}a^{22}-\frac{322047617119439}{12\!\cdots\!25}a^{21}-\frac{325656430370671}{64\!\cdots\!25}a^{20}+\frac{26694702100179}{25\!\cdots\!05}a^{19}-\frac{12\!\cdots\!03}{64\!\cdots\!25}a^{18}-\frac{280075592992714}{49\!\cdots\!25}a^{17}+\frac{85\!\cdots\!61}{64\!\cdots\!25}a^{16}-\frac{273375427853587}{64\!\cdots\!25}a^{15}-\frac{80\!\cdots\!08}{12\!\cdots\!25}a^{14}-\frac{11\!\cdots\!97}{13\!\cdots\!25}a^{13}+\frac{58\!\cdots\!01}{64\!\cdots\!25}a^{12}+\frac{942503246043277}{13\!\cdots\!25}a^{11}-\frac{25\!\cdots\!22}{12\!\cdots\!25}a^{10}-\frac{36\!\cdots\!79}{92\!\cdots\!75}a^{9}-\frac{24\!\cdots\!13}{64\!\cdots\!25}a^{8}-\frac{17\!\cdots\!39}{64\!\cdots\!25}a^{7}+\frac{50\!\cdots\!06}{64\!\cdots\!25}a^{6}+\frac{23\!\cdots\!86}{64\!\cdots\!25}a^{5}+\frac{98\!\cdots\!33}{64\!\cdots\!25}a^{4}+\frac{69\!\cdots\!07}{64\!\cdots\!25}a^{3}+\frac{13\!\cdots\!99}{12\!\cdots\!25}a^{2}-\frac{16\!\cdots\!13}{64\!\cdots\!25}a+\frac{70\!\cdots\!67}{25\!\cdots\!05}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, 1/35*a^11 - 13/35*a^10 - 13/35*a^9 - 13/35*a^8 - 13/35*a^7 + 1/35*a^6 + 13/35*a^5 - 1/35*a^4 + 6/35*a^3 - 3/7*a^2 + 6/35*a - 3/7, 1/35*a^12 - 1/5*a^10 - 1/5*a^9 - 1/5*a^8 + 1/5*a^7 - 9/35*a^6 - 1/5*a^5 - 1/5*a^4 - 1/5*a^3 - 2/5*a^2 - 1/5*a + 3/7, 1/35*a^13 + 1/5*a^10 + 1/5*a^9 - 2/5*a^8 + 1/7*a^7 + 2/5*a^5 - 2/5*a^4 - 1/5*a^3 - 1/5*a^2 - 13/35*a, 1/35*a^14 - 1/5*a^10 + 1/5*a^9 - 9/35*a^8 - 2/5*a^7 + 1/5*a^6 - 2/5*a^3 - 13/35*a^2 - 1/5*a, 1/35*a^15 - 2/5*a^10 + 1/7*a^9 - 2/5*a^7 + 1/5*a^6 - 2/5*a^5 + 2/5*a^4 - 6/35*a^3 - 1/5*a^2 + 1/5*a, 1/35*a^16 - 2/35*a^10 - 1/5*a^9 + 2/5*a^8 - 2/5*a^5 + 3/7*a^4 + 1/5*a^3 + 1/5*a^2 + 2/5*a, 1/175*a^17 - 1/175*a^16 + 1/175*a^15 + 1/175*a^13 + 1/175*a^12 - 2/175*a^11 - 54/175*a^10 + 61/175*a^9 + 1/5*a^8 - 2/175*a^7 - 16/175*a^6 + 57/175*a^5 + 4/35*a^4 - 11/35*a^3 - 3/25*a^2 - 62/175*a + 2/7, 1/175*a^18 + 1/175*a^15 + 1/175*a^14 + 2/175*a^13 - 1/175*a^12 - 1/175*a^11 - 8/175*a^10 + 81/175*a^9 + 18/175*a^8 - 33/175*a^7 - 79/175*a^6 - 83/175*a^5 + 17/35*a^4 + 79/175*a^3 - 33/175*a^2 - 32/175*a - 3/7, 1/1225*a^19 + 1/1225*a^18 + 3/1225*a^17 - 1/175*a^16 - 3/245*a^15 + 8/1225*a^14 - 6/1225*a^13 + 16/1225*a^12 + 3/245*a^11 - 82/175*a^10 + 37/1225*a^9 - 41/245*a^8 + 457/1225*a^7 + 3/35*a^6 - 32/1225*a^5 - 18/175*a^4 + 216/1225*a^3 - 328/1225*a^2 - 333/1225*a + 15/49, 1/1225*a^20 + 2/1225*a^18 - 3/1225*a^17 - 3/245*a^16 - 1/245*a^15 - 2/175*a^14 - 6/1225*a^13 + 6/1225*a^12 - 8/1225*a^11 + 93/1225*a^10 + 121/245*a^9 - 213/1225*a^8 - 436/1225*a^7 + 101/1225*a^6 - 107/245*a^5 - 113/1225*a^4 - 579/1225*a^3 - 12/1225*a^2 + 379/1225*a - 15/49, 1/1225*a^21 + 2/1225*a^18 - 12/1225*a^16 + 9/1225*a^15 - 3/245*a^14 - 17/1225*a^13 + 9/1225*a^12 + 2/175*a^11 + 318/1225*a^10 - 82/175*a^9 + 69/245*a^8 + 524/1225*a^7 + 256/1225*a^6 - 24/175*a^5 - 292/1225*a^4 - 591/1225*a^3 + 608/1225*a^2 + 82/245*a + 12/49, 1/1225*a^22 - 2/1225*a^18 + 3/1225*a^17 + 2/1225*a^16 + 1/1225*a^15 + 2/1225*a^14 + 1/175*a^13 + 3/1225*a^12 + 1/1225*a^11 + 1/7*a^10 - 338/1225*a^9 + 129/1225*a^8 - 4/35*a^7 + 38/175*a^6 + 234/1225*a^5 + 326/1225*a^4 - 34/1225*a^3 - 113/245*a^2 + 87/175*a + 12/49, 1/44032625*a^23 - 16643/44032625*a^22 - 1244/8806525*a^21 + 1332/44032625*a^20 + 916/44032625*a^19 + 65157/44032625*a^18 - 55838/44032625*a^17 - 430902/44032625*a^16 + 10729/898625*a^15 + 143903/44032625*a^14 + 116989/8806525*a^13 - 512223/44032625*a^12 - 415147/44032625*a^11 + 16666339/44032625*a^10 - 178393/3387125*a^9 + 392046/1258075*a^8 + 2794322/8806525*a^7 - 2967036/6290375*a^6 + 20112137/44032625*a^5 + 19998639/44032625*a^4 + 15494541/44032625*a^3 - 5663993/44032625*a^2 - 18964937/44032625*a + 146378/1761305, 1/1893402875*a^24 + 8/1893402875*a^23 - 24336/145646375*a^22 - 441683/1893402875*a^21 - 321522/1893402875*a^20 - 462382/1893402875*a^19 + 2892924/1893402875*a^18 + 178098/75736115*a^17 + 7531299/1893402875*a^16 + 17276354/1893402875*a^15 + 18880158/1893402875*a^14 + 6759047/1893402875*a^13 - 3906692/378680575*a^12 - 3374544/270486125*a^11 - 3714890/15147223*a^10 - 552542049/1893402875*a^9 + 4312746/75736115*a^8 + 701068378/1893402875*a^7 - 223197/677425*a^6 - 569685819/1893402875*a^5 - 3650397/54097225*a^4 + 789418808/1893402875*a^3 + 7744369/54097225*a^2 + 70928514/270486125*a + 29061023/75736115, 1/19310815922125*a^25 - 4408/19310815922125*a^24 - 23323/3862163184425*a^23 + 106054986/2758687988875*a^22 - 726922324/19310815922125*a^21 + 2721713062/19310815922125*a^20 - 6880779358/19310815922125*a^19 + 11722907778/19310815922125*a^18 - 14857757319/19310815922125*a^17 - 232017921272/19310815922125*a^16 - 4892462241/551737597775*a^15 + 1369907409/1485447378625*a^14 - 240544771552/19310815922125*a^13 - 66054933751/19310815922125*a^12 - 128836764934/19310815922125*a^11 - 133054197636/297089475725*a^10 + 591948599141/3862163184425*a^9 - 5240639872547/19310815922125*a^8 + 1151008062952/19310815922125*a^7 + 827061074447/2758687988875*a^6 - 7599442030379/19310815922125*a^5 - 1187173342723/19310815922125*a^4 + 2134991397143/19310815922125*a^3 - 201658403187/551737597775*a^2 + 452859215253/3862163184425*a + 76389771697/154486527377, 1/6455856803373375125*a^26 + 22836/922265257624767875*a^25 - 1021108381/6455856803373375125*a^24 + 20333032619/6455856803373375125*a^23 + 397398971682214/6455856803373375125*a^22 - 322047617119439/1291171360674675025*a^21 - 325656430370671/6455856803373375125*a^20 + 26694702100179/258234272134935005*a^19 - 12397429686334603/6455856803373375125*a^18 - 280075592992714/496604369490259625*a^17 + 85403018004263961/6455856803373375125*a^16 - 273375427853587/6455856803373375125*a^15 - 8032913868640408/1291171360674675025*a^14 - 1136420283964997/131752179660681125*a^13 + 58699501747074301/6455856803373375125*a^12 + 942503246043277/131752179660681125*a^11 - 255792108380049922/1291171360674675025*a^10 - 365166064235056979/922265257624767875*a^9 - 2463302219127411813/6455856803373375125*a^8 - 1780238315162754739/6455856803373375125*a^7 + 504155753554762306/6455856803373375125*a^6 + 2306167105234353286/6455856803373375125*a^5 + 985374921874962833/6455856803373375125*a^4 + 692404113641684207/6455856803373375125*a^3 + 13688155888648399/1291171360674675025*a^2 - 1601871731447581313/6455856803373375125*a + 70612416027932867/258234272134935005

