Properties

Label 16.16.369...296.1
Degree $16$
Signature $(16, 0)$
Discriminant $3.691\times 10^{30}$
Root discriminant \(81.37\)
Ramified primes $2,31$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_4^2:C_6$ (as 16T184)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 28*x^14 + 240*x^13 + 568*x^12 - 2656*x^11 - 7500*x^10 + 9240*x^9 + 42548*x^8 + 8184*x^7 - 94396*x^6 - 87088*x^5 + 52756*x^4 + 106544*x^3 + 38164*x^2 - 5048*x - 2661)
 
Copy content gp:K = bnfinit(y^16 - 8*y^15 - 28*y^14 + 240*y^13 + 568*y^12 - 2656*y^11 - 7500*y^10 + 9240*y^9 + 42548*y^8 + 8184*y^7 - 94396*y^6 - 87088*y^5 + 52756*y^4 + 106544*y^3 + 38164*y^2 - 5048*y - 2661, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 - 28*x^14 + 240*x^13 + 568*x^12 - 2656*x^11 - 7500*x^10 + 9240*x^9 + 42548*x^8 + 8184*x^7 - 94396*x^6 - 87088*x^5 + 52756*x^4 + 106544*x^3 + 38164*x^2 - 5048*x - 2661);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 8*x^15 - 28*x^14 + 240*x^13 + 568*x^12 - 2656*x^11 - 7500*x^10 + 9240*x^9 + 42548*x^8 + 8184*x^7 - 94396*x^6 - 87088*x^5 + 52756*x^4 + 106544*x^3 + 38164*x^2 - 5048*x - 2661)
 

\( x^{16} - 8 x^{15} - 28 x^{14} + 240 x^{13} + 568 x^{12} - 2656 x^{11} - 7500 x^{10} + 9240 x^{9} + \cdots - 2661 \) 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:  $16$
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:  $(16, 0)$
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:   \(3691277647829053341574465847296\) \(\medspace = 2^{52}\cdot 31^{10}\) 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:  \(81.37\)
Copy content comment:Root discriminant
 
