Properties

Label 27.3.346...625.1
Degree 2727
Signature [3,12][3, 12]
Discriminant 3.466×10423.466\times 10^{42}
Root discriminant 37.6337.63
Ramified primes 3,53,5
Class number 22 (GRH)
Class group [2] (GRH)
Galois group C33:S4C_3^3:S_4 (as 27T211)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^27 - 9*x^26 + 63*x^25 - 309*x^24 + 1278*x^23 - 4374*x^22 + 13038*x^21 - 33687*x^20 + 76365*x^19 - 150976*x^18 + 256509*x^17 - 363843*x^16 + 387384*x^15 - 193563*x^14 - 398313*x^13 + 1478712*x^12 - 3028527*x^11 + 4723794*x^10 - 6156003*x^9 + 6804108*x^8 - 6491583*x^7 + 5295510*x^6 - 3678588*x^5 + 2126304*x^4 - 986832*x^3 + 348192*x^2 - 79488*x + 8064)
 
Copy content gp:K = bnfinit(y^27 - 9*y^26 + 63*y^25 - 309*y^24 + 1278*y^23 - 4374*y^22 + 13038*y^21 - 33687*y^20 + 76365*y^19 - 150976*y^18 + 256509*y^17 - 363843*y^16 + 387384*y^15 - 193563*y^14 - 398313*y^13 + 1478712*y^12 - 3028527*y^11 + 4723794*y^10 - 6156003*y^9 + 6804108*y^8 - 6491583*y^7 + 5295510*y^6 - 3678588*y^5 + 2126304*y^4 - 986832*y^3 + 348192*y^2 - 79488*y + 8064, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 9*x^26 + 63*x^25 - 309*x^24 + 1278*x^23 - 4374*x^22 + 13038*x^21 - 33687*x^20 + 76365*x^19 - 150976*x^18 + 256509*x^17 - 363843*x^16 + 387384*x^15 - 193563*x^14 - 398313*x^13 + 1478712*x^12 - 3028527*x^11 + 4723794*x^10 - 6156003*x^9 + 6804108*x^8 - 6491583*x^7 + 5295510*x^6 - 3678588*x^5 + 2126304*x^4 - 986832*x^3 + 348192*x^2 - 79488*x + 8064);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^27 - 9*x^26 + 63*x^25 - 309*x^24 + 1278*x^23 - 4374*x^22 + 13038*x^21 - 33687*x^20 + 76365*x^19 - 150976*x^18 + 256509*x^17 - 363843*x^16 + 387384*x^15 - 193563*x^14 - 398313*x^13 + 1478712*x^12 - 3028527*x^11 + 4723794*x^10 - 6156003*x^9 + 6804108*x^8 - 6491583*x^7 + 5295510*x^6 - 3678588*x^5 + 2126304*x^4 - 986832*x^3 + 348192*x^2 - 79488*x + 8064)
 

x279x26+63x25309x24+1278x234374x22+13038x2133687x20++8064 x^{27} - 9 x^{26} + 63 x^{25} - 309 x^{24} + 1278 x^{23} - 4374 x^{22} + 13038 x^{21} - 33687 x^{20} + \cdots + 8064 Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  2727
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  [3,12][3, 12]
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   34659944178485905939573150277137756347656253465994417848590593957315027713775634765625 =354524\medspace = 3^{54}\cdot 5^{24} Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  37.6337.63
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  337/1858/939.9996246809048443^{37/18}5^{8/9}\approx 39.999624680904844
Ramified primes:   33, 55 Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  Q\Q
Aut(K/Q)\Aut(K/\Q):   C1C_1
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over Q\Q.
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator aa)

11, aa, a2a^{2}, a3a^{3}, a4a^{4}, a5a^{5}, a6a^{6}, a7a^{7}, a8a^{8}, a9a^{9}, a10a^{10}, a11a^{11}, a12a^{12}, a13a^{13}, 13a14+13a13+13a1213a1113a1013a9\frac{1}{3}a^{14}+\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{9}, 13a15+13a12+13a9\frac{1}{3}a^{15}+\frac{1}{3}a^{12}+\frac{1}{3}a^{9}, 13a16+13a13+13a10\frac{1}{3}a^{16}+\frac{1}{3}a^{13}+\frac{1}{3}a^{10}, 13a1713a1313a1213a11+13a10+13a9\frac{1}{3}a^{17}-\frac{1}{3}a^{13}-\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}, 19a18+13a12+29a9+13a613a3+13\frac{1}{9}a^{18}+\frac{1}{3}a^{12}+\frac{2}{9}a^{9}+\frac{1}{3}a^{6}-\frac{1}{3}a^{3}+\frac{1}{3}, 118a1916a1616a1512a13+13a12+49a1016a9+16a7+13a412a312a2+16a\frac{1}{18}a^{19}-\frac{1}{6}a^{16}-\frac{1}{6}a^{15}-\frac{1}{2}a^{13}+\frac{1}{3}a^{12}+\frac{4}{9}a^{10}-\frac{1}{6}a^{9}+\frac{1}{6}a^{7}+\frac{1}{3}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{1}{6}a, 118a2016a1716a1616a1413a13+13a12+19a1112a1013a9+16a8+13a512a412a3+16a2\frac{1}{18}a^{20}-\frac{1}{6}a^{17}-\frac{1}{6}a^{16}-\frac{1}{6}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{1}{9}a^{11}-\frac{1}{2}a^{10}-\frac{1}{3}a^{9}+\frac{1}{6}a^{8}+\frac{1}{3}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{6}a^{2}, 