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 $ -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:	$\frac{11\!\cdots\!59}{64\!\cdots\!25}a^{26}-\frac{15\!\cdots\!53}{92\!\cdots\!75}a^{25}+\frac{50\!\cdots\!66}{64\!\cdots\!25}a^{24}-\frac{20\!\cdots\!52}{12\!\cdots\!25}a^{23}+\frac{12\!\cdots\!44}{64\!\cdots\!25}a^{22}-\frac{23\!\cdots\!18}{64\!\cdots\!25}a^{21}+\frac{42\!\cdots\!47}{64\!\cdots\!25}a^{20}+\frac{22\!\cdots\!61}{64\!\cdots\!25}a^{19}-\frac{13\!\cdots\!27}{64\!\cdots\!25}a^{18}+\frac{76\!\cdots\!23}{64\!\cdots\!25}a^{17}+\frac{29\!\cdots\!64}{64\!\cdots\!25}a^{16}-\frac{69\!\cdots\!96}{64\!\cdots\!25}a^{15}+\frac{21\!\cdots\!61}{64\!\cdots\!25}a^{14}+\frac{15\!\cdots\!96}{92\!\cdots\!75}a^{13}-\frac{12\!\cdots\!77}{64\!\cdots\!25}a^{12}-\frac{64\!\cdots\!73}{92\!\cdots\!75}a^{11}+\frac{82\!\cdots\!92}{25\!\cdots\!05}a^{10}-\frac{74\!\cdots\!32}{92\!\cdots\!75}a^{9}-\frac{22\!\cdots\!93}{49\!\cdots\!25}a^{8}-\frac{27\!\cdots\!52}{25\!\cdots\!05}a^{7}+\frac{18\!\cdots\!13}{64\!\cdots\!25}a^{6}-\frac{23\!\cdots\!52}{64\!\cdots\!25}a^{5}-\frac{69\!\cdots\!91}{20\!\cdots\!75}a^{4}+\frac{10\!\cdots\!93}{12\!\cdots\!25}a^{3}-\frac{34\!\cdots\!52}{12\!\cdots\!25}a^{2}-\frac{15\!\cdots\!23}{64\!\cdots\!25}a-\frac{10\!\cdots\!83}{25\!\cdots\!05}$, $\frac{59389260354803}{49\!\cdots\!25}a^{26}-\frac{199457870579008}{18\!\cdots\!75}a^{25}+\frac{33\!\cdots\!11}{64\!\cdots\!25}a^{24}-\frac{77\!\cdots\!66}{64\!\cdots\!25}a^{23}+\frac{12\!\cdots\!87}{64\!\cdots\!25}a^{22}-\frac{22\!\cdots\!41}{64\!\cdots\!25}a^{21}+\frac{36\!\cdots\!33}{64\!\cdots\!25}a^{20}-\frac{23\!\cdots\!73}{64\!\cdots\!25}a^{19}-\frac{73\!\cdots\!02}{64\!\cdots\!25}a^{18}+\frac{11\!\cdots\!74}{64\!\cdots\!25}a^{17}+\frac{43\!\cdots\!29}{64\!\cdots\!25}a^{16}-\frac{46\!\cdots\!09}{10\!\cdots\!75}a^{15}+\frac{28\!\cdots\!47}{64\!\cdots\!25}a^{14}+\frac{74\!\cdots\!07}{18\!\cdots\!75}a^{13}-\frac{16\!\cdots\!83}{20\!\cdots\!75}a^{12}+\frac{38\!\cdots\!19}{13\!\cdots\!25}a^{11}+\frac{33\!\cdots\!36}{25\!\cdots\!05}a^{10}+\frac{16\!\cdots\!72}{92\!\cdots\!75}a^{9}-\frac{36\!\cdots\!26}{64\!\cdots\!25}a^{8}+\frac{49\!\cdots\!56}{64\!\cdots\!25}a^{7}+\frac{14\!\cdots\!22}{64\!\cdots\!25}a^{6}-\frac{13\!\cdots\!23}{64\!\cdots\!25}a^{5}+\frac{43\!\cdots\!87}{49\!\cdots\!25}a^{4}+\frac{28\!\cdots\!12}{64\!\cdots\!25}a^{3}-\frac{30\!\cdots\!44}{12\!\cdots\!25}a^{2}-\frac{27\!\cdots\!89}{64\!\cdots\!25}a+\frac{16\!\cdots\!56}{25\!\cdots\!05}$, $\frac{25\!\cdots\!79}{64\!\cdots\!25}a^{26}-\frac{35\!\cdots\!12}{92\!\cdots\!75}a^{25}+\frac{12\!\cdots\!57}{64\!\cdots\!25}a^{24}-\frac{34\!\cdots\!84}{64\!\cdots\!25}a^{23}+\frac{64\!\cdots\!87}{64\!\cdots\!25}a^{22}-\frac{12\!\cdots\!12}{64\!\cdots\!25}a^{21}+\frac{20\!\cdots\!32}{64\!\cdots\!25}a^{20}-\frac{33\!\cdots\!28}{15\!\cdots\!75}a^{19}-\frac{14\!\cdots\!38}{64\!\cdots\!25}a^{18}+\frac{79\!\cdots\!63}{99\!