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:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{27/8}31^{2/3}\approx 102.38052790852275$
Ramified primes:   \(2\), \(31\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{5}$, $\frac{1}{6}a^{14}+\frac{1}{6}a^{12}-\frac{1}{6}a^{11}+\frac{1}{6}a^{10}-\frac{1}{6}a^{9}-\frac{1}{3}a^{7}+\frac{1}{6}a^{6}-\frac{1}{3}a^{5}-\frac{1}{2}a^{4}+\frac{1}{6}a^{3}-\frac{1}{2}a^{2}+\frac{1}{6}a$, $\frac{1}{31044057187662}a^{15}+\frac{1941596506979}{31044057187662}a^{14}-\frac{1446686503780}{15522028593831}a^{13}-\frac{3533526296573}{31044057187662}a^{12}-\frac{149587552997}{15522028593831}a^{11}-\frac{2855452265647}{15522028593831}a^{10}-\frac{400313719415}{31044057187662}a^{9}+\frac{1636984074650}{15522028593831}a^{8}+\frac{2178923100355}{10348019062554}a^{7}+\frac{1002632576963}{10348019062554}a^{6}+\frac{6089704503895}{15522028593831}a^{5}+\frac{4499388641467}{31044057187662}a^{4}+\frac{180313875679}{500710599801}a^{3}+\frac{6578452486427}{15522028593831}a^{2}+\frac{3227880782711}{31044057187662}a+\frac{733986014275}{5174009531277}$ 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:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (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:  $C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  $15$
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:   \( -1 \)  (order $2$) 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:   $\frac{10532348892120}{1724669843759}a^{15}-\frac{100128233163679}{1724669843759}a^{14}-\frac{143963981253530}{1724669843759}a^{13}+\frac{27\cdots 68}{1724669843759}a^{12}+\frac{18\cdots 02}{1724669843759}a^{11}-\frac{30\cdots 89}{1724669843759}a^{10}-\frac{32\cdots 92}{1724669843759}a^{9}+\frac{29\cdots 11}{3449339687518}a^{8}+\frac{22\cdots 00}{1724669843759}a^{7}-\frac{25\cdots 59}{1724669843759}a^{6}-\frac{60\cdots 54}{1724669843759}a^{5}-\frac{23\cdots 83}{1724669843759}a^{4}+\frac{18\cdots 58}{55634511089}a^{3}+\frac{28\cdots 99}{1724669843759}a^{2}-\frac{22\cdots 80}{1724669843759}a-\frac{37\cdots 31}{3449339687518}$, $\frac{22613725523405}{1724669843759}a^{15}-\frac{215466832464400}{1724669843759}a^{14}-\frac{303807746295255}{1724669843759}a^{13}+\frac{11\cdots 47}{3449339687518}a^{12}+\frac{38\cdots 91}{1724669843759}a^{11}-\frac{65\cdots 11}{1724669843759}a^{10}-\frac{68\cdots 47}{1724669843759}a^{9}+\frac{31\cdots 84}{1724669843759}a^{8}+\frac{48\cdots 53}{1724669843759}a^{7}-\frac{55\cdots 50}{1724669843759}a^{6}-\frac{12\cdots 95}{1724669843759}a^{5}+\frac{60\cdots 29}{3449339687518}a^{4}+\frac{38\cdots 37}{55634511089}a^{3}+\frac{59\cdots 25}{1724669843759}a^{2}-\frac{48\cdots 11}{1724669843759}a-\frac{39\cdots 16}{1724669843759}$, $\frac{9477697172315}{15522028593831}a^{15}-\frac{92921118888920}{15522028593831}a^{14}-\frac{97871996433172}{15522028593831}a^{13}+\frac{24\cdots 66}{15522028593831}a^{12}+\frac{19\cdots 69}{31044057187662}a^{11}-\frac{26\cdots 40}{15522028593831}a^{10}-\frac{44\cdots 91}{31044057187662}a^{9}+\frac{25\cdots 47}{31044057187662}a^{8}+\frac{57\cdots 98}{5174009531277}a^{7}-\frac{77\cdots 74}{5174009531277}a^{6}-\frac{47\cdots 71}{15522028593831}a^{5}+\frac{32\cdots 60}{15522028593831}a^{4}+\frac{28\cdots 85}{1001421199602}a^{3}+\frac{20\cdots 05}{15522028593831}a^{2}-\frac{37\cdots 61}{31044057187662}a-\frac{92\cdots 49}{10348019062554}$, $\frac{31005308861651}{31044057187662}a^{15}-\frac{145543689801982}{15522028593831}a^{14}-\frac{231793636491314}{15522028593831}a^{13}+\frac{80\cdots 77}{31044057187662}a^{12}+\frac{31\cdots 33}{15522028593831}a^{11}-\frac{45\cdots 82}{15522028593831}a^{10}-\frac{10\cdots 81}{31044057187662}a^{9}+\frac{43\cdots 53}{31044057187662}a^{8}+\frac{23\cdots 