118a21118a1816a1716a1513a1329a12+16a11+13a10+118a913a612a512a416a3+13\frac{1}{18}a^{21}-\frac{1}{18}a^{18}-\frac{1}{6}a^{17}-\frac{1}{6}a^{15}-\frac{1}{3}a^{13}-\frac{2}{9}a^{12}+\frac{1}{6}a^{11}+\frac{1}{3}a^{10}+\frac{1}{18}a^{9}-\frac{1}{3}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{6}a^{3}+\frac{1}{3}, 136a22136a21136a20136a19118a1816a1716a16112a15112a1419a131136a12+1336a11+19a10+536a9+512a8+13a7112a616a5112a4+13a3112a2+16a+13\frac{1}{36}a^{22}-\frac{1}{36}a^{21}-\frac{1}{36}a^{20}-\frac{1}{36}a^{19}-\frac{1}{18}a^{18}-\frac{1}{6}a^{17}-\frac{1}{6}a^{16}-\frac{1}{12}a^{15}-\frac{1}{12}a^{14}-\frac{1}{9}a^{13}-\frac{11}{36}a^{12}+\frac{13}{36}a^{11}+\frac{1}{9}a^{10}+\frac{5}{36}a^{9}+\frac{5}{12}a^{8}+\frac{1}{3}a^{7}-\frac{1}{12}a^{6}-\frac{1}{6}a^{5}-\frac{1}{12}a^{4}+\frac{1}{3}a^{3}-\frac{1}{12}a^{2}+\frac{1}{6}a+\frac{1}{3}, 172a23172a22172a21172a20136a19+136a18112a17124a16124a15118a141172a133572a1249a113172a10+3172a9+16a8+1124a7+14a6124a513a438a3512a2+16a+13\frac{1}{72}a^{23}-\frac{1}{72}a^{22}-\frac{1}{72}a^{21}-\frac{1}{72}a^{20}-\frac{1}{36}a^{19}+\frac{1}{36}a^{18}-\frac{1}{12}a^{17}-\frac{1}{24}a^{16}-\frac{1}{24}a^{15}-\frac{1}{18}a^{14}-\frac{11}{72}a^{13}-\frac{35}{72}a^{12}-\frac{4}{9}a^{11}-\frac{31}{72}a^{10}+\frac{31}{72}a^{9}+\frac{1}{6}a^{8}+\frac{11}{24}a^{7}+\frac{1}{4}a^{6}-\frac{1}{24}a^{5}-\frac{1}{3}a^{4}-\frac{3}{8}a^{3}-\frac{5}{12}a^{2}+\frac{1}{6}a+\frac{1}{3}, 1432a241144a231144a22+148a21136a20172a19172a18116a17548a1635216a15+19144a14+25144a13172a1253144a11+47144a10+1372a91948a816a7+31144a618a51148a4+14a3+512a2+13a\frac{1}{432}a^{24}-\frac{1}{144}a^{23}-\frac{1}{144}a^{22}+\frac{1}{48}a^{21}-\frac{1}{36}a^{20}-\frac{1}{72}a^{19}-\frac{1}{72}a^{18}-\frac{1}{16}a^{17}-\frac{5}{48}a^{16}-\frac{35}{216}a^{15}+\frac{19}{144}a^{14}+\frac{25}{144}a^{13}-\frac{1}{72}a^{12}-\frac{53}{144}a^{11}+\frac{47}{144}a^{10}+\frac{13}{72}a^{9}-\frac{19}{48}a^{8}-\frac{1}{6}a^{7}+\frac{31}{144}a^{6}-\frac{1}{8}a^{5}-\frac{11}{48}a^{4}+\frac{1}{4}a^{3}+\frac{5}{12}a^{2}+\frac{1}{3}a, 1864a251864a24+1288a23196a221144a21148a20+1144a19+11288a181196a1731216a16119864a1525288a141172a1353288a12332a111172a1089288a9+1348a8+19288a73172a6+3796a51748a413a3+14a213a\frac{1}{864}a^{25}-\frac{1}{864}a^{24}+\frac{1}{288}a^{23}-\frac{1}{96}a^{22}-\frac{1}{144}a^{21}-\frac{1}{48}a^{20}+\frac{1}{144}a^{19}+\frac{11}{288}a^{18}-\frac{11}{96}a^{17}-\frac{31}{216}a^{16}-\frac{119}{864}a^{15}-\frac{25}{288}a^{14}-\frac{11}{72}a^{13}-\frac{53}{288}a^{12}-\frac{3}{32}a^{11}-\frac{11}{72}a^{10}-\frac{89}{288}a^{9}+\frac{13}{48}a^{8}+\frac{19}{288}a^{7}-\frac{31}{72}a^{6}+\frac{37}{96}a^{5}-\frac{17}{48}a^{4}-\frac{1}{3}a^{3}+\frac{1}{4}a^{2}-\frac{1}{3}a, 11784a2618495928a2519271784a24+19556692a2381637416a2277999988a2156372964a2013055928a19+68175928a18+99958992a1788375928a1614011784a1520632964a14+11035928a13+38715928a1279852964a1125975928a1065071432a951015928a888633396a763575928a670194112a513674944a469791236a389832006a247096218a+69253159\frac{1}{17\cdots 84}a^{26}-\frac{18\cdots 49}{59\cdots 28}a^{25}-\frac{19\cdots 27}{17\cdots 84}a^{24}+\frac{19\cdots 55}{66\cdots 92}a^{23}-\frac{81\cdots 63}{74\cdots 16}a^{22}-\frac{77\cdots 99}{99\cdots 88}a^{21}-\frac{56\cdots 37}{29\cdots 64}a^{20}-\frac{13\cdots 05}{59\cdots 28}a^{19}+\frac{68\cdots 17}{59\cdots 28}a^{18}+\frac{99\cdots 95}{89\cdots 92}a^{17}-\frac{88\cdots 37}{59\cdots 28}a^{16}-\frac{14\cdots 01}{17\cdots 84}a^{15}-\frac{20\cdots 63}{29\cdots 64}a^{14}+\frac{11\cdots 03}{59\cdots 28}a^{13}+\frac{38\cdots 71}{59\cdots 28}a^{12}-\frac{79\cdots 85}{29\cdots 64}a^{11}-\frac{25\cdots 97}{59\cdots 28}a^{10}-\frac{65\cdots 07}{14\cdots 32}a^{9}-\frac{51\cdots 01}{59\cdots 28}a^{8}-\frac{88\cdots 63}{33\cdots 96}a^{7}-\frac{63\cdots 57}{59\cdots 28}a^{6}-\frac{70\cdots 19}{41\cdots 12}a^{5}-\frac{13\cdots 67}{49\cdots 44}a^{4}-\frac{69\cdots 79}{12\cdots 36}a^{3}-\frac{89\cdots 83}{20\cdots 06}a^{2}-\frac{47\cdots 09}{62\cdots 18}a+\frac{69\cdots 25}{31\cdots 59} Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  11