\cdots\!25}a^{17}-\frac{97\!\cdots\!12}{20\!\cdots\!75}a^{16}-\frac{69\!\cdots\!78}{64\!\cdots\!25}a^{15}+\frac{14\!\cdots\!31}{64\!\cdots\!25}a^{14}-\frac{26\!\cdots\!59}{92\!\cdots\!75}a^{13}-\frac{14\!\cdots\!44}{64\!\cdots\!25}a^{12}+\frac{16\!\cdots\!34}{59\!\cdots\!25}a^{11}+\frac{29\!\cdots\!06}{12\!\cdots\!25}a^{10}-\frac{12\!\cdots\!06}{92\!\cdots\!75}a^{9}-\frac{14\!\cdots\!28}{64\!\cdots\!25}a^{8}+\frac{18\!\cdots\!22}{51\!\cdots\!01}a^{7}+\frac{34\!\cdots\!91}{64\!\cdots\!25}a^{6}-\frac{65\!\cdots\!98}{64\!\cdots\!25}a^{5}+\frac{79\!\cdots\!03}{64\!\cdots\!25}a^{4}+\frac{14\!\cdots\!79}{16\!\cdots\!75}a^{3}-\frac{16\!\cdots\!89}{25\!\cdots\!05}a^{2}-\frac{52\!\cdots\!62}{64\!\cdots\!25}a+\frac{19\!\cdots\!06}{19\!\cdots\!85}$, $\frac{54298416131889}{49\!\cdots\!25}a^{26}-\frac{902897548145063}{92\!\cdots\!75}a^{25}+\frac{59\!\cdots\!16}{12\!\cdots\!25}a^{24}-\frac{64\!\cdots\!54}{64\!\cdots\!25}a^{23}+\frac{89\!\cdots\!06}{64\!\cdots\!25}a^{22}-\frac{17\!\cdots\!21}{64\!\cdots\!25}a^{21}+\frac{35\!\cdots\!72}{81\!\cdots\!75}a^{20}+\frac{74\!\cdots\!28}{64\!\cdots\!25}a^{19}-\frac{70\!\cdots\!04}{64\!\cdots\!25}a^{18}+\frac{34\!\cdots\!19}{30\!\cdots\!75}a^{17}+\frac{79\!\cdots\!96}{64\!\cdots\!25}a^{16}-\frac{30\!\cdots\!74}{64\!\cdots\!25}a^{15}+\frac{17\!\cdots\!42}{64\!\cdots\!25}a^{14}+\frac{58\!\cdots\!69}{92\!\cdots\!75}a^{13}-\frac{48\!\cdots\!49}{64\!\cdots\!25}a^{12}-\frac{48\!\cdots\!87}{92\!\cdots\!75}a^{11}+\frac{82\!\cdots\!13}{64\!\cdots\!25}a^{10}-\frac{38\!\cdots\!07}{21\!\cdots\!25}a^{9}-\frac{92\!\cdots\!61}{64\!\cdots\!25}a^{8}+\frac{50\!\cdots\!53}{64\!\cdots\!25}a^{7}+\frac{11\!\cdots\!38}{64\!\cdots\!25}a^{6}-\frac{10\!\cdots\!37}{64\!\cdots\!25}a^{5}-\frac{26\!\cdots\!16}{64\!\cdots\!25}a^{4}+\frac{25\!\cdots\!12}{64\!\cdots\!25}a^{3}-\frac{18\!\cdots\!11}{64\!\cdots\!25}a^{2}-\frac{45\!\cdots\!29}{64\!\cdots\!25}a-\frac{10\!\cdots\!39}{25\!\cdots\!05}$, $\frac{43700351657934}{92\!\cdots\!75}a^{26}-\frac{30103309449026}{70\!\cdots\!75}a^{25}+\frac{18\!\cdots\!56}{92\!\cdots\!75}a^{24}-\frac{155960005282377}{36\!\cdots\!15}a^{23}+\frac{50\!\cdots\!39}{92\!\cdots\!75}a^{22}-\frac{89\!\cdots\!22}{92\!\cdots\!75}a^{21}+\frac{13\!\cdots\!96}{92\!\cdots\!75}a^{20}+\frac{18\!\cdots\!06}{13\!\cdots\!25}a^{19}-\frac{59\!\cdots\!21}{92\!\cdots\!75}a^{18}+\frac{71\!\cdots\!97}{92\!\cdots\!75}a^{17}+\frac{30\!\cdots\!18}{92\!\cdots\!75}a^{16}-\frac{17\!\cdots\!82}{92\!\cdots\!75}a^{15}+\frac{10\!\cdots\!53}{92\!\cdots\!75}a^{14}+\frac{32\!\cdots\!29}{92\!\cdots\!75}a^{13}-\frac{47\!\cdots\!04}{92\!\cdots\!75}a^{12}+\frac{24\!\cdots\!59}{92\!\cdots\!75}a^{11}+\frac{50\!\cdots\!02}{92\!\cdots\!75}a^{10}+\frac{29\!\cdots\!43}{92\!\cdots\!75}a^{9}-\frac{16\!\cdots\!97}{26\!\cdots\!25}a^{8}+\frac{44\!\cdots\!33}{92\!\cdots\!75}a^{7}+\frac{38\!\cdots\!64}{92\!\cdots\!75}a^{6}-\frac{29\!\cdots\!54}{92\!\cdots\!75}a^{5}+\frac{19\!\cdots\!47}{92\!\cdots\!75}a^{4}+\frac{25\!\cdots\!99}{92\!\cdots\!75}a^{3}+\frac{17\!\cdots\!21}{92\!\cdots\!75}a^{2}-\frac{17\!\cdots\!67}{92\!