11}{10348019062554}a^{7}-\frac{12\cdots 32}{5174009531277}a^{6}-\frac{94\cdots 39}{15522028593831}a^{5}-\frac{74\cdots 79}{31044057187662}a^{4}+\frac{27\cdots 13}{500710599801}a^{3}+\frac{45\cdots 57}{15522028593831}a^{2}-\frac{69\cdots 09}{31044057187662}a-\frac{19\cdots 41}{10348019062554}$, $\frac{30331065217126}{15522028593831}a^{15}-\frac{562027831965023}{31044057187662}a^{14}-\frac{493671139048112}{15522028593831}a^{13}+\frac{15\cdots 53}{31044057187662}a^{12}+\frac{72\cdots 89}{15522028593831}a^{11}-\frac{17\cdots 45}{31044057187662}a^{10}-\frac{11\cdots 19}{15522028593831}a^{9}+\frac{42\cdots 35}{15522028593831}a^{8}+\frac{83\cdots 80}{1724669843759}a^{7}-\frac{46\cdots 23}{10348019062554}a^{6}-\frac{19\cdots 68}{15522028593831}a^{5}-\frac{30\cdots 65}{31044057187662}a^{4}+\frac{57\cdots 24}{500710599801}a^{3}+\frac{19\cdots 91}{31044057187662}a^{2}-\frac{70\cdots 84}{15522028593831}a-\frac{21\cdots 02}{5174009531277}$, $\frac{194657796285719}{31044057187662}a^{15}-\frac{18\cdots 15}{31044057187662}a^{14}-\frac{27\cdots 11}{31044057187662}a^{13}+\frac{25\cdots 93}{15522028593831}a^{12}+\frac{36\cdots 67}{31044057187662}a^{11}-\frac{56\cdots 51}{31044057187662}a^{10}-\frac{62\cdots 27}{31044057187662}a^{9}+\frac{27\cdots 59}{31044057187662}a^{8}+\frac{47\cdots 57}{3449339687518}a^{7}-\frac{15\cdots 59}{10348019062554}a^{6}-\frac{11\cdots 83}{31044057187662}a^{5}-\frac{88\cdots 02}{15522028593831}a^{4}+\frac{33\cdots 67}{1001421199602}a^{3}+\frac{53\cdots 55}{31044057187662}a^{2}-\frac{42\cdots 99}{31044057187662}a-\frac{11\cdots 05}{10348019062554}$, $\frac{8053295037808}{15522028593831}a^{15}-\frac{73141322991709}{15522028593831}a^{14}-\frac{293271447554983}{31044057187662}a^{13}+\frac{41\cdots 33}{31044057187662}a^{12}+\frac{23\cdots 15}{15522028593831}a^{11}-\frac{47\cdots 27}{31044057187662}a^{10}-\frac{68\cdots 87}{31044057187662}a^{9}+\frac{22\cdots 27}{31044057187662}a^{8}+\frac{73\cdots 68}{5174009531277}a^{7}-\frac{59\cdots 41}{5174009531277}a^{6}-\frac{11\cdots 55}{31044057187662}a^{5}-\frac{14\cdots 59}{31044057187662}a^{4}+\frac{16\cdots 84}{500710599801}a^{3}+\frac{58\cdots 25}{31044057187662}a^{2}-\frac{39\cdots 41}{31044057187662}a-\frac{12\cdots 63}{10348019062554}$, $\frac{154453535311057}{31044057187662}a^{15}-\frac{721899990656912}{15522028593831}a^{14}-\frac{11\cdots 14}{15522028593831}a^{13}+\frac{20\cdots 55}{15522028593831}a^{12}+\frac{16\cdots 89}{15522028593831}a^{11}-\frac{45\cdots 03}{31044057187662}a^{10}-\frac{54\cdots 73}{31044057187662}a^{9}+\frac{10\cdots 28}{15522028593831}a^{8}+\frac{12\cdots 65}{10348019062554}a^{7}-\frac{20\cdots 69}{1724669843759}a^{6}-\frac{48\cdots 18}{15522028593831}a^{5}-\frac{25\cdots 45}{15522028593831}a^{4}+\frac{14\cdots 67}{500710599801}a^{3}+\frac{46\cdots 75}{31044057187662}a^{2}-\frac{35\cdots 79}{31044057187662}a-\frac{51\cdots 93}{5174009531277}$, $\frac{35762199516892}{15522028593831}a^{15}-\frac{344689690036423}{15522028593831}a^{14}-\frac{872703404314063}{31044057187662}a^{13}+\frac{92\cdots 38}{15522028593831}a^{12}+\frac{50\cdots 09}{15522028593831}a^{11}-\frac{20\cdots 41}{31044057187662}a^{10}-\frac{19\cdots 31}{31044057187662}a^{9}+\frac{49\cdots 85}{15522028593831}a^{8}+\frac{79\cdots 23}{1724669843759}a^{7}-\frac{29\cdots 61}{5174009531277}a^{6}-\frac{38\cdots 11}{31044057187662}a^{5}+\frac{55\cdots 30}{15522028593831}a^{4}+\frac{57\cdots 62}{500710599801}a^{3}+\frac{17\cdots 49}{31044057187662}a^{2}-\frac{14\cdots 55}{31044057187662}a-\frac{19\cdots 94}{5174009531277}$, $\frac{270328706830223}{31044057187662}a^{15}-\frac{25\cdots 95}{31044057187662}a^{14}-\frac{17\cdots 84}{15522028593831}a^{13}+\frac{35\cdots 25}{15522028593831}a^{12}+\frac{41\cdots 63}{31044057187662}a^{11}-\frac{78\cdots 