Inessential primes:  None

Class group and class number

Ideal class group:  C2C_{2}, which has order 22 (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  C2C_{2}, which has order 22 (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  1414
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   1 -1  (order 22) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   19432964a2657298224a25+11212472a2491953708a23+30892964a2251011432a21+94609377a2077432964a19+28914944a1832832964a17+17959988a1638451648a15+62012964a1425672964a1334017416a12+36592964a1165792964a10+31559988a936439988a8+12073396a731379988a6+75173396a511698224a4+15532472a328971236a2+31276218a47671053\frac{19\cdots 43}{29\cdots 64}a^{26}-\frac{57\cdots 29}{82\cdots 24}a^{25}+\frac{11\cdots 21}{24\cdots 72}a^{24}-\frac{91\cdots 95}{37\cdots 08}a^{23}+\frac{30\cdots 89}{29\cdots 64}a^{22}-\frac{51\cdots 01}{14\cdots 32}a^{21}+\frac{94\cdots 60}{93\cdots 77}a^{20}-\frac{77\cdots 43}{29\cdots 64}a^{19}+\frac{28\cdots 91}{49\cdots 44}a^{18}-\frac{32\cdots 83}{29\cdots 64}a^{17}+\frac{17\cdots 95}{99\cdots 88}a^{16}-\frac{38\cdots 45}{16\cdots 48}a^{15}+\frac{62\cdots 01}{29\cdots 64}a^{14}-\frac{25\cdots 67}{29\cdots 64}a^{13}-\frac{34\cdots 01}{74\cdots 16}a^{12}+\frac{36\cdots 59}{29\cdots 64}a^{11}-\frac{65\cdots 79}{29\cdots 64}a^{10}+\frac{31\cdots 55}{99\cdots 88}a^{9}-\frac{36\cdots 43}{99\cdots 88}a^{8}+\frac{12\cdots 07}{33\cdots 96}a^{7}-\frac{31\cdots 37}{99\cdots 88}a^{6}+\frac{75\cdots 17}{33\cdots 96}a^{5}-\frac{11\cdots 69}{82\cdots 24}a^{4}+\frac{15\cdots 53}{24\cdots 72}a^{3}-\frac{28\cdots 97}{12\cdots 36}a^{2}+\frac{31\cdots 27}{62\cdots 18}a-\frac{47\cdots 67}{10\cdots 53}, 18091976a26+43015928a2529395928a24+44911976a2314551648a22+27899988a2123112964a20+11415928a1926776692a18+24973396a1766355928a16+81955928a1534153396a1423036692a13+68791976a1223572964a11+82695928a1094354944a9+44751976a874533396a7+39791976a618351236a5+23872472a439678224a3+12956218a235176218a+83101053\frac{18\cdots 09}{19\cdots 76}a^{26}+\frac{43\cdots 01}{59\cdots 28}a^{25}-\frac{29\cdots 39}{59\cdots 28}a^{24}+\frac{44\cdots 91}{19\cdots 76}a^{23}-\frac{14\cdots 55}{16\cdots 48}a^{22}+\frac{27\cdots 89}{99\cdots 88}a^{21}-\frac{23\cdots 11}{29\cdots 64}a^{20}+\frac{11\cdots 41}{59\cdots 28}a^{19}-\frac{26\cdots 77}{66\cdots 92}a^{18}+\frac{24\cdots 97}{33\cdots 96}a^{17}-\frac{66\cdots 35}{59\cdots 28}a^{16}+\frac{81\cdots 95}{59\cdots 28}a^{15}-\frac{34\cdots 15}{33\cdots 96}a^{14}-\frac{23\cdots 03}{66\cdots 92}a^{13}+\frac{68\cdots 79}{19\cdots 76}a^{12}-\frac{23\cdots 57}{29\cdots 64}a^{11}+\frac{82\cdots 69}{59\cdots 28}a^{10}-\frac{94\cdots 35}{49\cdots 44}a^{9}+\frac{44\cdots 75}{19\cdots 76}a^{8}-\frac{74\cdots 53}{33\cdots 96}a^{7}+\frac{39\cdots 79}{19\cdots 76}a^{6}-\frac{18\cdots 35}{12\cdots 36}a^{5}+\frac{23\cdots 87}{24\cdots 72}a^{4}-\frac{39\cdots 67}{82\cdots 24}a^{3}+\frac{12\cdots 95}{62\cdots 18}a^{2}-\frac{35\cdots 17}{62\cdots 18}a+\frac{83\cdots 10}{10\cdots 53}, 27911784a26+10591784a2545571784a2434275928a23+96212964a2270112964a21+29692964a2020575928a19+56075928a1824391124a17+79891784a1613671784a15+25412472a1462096692a13+93055928a12+68193708a1130735928a10+27532964a975455928a8+11437416a786255928a6+39833396a520432472a4+11132472a324791236a2+59931053a76911053\frac{27\cdots 91}{17\cdots 84}a^{26}+\frac{10\cdots 59}{17\cdots 84}a^{25}-\frac{45\cdots 57}{17\cdots 84}a^{24}-\frac{34\cdots 27}{59\cdots 28}a^{23}+\frac{96\cdots 21}{29\cdots 64}a^{22}-\frac{70\cdots 11}{29\cdots 64}a^{21}+\frac{29\cdots 69}{29\cdots 64}a^{20}-\frac{20\cdots 57}{59\cdots 28}a^{19}+\frac{56\cdots 07}{59\cdots 28}a^{18}-\frac{24\cdots 