\cdots\!75}a+\frac{408686444194319}{784906602233845}$, $\frac{135944007}{93760174328275}a^{26}-\frac{230580102}{9567364727375}a^{25}+\frac{83112830599}{468800871641375}a^{24}-\frac{350455873827}{468800871641375}a^{23}+\frac{36240216204}{18752034865655}a^{22}-\frac{1709470745288}{468800871641375}a^{21}+\frac{678130683444}{93760174328275}a^{20}-\frac{426325675136}{36061605510875}a^{19}+\frac{3819732365742}{468800871641375}a^{18}+\frac{233401371461}{36061605510875}a^{17}-\frac{1392268980078}{93760174328275}a^{16}+\frac{7434794265868}{468800871641375}a^{15}+\frac{5044772262808}{468800871641375}a^{14}-\frac{1950127818602}{66971553091625}a^{13}+\frac{18249488499754}{468800871641375}a^{12}+\frac{697847176928}{66971553091625}a^{11}+\frac{234540489}{1163277597125}a^{10}+\frac{4042785121189}{66971553091625}a^{9}-\frac{5659484479709}{468800871641375}a^{8}+\frac{7275729022973}{36061605510875}a^{7}+\frac{129468758084327}{468800871641375}a^{6}+\frac{174464049770663}{468800871641375}a^{5}+\frac{305232529877461}{468800871641375}a^{4}-\frac{172567854271581}{468800871641375}a^{3}+\frac{221635898507421}{468800871641375}a^{2}+\frac{398467816665794}{468800871641375}a+\frac{23685993561059}{18752034865655}$, $\frac{78378804390422}{25\!\cdots\!05}a^{26}-\frac{36121310865336}{14\!\cdots\!75}a^{25}+\frac{74\!\cdots\!94}{64\!\cdots\!25}a^{24}-\frac{28\!\cdots\!52}{12\!\cdots\!25}a^{23}+\frac{20\!\cdots\!19}{64\!\cdots\!25}a^{22}-\frac{42\!\cdots\!17}{64\!\cdots\!25}a^{21}+\frac{48\!\cdots\!71}{49\!\cdots\!25}a^{20}+\frac{29\!\cdots\!90}{51\!\cdots\!01}a^{19}-\frac{15\!\cdots\!03}{64\!\cdots\!25}a^{18}+\frac{15\!\cdots\!21}{64\!\cdots\!25}a^{17}+\frac{51\!\cdots\!38}{12\!\cdots\!25}a^{16}-\frac{61\!\cdots\!91}{64\!\cdots\!25}a^{15}+\frac{23\!\cdots\!81}{64\!\cdots\!25}a^{14}+\frac{13\!\cdots\!14}{92\!\cdots\!75}a^{13}-\frac{57\!\cdots\!04}{64\!\cdots\!25}a^{12}-\frac{23\!\cdots\!09}{92\!\cdots\!75}a^{11}+\frac{20\!\cdots\!17}{64\!\cdots\!25}a^{10}+\frac{38\!\cdots\!93}{13\!\cdots\!25}a^{9}-\frac{48\!\cdots\!06}{25\!\cdots\!05}a^{8}-\frac{55\!\cdots\!38}{64\!\cdots\!25}a^{7}+\frac{45\!\cdots\!09}{64\!\cdots\!25}a^{6}-\frac{15\!\cdots\!24}{12\!\cdots\!25}a^{5}-\frac{31\!\cdots\!48}{64\!\cdots\!25}a^{4}+\frac{21\!\cdots\!12}{12\!\cdots\!25}a^{3}+\frac{60\!\cdots\!36}{64\!\cdots\!25}a^{2}-\frac{65\!\cdots\!64}{64\!\cdots\!25}a-\frac{21\!\cdots\!84}{25\!\cdots\!05}$, $\frac{140967034366906}{64\!\cdots\!25}a^{26}-\frac{28506386777677}{13\!\cdots\!25}a^{25}+\frac{58178315852217}{51\!\cdots\!01}a^{24}-\frac{14\!\cdots\!77}{49\!\cdots\!25}a^{23}+\frac{204615166494356}{41\!\cdots\!75}a^{22}-\frac{46\!\cdots\!03}{64\!\cdots\!25}a^{21}+\frac{74\!\cdots\!36}{64\!\cdots\!25}a^{20}-\frac{10\!\cdots\!09}{25\!\cdots\!05}a^{19}-\frac{85\!\cdots\!63}{25\!\cdots\!05}a^{18}+\frac{44\!\cdots\!52}{64\!\cdots\!25}a^{17}+\frac{28\!\cdots\!21}{64\!\cdots\!25}a^{16}-\frac{91\!\cdots\!86}{64\!\cdots\!25}a^{15}+\frac{12\!\cdots\!34}{64\!\cdots\!25}a^{14}-\frac{60217690714042}{11\!\cdots\!25}a^{13}-\frac{46\!\cdots\!82}{12\!