31}{31044057187662}a^{10}-\frac{38\cdots 73}{15522028593831}a^{9}+\frac{37\cdots 69}{31044057187662}a^{8}+\frac{61\cdots 11}{3449339687518}a^{7}-\frac{21\cdots 21}{10348019062554}a^{6}-\frac{74\cdots 22}{15522028593831}a^{5}+\frac{12\cdots 45}{15522028593831}a^{4}+\frac{44\cdots 21}{1001421199602}a^{3}+\frac{68\cdots 99}{31044057187662}a^{2}-\frac{28\cdots 17}{15522028593831}a-\frac{15\cdots 65}{10348019062554}$, $\frac{595579102475}{31044057187662}a^{15}-\frac{1334532935023}{15522028593831}a^{14}-\frac{20530394387255}{15522028593831}a^{13}+\frac{81831562007671}{15522028593831}a^{12}+\frac{867732804783343}{31044057187662}a^{11}-\frac{11\cdots 43}{15522028593831}a^{10}-\frac{48\cdots 32}{15522028593831}a^{9}+\frac{10\cdots 79}{31044057187662}a^{8}+\frac{16\cdots 51}{10348019062554}a^{7}-\frac{838324178551279}{5174009531277}a^{6}-\frac{59\cdots 03}{15522028593831}a^{5}-\frac{25\cdots 28}{15522028593831}a^{4}+\frac{32\cdots 77}{1001421199602}a^{3}+\frac{37\cdots 56}{15522028593831}a^{2}-\frac{13\cdots 97}{15522028593831}a-\frac{16\cdots 57}{10348019062554}$, $\frac{53727007961423}{15522028593831}a^{15}-\frac{509851799518148}{15522028593831}a^{14}-\frac{744760702153510}{15522028593831}a^{13}+\frac{28\cdots 07}{31044057187662}a^{12}+\frac{19\cdots 93}{31044057187662}a^{11}-\frac{31\cdots 61}{31044057187662}a^{10}-\frac{33\cdots 91}{31044057187662}a^{9}+\frac{14\cdots 65}{31044057187662}a^{8}+\frac{13\cdots 33}{1724669843759}a^{7}-\frac{43\cdots 58}{5174009531277}a^{6}-\frac{31\cdots 04}{15522028593831}a^{5}-\frac{42\cdots 95}{31044057187662}a^{4}+\frac{18\cdots 85}{1001421199602}a^{3}+\frac{29\cdots 07}{31044057187662}a^{2}-\frac{23\cdots 53}{31044057187662}a-\frac{63\cdots 49}{10348019062554}$, $\frac{13303304750612}{5174009531277}a^{15}-\frac{85194044882591}{3449339687518}a^{14}-\frac{335451218602229}{10348019062554}a^{13}+\frac{34\cdots 36}{5174009531277}a^{12}+\frac{19\cdots 98}{5174009531277}a^{11}-\frac{77\cdots 21}{10348019062554}a^{10}-\frac{25\cdots 73}{3449339687518}a^{9}+\frac{36\cdots 55}{10348019062554}a^{8}+\frac{26\cdots 85}{5174009531277}a^{7}-\frac{65\cdots 13}{10348019062554}a^{6}-\frac{48\cdots 27}{3449339687518}a^{5}+\frac{23\cdots 57}{5174009531277}a^{4}+\frac{72\cdots 24}{55634511089}a^{3}+\frac{65\cdots 97}{10348019062554}a^{2}-\frac{18\cdots 15}{3449339687518}a-\frac{14\cdots 61}{3449339687518}$, $\frac{242544641912467}{31044057187662}a^{15}-\frac{22\cdots 25}{31044057187662}a^{14}-\frac{17\cdots 50}{15522028593831}a^{13}+\frac{31\cdots 25}{15522028593831}a^{12}+\frac{49\cdots 11}{31044057187662}a^{11}-\frac{71\cdots 83}{31044057187662}a^{10}-\frac{41\cdots 44}{15522028593831}a^{9}+\frac{33\cdots 59}{31044057187662}a^{8}+\frac{18\cdots 47}{10348019062554}a^{7}-\frac{64\cdots 87}{3449339687518}a^{6}-\frac{73\cdots 34}{15522028593831}a^{5}-\frac{27\cdots 35}{15522028593831}a^{4}+\frac{43\cdots 51}{1001421199602}a^{3}+\frac{70\cdots 89}{31044057187662}a^{2}-\frac{27\cdots 34}{15522028593831}a-\frac{15\cdots 35}{10348019062554}$, $\frac{171360696305531}{5174009531277}a^{15}-\frac{32\cdots 97}{10348019062554}a^{14}-\frac{24\cdots 06}{5174009531277}a^{13}+\frac{14\cdots 42}{1724669843759}a^{12}+\frac{10\cdots 76}{1724669843759}a^{11}-\frac{33\cdots 97}{3449339687518}a^{10}-\frac{55\cdots 76}{5174009531277}a^{9}+\frac{47\cdots 79}{10348019062554}a^{8}+\frac{38\cdots 67}{5174009531277}a^{7}-\frac{82\cdots 17}{10348019062554}a^{6}-\frac{10\cdots 81}{5174009531277}a^{5}-\frac{16\cdots 47}{5174009531277}a^{4}+\frac{29\cdots 90}{166903533267}a^{3}+\frac{95\cdots 53}{10348019062554}a^{2}-\frac{37\cdots 65}{5174009531277}a-\frac{20\cdots 41}{3449339687518}$ 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:  \( 40690835746.2 \) (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)
 