39}{11\cdots 24}a^{17}+\frac{79\cdots 89}{17\cdots 84}a^{16}-\frac{13\cdots 67}{17\cdots 84}a^{15}+\frac{25\cdots 41}{24\cdots 72}a^{14}-\frac{62\cdots 09}{66\cdots 92}a^{13}+\frac{93\cdots 05}{59\cdots 28}a^{12}+\frac{68\cdots 19}{37\cdots 08}a^{11}-\frac{30\cdots 73}{59\cdots 28}a^{10}+\frac{27\cdots 53}{29\cdots 64}a^{9}-\frac{75\cdots 45}{59\cdots 28}a^{8}+\frac{11\cdots 43}{74\cdots 16}a^{7}-\frac{86\cdots 25}{59\cdots 28}a^{6}+\frac{39\cdots 83}{33\cdots 96}a^{5}-\frac{20\cdots 43}{24\cdots 72}a^{4}+\frac{11\cdots 13}{24\cdots 72}a^{3}-\frac{24\cdots 79}{12\cdots 36}a^{2}+\frac{59\cdots 93}{10\cdots 53}a-\frac{76\cdots 91}{10\cdots 53}, 28691784a26+24291784a2516791784a24+86411976a2351952964a22+16472964a2148492964a20+23575928a1951995928a18+71594496a1744011784a16+55471784a1536651432a1442075928a13+45635928a1227531432a11+18335928a1012392964a9+29055928a871571432a7+24535928a694093396a5+51543159a419652472a3+11974112a212872006a+67741053\frac{28\cdots 69}{17\cdots 84}a^{26}+\frac{24\cdots 29}{17\cdots 84}a^{25}-\frac{16\cdots 79}{17\cdots 84}a^{24}+\frac{86\cdots 41}{19\cdots 76}a^{23}-\frac{51\cdots 95}{29\cdots 64}a^{22}+\frac{16\cdots 47}{29\cdots 64}a^{21}-\frac{48\cdots 49}{29\cdots 64}a^{20}+\frac{23\cdots 57}{59\cdots 28}a^{19}-\frac{51\cdots 99}{59\cdots 28}a^{18}+\frac{71\cdots 59}{44\cdots 96}a^{17}-\frac{44\cdots 01}{17\cdots 84}a^{16}+\frac{55\cdots 47}{17\cdots 84}a^{15}-\frac{36\cdots 65}{14\cdots 32}a^{14}-\frac{42\cdots 07}{59\cdots 28}a^{13}+\frac{45\cdots 63}{59\cdots 28}a^{12}-\frac{27\cdots 53}{14\cdots 32}a^{11}+\frac{18\cdots 33}{59\cdots 28}a^{10}-\frac{12\cdots 39}{29\cdots 64}a^{9}+\frac{29\cdots 05}{59\cdots 28}a^{8}-\frac{71\cdots 57}{14\cdots 32}a^{7}+\frac{24\cdots 53}{59\cdots 28}a^{6}-\frac{94\cdots 09}{33\cdots 96}a^{5}+\frac{51\cdots 54}{31\cdots 59}a^{4}-\frac{19\cdots 65}{24\cdots 72}a^{3}+\frac{11\cdots 97}{41\cdots 12}a^{2}-\frac{12\cdots 87}{20\cdots 06}a+\frac{67\cdots 74}{10\cdots 53}, 20431784a26+12191784a2512091784a24+35975928a2321937416a22+39313396a2113153396a20+67895928a1955511976a18+52298992a1718051784a16+28051784a1518359988a14+71216692a13+77375928a1260699988a11+74495928a1016898224a9+15135928a881272964a7+14615928a693754944a5+36973159a418733159a3+28571236a258971053a+18653159\frac{20\cdots 43}{17\cdots 84}a^{26}+\frac{12\cdots 19}{17\cdots 84}a^{25}-\frac{12\cdots 09}{17\cdots 84}a^{24}+\frac{35\cdots 97}{59\cdots 28}a^{23}-\frac{21\cdots 93}{74\cdots 16}a^{22}+\frac{39\cdots 31}{33\cdots 96}a^{21}-\frac{13\cdots 15}{33\cdots 96}a^{20}+\frac{67\cdots 89}{59\cdots 28}a^{19}-\frac{55\cdots 51}{19\cdots 76}a^{18}+\frac{52\cdots 29}{89\cdots 92}a^{17}-\frac{18\cdots 05}{17\cdots 84}a^{16}+\frac{28\cdots 05}{17\cdots 84}a^{15}-\frac{18\cdots 35}{99\cdots 88}a^{14}+\frac{71\cdots 21}{66\cdots 92}a^{13}+\frac{77\cdots 37}{59\cdots 28}a^{12}-\frac{60\cdots 69}{99\cdots 88}a^{11}+\frac{74\cdots 49}{59\cdots 28}a^{10}-\frac{16\cdots 89}{82\cdots 24}a^{9}+\frac{15\cdots 13}{59\cdots 28}a^{8}-\frac{81\cdots 27}{29\cdots 64}a^{7}+\frac{14\cdots 61}{59\cdots 28}a^{6}-\frac{93\cdots 75}{49\cdots 44}a^{5}+\frac{36\cdots 97}{31\cdots 59}a^{4}-\frac{18\cdots 73}{31\cdots 59}a^{3}+\frac{28\cdots 57}{12\cdots 36}a^{2}-\frac{58\cdots 97}{10\cdots 53}a+\frac{18\cdots 65}{31\cdots 59}, 12459988a2635892964a25+79099988a2412813396a23+58813708a2277511432a21+22051432a2011092964a19+81159988a1825731648a17+73712964a1631059988a15+13854944a14+24792964a1319092964a12+25451432a1191812964a10+35738224a950799988a8+25634944a744999988a6+40351236a548512472a4+23132472a344591236a2+27373159a10671053\frac{12\cdots 45}{99\cdots 88}a^{26}-\frac{35\cdots 89}{29\cdots 64}a^{25}+\frac{79\cdots 09}{99\cdots 88}a^{24}-\frac{12\cdots 81}{33\cdots 96}a^{23}+\frac{58\cdots 81}{37\cdots 08}a^{22}-\frac{77\cdots 51}{14\cdots 32}a^{21}+\frac{22\cdots 05}{14\cdots 32}a^{20}-\frac{11\cdots 09}{29\cdots 64}a^{19}+\frac{81\cdots 15}{99\cdots 