\cdots\!25}a^{12}+\frac{29\!\cdots\!78}{92\!\cdots\!75}a^{11}+\frac{32\!\cdots\!88}{64\!\cdots\!25}a^{10}-\frac{18\!\cdots\!51}{73\!\cdots\!43}a^{9}-\frac{91\!\cdots\!48}{64\!\cdots\!25}a^{8}+\frac{10\!\cdots\!19}{64\!\cdots\!25}a^{7}+\frac{14\!\cdots\!62}{64\!\cdots\!25}a^{6}-\frac{54\!\cdots\!94}{64\!\cdots\!25}a^{5}+\frac{68\!\cdots\!01}{64\!\cdots\!25}a^{4}+\frac{18\!\cdots\!78}{81\!\cdots\!75}a^{3}-\frac{75\!\cdots\!71}{64\!\cdots\!25}a^{2}-\frac{97\!\cdots\!59}{64\!\cdots\!25}a+\frac{29\!\cdots\!71}{25\!\cdots\!05}$, $\frac{30\!\cdots\!88}{64\!\cdots\!25}a^{26}-\frac{116924489840968}{29\!\cdots\!25}a^{25}+\frac{11\!\cdots\!99}{64\!\cdots\!25}a^{24}-\frac{19\!\cdots\!04}{64\!\cdots\!25}a^{23}+\frac{51\!\cdots\!99}{15\!\cdots\!75}a^{22}-\frac{73\!\cdots\!06}{99\!\cdots\!25}a^{21}+\frac{67\!\cdots\!13}{64\!\cdots\!25}a^{20}+\frac{21\!\cdots\!23}{12\!\cdots\!25}a^{19}-\frac{29\!\cdots\!94}{64\!\cdots\!25}a^{18}+\frac{17\!\cdots\!14}{64\!\cdots\!25}a^{17}+\frac{64\!\cdots\!49}{64\!\cdots\!25}a^{16}-\frac{12\!\cdots\!46}{64\!\cdots\!25}a^{15}+\frac{27\!\cdots\!02}{64\!\cdots\!25}a^{14}+\frac{99\!\cdots\!56}{29\!\cdots\!25}a^{13}-\frac{15\!\cdots\!53}{64\!\cdots\!25}a^{12}-\frac{78\!\cdots\!19}{92\!\cdots\!75}a^{11}+\frac{35\!\cdots\!13}{49\!\cdots\!25}a^{10}+\frac{23\!\cdots\!74}{92\!\cdots\!75}a^{9}-\frac{98\!\cdots\!33}{25\!\cdots\!05}a^{8}+\frac{15\!\cdots\!82}{64\!\cdots\!25}a^{7}+\frac{58\!\cdots\!78}{64\!\cdots\!25}a^{6}-\frac{41\!\cdots\!71}{64\!\cdots\!25}a^{5}-\frac{51\!\cdots\!76}{64\!\cdots\!25}a^{4}+\frac{16\!\cdots\!54}{64\!\cdots\!25}a^{3}+\frac{97\!\cdots\!72}{64\!\cdots\!25}a^{2}-\frac{13\!\cdots\!88}{64\!\cdots\!25}a-\frac{41\!\cdots\!28}{25\!\cdots\!05}$, $\frac{18\!\cdots\!89}{64\!\cdots\!25}a^{26}-\frac{382502559002826}{13\!\cdots\!25}a^{25}+\frac{99\!\cdots\!74}{64\!\cdots\!25}a^{24}-\frac{28\!\cdots\!51}{64\!\cdots\!25}a^{23}+\frac{22\!\cdots\!61}{27\!\cdots\!75}a^{22}-\frac{89\!\cdots\!63}{64\!\cdots\!25}a^{21}+\frac{13\!\cdots\!62}{64\!\cdots\!25}a^{20}-\frac{73\!\cdots\!01}{64\!\cdots\!25}a^{19}-\frac{23\!\cdots\!31}{64\!\cdots\!25}a^{18}+\frac{15\!\cdots\!03}{15\!\cdots\!75}a^{17}-\frac{43\!\cdots\!56}{49\!\cdots\!25}a^{16}-\frac{43\!\cdots\!43}{64\!\cdots\!25}a^{15}+\frac{15\!\cdots\!63}{64\!\cdots\!25}a^{14}-\frac{13\!\cdots\!77}{92\!\cdots\!75}a^{13}-\frac{19\!\cdots\!72}{12\!\cdots\!25}a^{12}+\frac{65\!\cdots\!53}{18\!\cdots\!75}a^{11}-\frac{28\!\cdots\!69}{49\!\cdots\!25}a^{10}-\frac{13\!\cdots\!54}{92\!\cdots\!75}a^{9}+\frac{87\!\cdots\!27}{64\!\cdots\!25}a^{8}+\frac{86\!\cdots\!01}{64\!\cdots\!25}a^{7}+\frac{25\!\cdots\!22}{64\!\cdots\!25}a^{6}-\frac{61\!\cdots\!67}{64\!\cdots\!25}a^{5}+\frac{79\!\cdots\!11}{64\!\cdots\!25}a^{4}+\frac{55\!\cdots\!52}{12\!\cdots\!25}a^{3}-\frac{98\!\cdots\!46}{64\!\cdots\!25}a^{2}+\frac{11\!\cdots\!06}{12\!\cdots\!25}a-\frac{47\!\cdots\!93}{51\!