88}a^{18}-\frac{25\cdots 73}{16\cdots 48}a^{17}+\frac{73\cdots 71}{29\cdots 64}a^{16}-\frac{31\cdots 05}{99\cdots 88}a^{15}+\frac{13\cdots 85}{49\cdots 44}a^{14}+\frac{24\cdots 79}{29\cdots 64}a^{13}-\frac{19\cdots 09}{29\cdots 64}a^{12}+\frac{25\cdots 45}{14\cdots 32}a^{11}-\frac{91\cdots 81}{29\cdots 64}a^{10}+\frac{35\cdots 73}{82\cdots 24}a^{9}-\frac{50\cdots 79}{99\cdots 88}a^{8}+\frac{25\cdots 63}{49\cdots 44}a^{7}-\frac{44\cdots 99}{99\cdots 88}a^{6}+\frac{40\cdots 35}{12\cdots 36}a^{5}-\frac{48\cdots 51}{24\cdots 72}a^{4}+\frac{23\cdots 13}{24\cdots 72}a^{3}-\frac{44\cdots 59}{12\cdots 36}a^{2}+\frac{27\cdots 37}{31\cdots 59}a-\frac{10\cdots 67}{10\cdots 53}, 18831784a2614191784a25+94491784a2414415928a23+27572964a2287752964a21+80459988a2011515928a19+23695928a1830274496a17+17911784a1618011784a15+21754944a14+24431976a1326895928a12+41734944a1175635928a10+45472964a993555928a8+97717416a753275928a6+46999988a581214944a4+36772472a3+32851236a229992006a+22871053\frac{18\cdots 83}{17\cdots 84}a^{26}-\frac{14\cdots 19}{17\cdots 84}a^{25}+\frac{94\cdots 49}{17\cdots 84}a^{24}-\frac{14\cdots 41}{59\cdots 28}a^{23}+\frac{27\cdots 57}{29\cdots 64}a^{22}-\frac{87\cdots 75}{29\cdots 64}a^{21}+\frac{80\cdots 45}{99\cdots 88}a^{20}-\frac{11\cdots 51}{59\cdots 28}a^{19}+\frac{23\cdots 69}{59\cdots 28}a^{18}-\frac{30\cdots 27}{44\cdots 96}a^{17}+\frac{17\cdots 91}{17\cdots 84}a^{16}-\frac{18\cdots 01}{17\cdots 84}a^{15}+\frac{21\cdots 75}{49\cdots 44}a^{14}+\frac{24\cdots 43}{19\cdots 76}a^{13}-\frac{26\cdots 89}{59\cdots 28}a^{12}+\frac{41\cdots 73}{49\cdots 44}a^{11}-\frac{75\cdots 63}{59\cdots 28}a^{10}+\frac{45\cdots 47}{29\cdots 64}a^{9}-\frac{93\cdots 55}{59\cdots 28}a^{8}+\frac{97\cdots 71}{74\cdots 16}a^{7}-\frac{53\cdots 27}{59\cdots 28}a^{6}+\frac{46\cdots 99}{99\cdots 88}a^{5}-\frac{81\cdots 21}{49\cdots 44}a^{4}+\frac{36\cdots 77}{24\cdots 72}a^{3}+\frac{32\cdots 85}{12\cdots 36}a^{2}-\frac{29\cdots 99}{20\cdots 06}a+\frac{22\cdots 87}{10\cdots 53}, 43455928a2642375928a25+12931784a2427015928a23+82773708a2287399988a21+86312964a2049415928a19+12655928a1812072964a17+46715928a1621731784a15+42412964a1459815928a1341735928a12+12512964a1155495928a10+22591432a913456692a8+72653396a712215928a6+40152472a518171648a4+72791236a330931236a2+42336218a26373159\frac{43\cdots 45}{59\cdots 28}a^{26}-\frac{42\cdots 37}{59\cdots 28}a^{25}+\frac{12\cdots 93}{17\cdots 84}a^{24}-\frac{27\cdots 01}{59\cdots 28}a^{23}+\frac{82\cdots 77}{37\cdots 08}a^{22}-\frac{87\cdots 39}{99\cdots 88}a^{21}+\frac{86\cdots 31}{29\cdots 64}a^{20}-\frac{49\cdots 41}{59\cdots 28}a^{19}+\frac{12\cdots 65}{59\cdots 28}a^{18}-\frac{12\cdots 07}{29\cdots 64}a^{17}+\frac{46\cdots 71}{59\cdots 28}a^{16}-\frac{21\cdots 73}{17\cdots 84}a^{15}+\frac{42\cdots 41}{29\cdots 64}a^{14}-\frac{59\cdots 81}{59\cdots 28}a^{13}-\frac{41\cdots 73}{59\cdots 28}a^{12}+\frac{12\cdots 51}{29\cdots 64}a^{11}-\frac{55\cdots 49}{59\cdots 28}a^{10}+\frac{22\cdots 59}{14\cdots 32}a^{9}-\frac{13\cdots 45}{66\cdots 92}a^{8}+\frac{72\cdots 65}{33\cdots 96}a^{7}-\frac{12\cdots 21}{59\cdots 28}a^{6}+\frac{40\cdots 15}{24\cdots 72}a^{5}-\frac{18\cdots 17}{16\cdots 48}a^{4}+\frac{72\cdots 79}{12\cdots 36}a^{3}-\frac{30\cdots 93}{12\cdots 36}a^{2}+\frac{42\cdots 33}{62\cdots 18}a-\frac{26\cdots 37}{31\cdots 59}, 48399988a26+45434496a2533574496a24+70311432a2360172964a22+37934944a2157172472a20+18492964a1986836218a18+27899988a1740698992a16+13832248a1516692964a14+15052964a13+16291432a1210033396a11+16692964a1025613396a9+91659988a827592964a7+22372964a654559988a5+78412472a436932472a3+10132006a268556218a+38183159\frac{48\cdots 39}{99\cdots 88}a^{26}+\frac{45\cdots 43}{44\cdots 96}a^{25}-\frac{33\cdots 57}{44\cdots 96}a^{24}+\frac{70\cdots 31}{14\cdots 32}a^{23}-\frac{60\cdots 17}{29\cdots 64}a^{22}+\frac{37\cdots 93}{49\cdots 44}a^{21}-\frac{57\cdots 17}{24\cdots 72}a^{20}+\frac{18\cdots 49}{29\cdots 64}a^{19}-\frac{86\cdots 83}{62\cdots 18}a^{18}+\frac{27\cdots 89}{99\cdots 88}a^{17}-\frac{40\cdots 69}{89\cdots 92}a^{16}+\frac{13\cdots 83}{22\cdots 48}a^{15}-\frac{16\cdots 69}{29\cdots 64}a^{14}+\frac{15\cdots 05}{29\cdots 64}a^{13}+\frac{16\cdots 29}{14\cdots 32}a^{12}-\frac{10\cdots 03}{33\cdots 96}a^{11}+\frac{16\cdots 69}{29\cdots 64}a^{10}-\frac{25\cdots 61}{33\cdots 96}a^{9}+\frac{91\cdots 65}{99\cdots 88}a^{8}-\frac{27\cdots 59}{29\cdots 64}a^{7}+\frac{22\cdots 37}{29\cdots 64}a^{6}-\frac{54\cdots 55}{99\cdots 88}a^{5}+\frac{78\cdots 41}{24\cdots 72}a^{4}-\frac{36\cdots 93}{24\cdots 72}a^{3}+\frac{10\cdots 13}{20\cdots 06}a^{2}-\frac{68\cdots 55}{62\cdots 18}a+\frac{38\cdots 18}{31\cdots 59}, 21171784a26+19471784a2513331784a24+73311976a2355893708a22+14412964a2143632964a20+24296692a1947295928a18+13258992a1743371784a16+56291784a1584812964a14+34391976a13+37975928a1249632964a11+59671976a1063171432a9+30855928a815132964a7+27015928a610293159a5+10454944a426832472a3+51811236a268236218a+13441053\frac{21\cdots 17}{17\cdots 84}a^{26}+\frac{19\cdots 47}{17\cdots 84}a^{25}-\frac{13\cdots 33}{17\cdots 84}a^{24}+\frac{73\cdots 31}{19\cdots 76}a^{23}-\frac{55\cdots 89}{37\cdots 08}a^{22}+\frac{14\cdots 41}{29\cdots 64}a^{21}-\frac{43\cdots 63}{29\cdots 64}a^{20}+\frac{24\cdots 29}{66\cdots 92}a^{19}-\frac{47\cdots 29}{59\cdots 28}a^{18}+\frac{13\cdots 25}{89\cdots 92}a^{17}-\frac{43\cdots 37}{17\cdots 84}a^{16}+\frac{56\cdots 29}{17\cdots 84}a^{15}-\frac{84\cdots 81}{29\cdots 64}a^{14}+\frac{34\cdots 39}{19\cdots 76}a^{13}+\frac{37\cdots 97}{59\cdots 28}a^{12}-\frac{49\cdots 63}{29\cdots 64}a^{11}+\frac{59\cdots 67}{19\cdots 76}a^{10}-\frac{63\cdots 17}{14\cdots 32}a^{9}+\frac{30\cdots 85}{59\cdots 28}a^{8}-\frac{15\cdots 13}{29\cdots 64}a^{7}+\frac{27\cdots 01}{59\cdots 28}a^{6}-\frac{10\cdots 29}{31\cdots 59}a^{5}+\frac{10\cdots 45}{49\cdots 44}a^{4}-\frac{26\cdots 83}{24\cdots 72}a^{3}+\frac{51\cdots 81}{12\cdots 36}a^{2}-\frac{68\cdots 23}{62\cdots 18}a+\frac{13\cdots 44}{10\cdots 53}, 95754944a26+52652964a2536972964a24+17152964a2378393396a22+11891432a2111094944a20+21313708a1912299988a18+81513396a1759831432a16+15452964a1514932964a14+29354944a13+29372964a1226579988a11+73491432a1070919988a9+10791236a829053396a7+38934944a657199988a5+18894944a445412472a3+29314112a258053159a+23411053\frac{95\cdots 75}{49\cdots 44}a^{26}+\frac{52\cdots 65}{29\cdots 64}a^{25}-\frac{36\cdots 97}{29\cdots 64}a^{24}+\frac{17\cdots 15}{29\cdots 64}a^{23}-\frac{78\cdots 39}{33\cdots 96}a^{22}+\frac{11\cdots 89}{14\cdots 32}a^{21}-\frac{11\cdots 09}{49\cdots 44}a^{20}+\frac{21\cdots 31}{37\cdots 08}a^{19}-\frac{12\cdots 29}{99\cdots 88}a^{18}+\frac{81\cdots 51}{33\cdots 96}a^{17}-\frac{59\cdots 83}{14\cdots 32}a^{16}+\frac{15\cdots 45}{29\cdots 64}a^{15}-\frac{14\cdots 93}{29\cdots 64}a^{14}+\frac{29\cdots 35}{49\cdots 44}a^{13}+\frac{29\cdots 37}{29\cdots 64}a^{12}-\frac{26\cdots 57}{99\cdots 88}a^{11}+\frac{73\cdots 49}{14\cdots 32}a^{10}-\frac{70\cdots 91}{99\cdots 88}a^{9}+\frac{10\cdots 79}{12\cdots 36}a^{8}-\frac{29\cdots 05}{33\cdots 96}a^{7}+\frac{38\cdots 93}{49\cdots 44}a^{6}-\frac{57\cdots 19}{99\cdots 88}a^{5}+\frac{18\cdots 89}{49\cdots 44}a^{4}-\frac{45\cdots 41}{24\cdots 72}a^{3}+\frac{29\cdots 31}{41\cdots 12}a^{2}-\frac{58\cdots 05}{31\cdots 59}a+\frac{23\cdots 41}{10\cdots 53}, 32371784a2627491784a25+18231784a2429195928a23+58032964a2221293396a21+54612964a2026975928a19+57755928a1880434496a17+50771784a1662191784a15+41571432a14+45435928a1351995928a12+30831432a1120655928a10+14212964a933295928a8+20413708a727655928a6+32099988a594734944a4+11191236a365472006a2+76001053a24223159\frac{32\cdots 37}{17\cdots 84}a^{26}-\frac{27\cdots 49}{17\cdots 84}a^{25}+\frac{18\cdots 23}{17\cdots 84}a^{24}-\frac{29\cdots 19}{59\cdots 28}a^{23}+\frac{58\cdots 03}{29\cdots 64}a^{22}-\frac{21\cdots 29}{33\cdots 96}a^{21}+\frac{54\cdots 61}{29\cdots 64}a^{20}-\frac{26\cdots 