\cdots\!01}$, $\frac{6695897567}{468800871641375}a^{26}-\frac{9863796}{76538917819}a^{25}+\frac{280571090011}{468800871641375}a^{24}-\frac{565231937353}{468800871641375}a^{23}+\frac{550681081744}{468800871641375}a^{22}-\frac{700043322702}{468800871641375}a^{21}+\frac{182154983326}{93760174328275}a^{20}+\frac{3993167192876}{468800871641375}a^{19}-\frac{13513183851582}{468800871641375}a^{18}+\frac{13784719176629}{468800871641375}a^{17}+\frac{10977972577042}{468800871641375}a^{16}-\frac{38802159632914}{468800871641375}a^{15}+\frac{20179394499148}{468800871641375}a^{14}+\frac{7324795530112}{66971553091625}a^{13}-\frac{78682074424322}{468800871641375}a^{12}+\frac{2583525605666}{66971553091625}a^{11}+\frac{116758789216679}{468800871641375}a^{10}-\frac{909268618509}{5151657930125}a^{9}-\frac{49883700126303}{468800871641375}a^{8}+\frac{59603271271527}{468800871641375}a^{7}+\frac{146202608706659}{468800871641375}a^{6}-\frac{280605849127894}{468800871641375}a^{5}+\frac{15094996466619}{36061605510875}a^{4}+\frac{157790698152551}{468800871641375}a^{3}-\frac{90872287756098}{468800871641375}a^{2}-\frac{2305658274558}{3750406973131}a+\frac{4786241078783}{3750406973131}$, $\frac{5076242311062}{49\!\cdots\!25}a^{26}-\frac{159918308489664}{92\!\cdots\!75}a^{25}+\frac{293449616015516}{25\!\cdots\!05}a^{24}-\frac{26\!\cdots\!74}{64\!\cdots\!25}a^{23}+\frac{47\!\cdots\!09}{64\!\cdots\!25}a^{22}-\frac{63\!\cdots\!23}{64\!\cdots\!25}a^{21}+\frac{27\!\cdots\!04}{12\!\cdots\!25}a^{20}-\frac{20\!\cdots\!08}{64\!\cdots\!25}a^{19}-\frac{81\!\cdots\!87}{49\!\cdots\!25}a^{18}+\frac{51\!\cdots\!91}{64\!\cdots\!25}a^{17}+\frac{41\!\cdots\!87}{64\!\cdots\!25}a^{16}-\frac{38\!\cdots\!57}{13\!\cdots\!75}a^{15}+\frac{10\!\cdots\!68}{41\!\cdots\!75}a^{14}+\frac{22\!\cdots\!04}{70\!\cdots\!75}a^{13}-\frac{48\!\cdots\!21}{64\!\cdots\!25}a^{12}-\frac{43\!\cdots\!44}{92\!\cdots\!75}a^{11}+\frac{55\!\cdots\!86}{64\!\cdots\!25}a^{10}-\frac{16\!\cdots\!43}{16\!\cdots\!75}a^{9}-\frac{45\!\cdots\!36}{49\!\cdots\!25}a^{8}+\frac{61\!\cdots\!59}{64\!\cdots\!25}a^{7}+\frac{10\!\cdots\!16}{49\!\cdots\!25}a^{6}-\frac{10\!\cdots\!60}{51\!\cdots\!01}a^{5}-\frac{20\!\cdots\!16}{64\!\cdots\!25}a^{4}+\frac{70\!\cdots\!99}{64\!\cdots\!25}a^{3}-\frac{41\!\cdots\!02}{64\!\cdots\!25}a^{2}-\frac{24\!\cdots\!13}{64\!\cdots\!25}a+\frac{36\!\cdots\!72}{25\!\cdots\!05}$, $\frac{62098536544473}{49\!\cdots\!25}a^{26}-\frac{10\!\cdots\!54}{92\!\cdots\!75}a^{25}+\frac{33\!\cdots\!74}{64\!\cdots\!25}a^{24}-\frac{71\!\cdots\!08}{64\!\cdots\!25}a^{23}+\frac{10\!\cdots\!76}{64\!\cdots\!25}a^{22}-\frac{14\!\cdots\!07}{49\!\cdots\!25}a^{21}+\frac{27\!\cdots\!66}{64\!\cdots\!25}a^{20}+\frac{13\!\cdots\!99}{64\!\cdots\!25}a^{19}-\frac{65\!\cdots\!61}{49\!\cdots\!25}a^{18}+\frac{11\!\cdots\!43}{64\!\cdots\!25}a^{17}+\frac{51\!\cdots\!48}{49\!\cdots\!25}a^{16}-\frac{34\!\cdots\!47}{64\!\cdots\!25}a^{15}+\frac{32\!\cdots\!36}{64\!\cdots\!25}a^{14}+\frac{18\!\cdots\!08}{39\!\cdots\!25}a^{13}-\frac{59\!\cdots\!18}{64\!\cdots\!25}a^{12}+\frac{52\!\cdots\!42}{92\!\cdots\!75}a^{11}+\frac{95\!\cdots\!14}{64\!\cdots\!25}a^{10}-\frac{98\!\cdots\!83}{92\!\cdots\!75}a^{9}-\frac{67\!\cdots\!46}{12\!\cdots\!25}a^{8}+\frac{33\!\cdots\!04}{49\!\cdots\!25}a^{7}+\frac{13\!\cdots\!33}{64\!\cdots\!25}a^{6}-\frac{11\!\cdots\!63}{60\!\cdots\!35}a^{5}+\frac{40\!\cdots\!19}{64\!\cdots\!25}a^{4}+\frac{47\!\cdots\!41}{64\!\cdots\!25}a^{3}+\frac{45\!\cdots\!14}{49\!\cdots\!25}a^{2}-\frac{64\!\cdots\!92}{13\!\cdots\!75}a+\frac{58\!\cdots\!16}{25\!\cdots\!05}$ 1195802359071759/6455856803373375125a^26 - 1549888788736653/922265257624767875a^25 + 50496179184185866/6455856803373375125a^24 - 20983520568086352/1291171360674675025a^23 + 121752420396567944/6455856803373375125a^22 - 234659187811538018/6455856803373375125a^21 + 425834309204315447/6455856803373375125a^20 + 222904626023410561/6455856803373375125a^19 - 1373348085605849827/6455856803373375125a^18 + 768347735957836123/6455856803373375125a^17 + 2906286889759441564/6455856803373375125a^16 - 6933851347927848996/6455856803373375125a^15 + 2123270459315984661/6455856803373375125a^14 + 1545351543793501896/922265257624767875a^13 - 12128516560278495477/6455856803373375125a^12 - 642024366379079973/922265257624767875a^11 + 828798306719661892/258234272134935005a^10 - 748416866291597032/922265257624767875a^9 - 2293486487565436493/496604369490259625a^8 - 276468659767098352/258234272134935005a^7 + 18621634505708317313/6455856803373375125a^6 - 23034745874619833652/6455856803373375125a^5 - 697834196620636591/208253445270108875a^4 + 10719830016825765893/1291171360674675025a^3 - 3445410890359969952/1291171360674675025a^2 - 157255131903038912423/6455856803373375125a - 1078944005846449583/258234272134935005, 59389260354803/496604369490259625a^26 - 199457870579008/184453051524953575a^25 + 33701456853627611/6455856803373375125a^24 - 77218904289737466/6455856803373375125a^23 + 122018411021089487/6455856803373375125a^22 - 225417906963985941/6455856803373375125a^21 + 360077797750528033/6455856803373375125a^20 - 23352789702791973/6455856803373375125a^19 - 734773568259918102/6455856803373375125a^18 + 1143039100362644974/6455856803373375125a^17 + 432642088947280729/6455856803373375125a^16 - 4617873311627809/10566050414686375a^15 + 2841423860469499447/6455856803373375125a^14 + 7454467892072507/18821739951525875a^13 - 163479996416175983/208253445270108875a^12 + 38559548039105819/131752179660681125a^11 + 332149713109356736/258234272134935005a^10 + 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13982429992065839233/6455856803373375125a^6 - 11074655313499163/6005448189184535a^5 + 4051374673653030219/6455856803373375125a^4 + 47120596951279073241/6455856803373375125a^3 + 456289036384417514/496604369490259625a^2 - 644081867432740092/137358655390922875a + 586353438390235816/258234272134935005 (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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$7$ 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	$x^{2} + 21$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$521$ 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 $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

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 *.

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