97}{59\cdots 28}a^{19}+\frac{57\cdots 75}{59\cdots 28}a^{18}-\frac{80\cdots 43}{44\cdots 96}a^{17}+\frac{50\cdots 77}{17\cdots 84}a^{16}-\frac{62\cdots 19}{17\cdots 84}a^{15}+\frac{41\cdots 57}{14\cdots 32}a^{14}+\frac{45\cdots 43}{59\cdots 28}a^{13}-\frac{51\cdots 99}{59\cdots 28}a^{12}+\frac{30\cdots 83}{14\cdots 32}a^{11}-\frac{20\cdots 65}{59\cdots 28}a^{10}+\frac{14\cdots 21}{29\cdots 64}a^{9}-\frac{33\cdots 29}{59\cdots 28}a^{8}+\frac{20\cdots 41}{37\cdots 08}a^{7}-\frac{27\cdots 65}{59\cdots 28}a^{6}+\frac{32\cdots 09}{99\cdots 88}a^{5}-\frac{94\cdots 73}{49\cdots 44}a^{4}+\frac{11\cdots 19}{12\cdots 36}a^{3}-\frac{65\cdots 47}{20\cdots 06}a^{2}+\frac{76\cdots 00}{10\cdots 53}a-\frac{24\cdots 22}{31\cdots 59}, 96016692a26+16671784a2533035928a24+12295928a2322372964a22+60332964a2114632964a20+58716692a1982816692a18+46394944a17+27051784a1644095928a15+27831432a1418855928a13+24855928a1248291432a1138151976a10+22873396a929591976a8+32451432a716256692a6+75093396a570174112a4+12751236a354411236a2+81536218a17491053\frac{96\cdots 01}{66\cdots 92}a^{26}+\frac{16\cdots 67}{17\cdots 84}a^{25}-\frac{33\cdots 03}{59\cdots 28}a^{24}+\frac{12\cdots 29}{59\cdots 28}a^{23}-\frac{22\cdots 37}{29\cdots 64}a^{22}+\frac{60\cdots 33}{29\cdots 64}a^{21}-\frac{14\cdots 63}{29\cdots 64}a^{20}+\frac{58\cdots 71}{66\cdots 92}a^{19}-\frac{82\cdots 81}{66\cdots 92}a^{18}+\frac{46\cdots 39}{49\cdots 44}a^{17}+\frac{27\cdots 05}{17\cdots 84}a^{16}-\frac{44\cdots 09}{59\cdots 28}a^{15}+\frac{27\cdots 83}{14\cdots 32}a^{14}-\frac{18\cdots 85}{59\cdots 28}a^{13}+\frac{24\cdots 85}{59\cdots 28}a^{12}-\frac{48\cdots 29}{14\cdots 32}a^{11}-\frac{38\cdots 15}{19\cdots 76}a^{10}+\frac{22\cdots 87}{33\cdots 96}a^{9}-\frac{29\cdots 59}{19\cdots 76}a^{8}+\frac{32\cdots 45}{14\cdots 32}a^{7}-\frac{16\cdots 25}{66\cdots 92}a^{6}+\frac{75\cdots 09}{33\cdots 96}a^{5}-\frac{70\cdots 17}{41\cdots 12}a^{4}+\frac{12\cdots 75}{12\cdots 36}a^{3}-\frac{54\cdots 41}{12\cdots 36}a^{2}+\frac{81\cdots 53}{62\cdots 18}a-\frac{17\cdots 49}{10\cdots 53}, 11014496a2619658992a25+43612964a2420112964a23+26499988a2216051854a21+37391432a2045597416a19+38652964a1821338992a17+82052248a1613792964a15+32679988a14+11217416a1335592964a12+81992964a1117753708a10+18152964a910071432a8+20192964a771271236a6+39099988a514916218a4+43094112a372692006a2+41196218a15213159\frac{11\cdots 01}{44\cdots 96}a^{26}-\frac{19\cdots 65}{89\cdots 92}a^{25}+\frac{43\cdots 61}{29\cdots 64}a^{24}-\frac{20\cdots 11}{29\cdots 64}a^{23}+\frac{26\cdots 49}{99\cdots 88}a^{22}-\frac{16\cdots 05}{18\cdots 54}a^{21}+\frac{37\cdots 39}{14\cdots 32}a^{20}-\frac{45\cdots 59}{74\cdots 16}a^{19}+\frac{38\cdots 65}{29\cdots 64}a^{18}-\frac{21\cdots 33}{89\cdots 92}a^{17}+\frac{82\cdots 05}{22\cdots 48}a^{16}-\frac{13\cdots 79}{29\cdots 64}a^{15}+\frac{32\cdots 67}{99\cdots 88}a^{14}+\frac{11\cdots 21}{74\cdots 16}a^{13}-\frac{35\cdots 59}{29\cdots 64}a^{12}+\frac{81\cdots 99}{29\cdots 64}a^{11}-\frac{17\cdots 75}{37\cdots 08}a^{10}+\frac{18\cdots 15}{29\cdots 64}a^{9}-\frac{10\cdots 07}{14\cdots 32}a^{8}+\frac{20\cdots 19}{29\cdots 64}a^{7}-\frac{71\cdots 27}{12\cdots 36}a^{6}+\frac{39\cdots 09}{99\cdots 88}a^{5}-\frac{14\cdots 91}{62\cdots 18}a^{4}+\frac{43\cdots 09}{41\cdots 12}a^{3}-\frac{72\cdots 69}{20\cdots 06}a^{2}+\frac{41\cdots 19}{62\cdots 18}a-\frac{15\cdots 21}{31\cdots 59} Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  102814738585.40065 102814738585.40065 (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

lims1(s1)ζK(s)=(2r1(2π)r2RhwD(23(2π)12102814738585.40065223465994417848590593957315027713775634765625(1.67259137697075 \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^{3}\cdot(2\pi)^{12}\cdot 102814738585.40065 \cdot 2}{2\cdot\sqrt{3465994417848590593957315027713775634765625}}\cr\approx \mathstrut & 1.67259137697075 \end{aligned} (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^27 - 9*x^26 + 63*x^25 - 309*x^24 + 1278*x^23 - 4374*x^22 + 13038*x^21 - 33687*x^20 + 76365*x^19 - 150976*x^18 + 256509*x^17 - 363843*x^16 + 387384*x^15 - 193563*x^14 - 398313*x^13 + 1478712*x^12 - 3028527*x^11 + 4723794*x^10 - 6156003*x^9 + 6804108*x^8 - 6491583*x^7 + 5295510*x^6 - 3678588*x^5 + 2126304*x^4 - 986832*x^3 + 348192*x^2 - 79488*x + 8064) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^27 - 9*x^26 + 63*x^25 - 309*x^24 + 1278*x^23 - 4374*x^22 + 13038*x^21 - 33687*x^20 + 76365*x^19 - 150976*x^18 + 256509*x^17 - 363843*x^16 + 387384*x^15 - 193563*x^14 - 398313*x^13 + 1478712*x^12 - 3028527*x^11 + 4723794*x^10 - 6156003*x^9 + 6804108*x^8 - 6491583*x^7 + 5295510*x^6 - 3678588*x^5 + 2126304*x^4 - 986832*x^3 + 348192*x^2 - 79488*x + 8064, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 9*x^26 + 63*x^25 - 309*x^24 + 1278*x^23 - 4374*x^22 + 13038*x^21 - 33687*x^20 + 76365*x^19 - 150976*x^18 + 256509*x^17 - 363843*x^16 + 387384*x^15 - 193563*x^14 - 398313*x^13 + 1478712*x^12 - 3028527*x^11 + 4723794*x^10 - 6156003*x^9 + 6804108*x^8 - 6491583*x^7 + 5295510*x^6 - 3678588*x^5 + 2126304*x^4 - 986832*x^3 + 348192*x^2 - 79488*x + 8064); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 9*x^26 + 63*x^25 - 309*x^24 + 1278*x^23 - 4374*x^22 + 13038*x^21 - 33687*x^20 + 76365*x^19 - 150976*x^18 + 256509*x^17 - 363843*x^16 + 387384*x^15 - 193563*x^14 - 398313*x^13 + 1478712*x^12 - 3028527*x^11 + 4723794*x^10 - 6156003*x^9 + 6804108*x^8 - 6491583*x^7 + 5295510*x^6 - 3678588*x^5 + 2126304*x^4 - 986832*x^3 + 348192*x^2 - 79488*x + 8064); 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

C33:S4C_3^3:S_4 (as 27T211):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 648
The 14 conjugacy class representatives for C33:S4C_3^3:S_4
Character table for C33:S4C_3^3:S_4

Intermediate fields

3.1.6075.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 9 sibling: data not computed
Degree 12 siblings: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 27 sibling: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 9.1.151336128515625.2

Frobenius cycle types

pp 22 33 55 77 1111 1313 1717 1919 2323 2929 3131 3737 4141 4343 4747 5353 5959
Cycle type 12,6,42,1{\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} } R R 62,32,24,1{\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} } 12,6,42,1{\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} } 93{\href{/padicField/13.9.0.1}{9} }^{3} 45,23,1{\href{/padicField/17.4.0.1}{4} }^{5}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} } 39{\href{/padicField/19.3.0.1}{3} }^{9} 64,3{\href{/padicField/23.6.0.1}{6} }^{4}{,}\,{\href{/padicField/23.3.0.1}{3} } 12,6,42,1{\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} } 62,32,24,1{\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} } 93{\href{/padicField/37.9.0.1}{9} }^{3} 45,23,1{\href{/padicField/41.4.0.1}{4} }^{5}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} } 93{\href{/padicField/43.9.0.1}{9} }^{3} 12,6,42,1{\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} } 12,6,42,1{\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} } 12,6,42,1{\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\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.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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

ppLabelPolynomial ee ff cc Galois group Slope content
33 Copy content Toggle raw display 3.1.9.18b1.4x9+3x3+9x2+9x+3x^{9} + 3 x^{3} + 9 x^{2} + 9 x + 399111818C32:C2C_3^2:C_2[32,52]2[\frac{3}{2}, \frac{5}{2}]_{2}
3.1.9.18b1.4x9+3x3+9x2+9x+3x^{9} + 3 x^{3} + 9 x^{2} + 9 x + 399111818C32:C2C_3^2:C_2[32,52]2[\frac{3}{2}, \frac{5}{2}]_{2}
3.1.9.18b1.4x9+3x3+9x2+9x+3x^{9} + 3 x^{3} + 9 x^{2} + 9 x + 399111818C32:C2C_3^2:C_2[32,52]2[\frac{3}{2}, \frac{5}{2}]_{2}
55 Copy content Toggle raw display Deg 272799332424

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)