Unit signature rank:  \( 13 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 40690835746.2 \cdot 1}{2\cdot\sqrt{3691277647829053341574465847296}}\cr\approx \mathstrut & 0.69399795131 \end{aligned}\] (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^16 - 8*x^15 - 28*x^14 + 240*x^13 + 568*x^12 - 2656*x^11 - 7500*x^10 + 9240*x^9 + 42548*x^8 + 8184*x^7 - 94396*x^6 - 87088*x^5 + 52756*x^4 + 106544*x^3 + 38164*x^2 - 5048*x - 2661) 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^16 - 8*x^15 - 28*x^14 + 240*x^13 + 568*x^12 - 2656*x^11 - 7500*x^10 + 9240*x^9 + 42548*x^8 + 8184*x^7 - 94396*x^6 - 87088*x^5 + 52756*x^4 + 106544*x^3 + 38164*x^2 - 5048*x - 2661, 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^16 - 8*x^15 - 28*x^14 + 240*x^13 + 568*x^12 - 2656*x^11 - 7500*x^10 + 9240*x^9 + 42548*x^8 + 8184*x^7 - 94396*x^6 - 87088*x^5 + 52756*x^4 + 106544*x^3 + 38164*x^2 - 5048*x - 2661); 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 = polynomial_ring(QQ); K, a = number_field(x^16 - 8*x^15 - 28*x^14 + 240*x^13 + 568*x^12 - 2656*x^11 - 7500*x^10 + 9240*x^9 + 42548*x^8 + 8184*x^7 - 94396*x^6 - 87088*x^5 + 52756*x^4 + 106544*x^3 + 38164*x^2 - 5048*x - 2661); 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

$C_4^2:C_6$ (as 16T184):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 96
The 10 conjugacy class representatives for $C_4^2:C_6$
Character table for $C_4^2:C_6$

Intermediate fields

4.4.61504.1

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(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 24 siblings: data not computed
Degree 32 sibling: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.3.0.1}{3} }^{5}{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.3.0.1}{3} }^{5}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ R ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.3.0.1}{3} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.2.8.52c1.272$x^{16} + 8 x^{15} + 44 x^{14} + 168 x^{13} + 494 x^{12} + 1152 x^{11} + 2196 x^{10} + 3472 x^{9} + 4599 x^{8} + 5120 x^{7} + 4780 x^{6} + 3720 x^{5} + 2374 x^{4} + 1216 x^{3} + 472 x^{2} + 144 x + 19$$8$$2$$52$16T27$$[2, 3, \frac{7}{2}, 4]^{2}$$
\(31\) Copy content Toggle raw display $\Q_{31}$$x + 28$$1$$1$$0$Trivial$$[\ ]$$
31.1.3.2a1.1$x^{3} + 31$$3$$1$$2$$C_3$$$[\ ]_{3}$$
31.1.3.2a1.1$x^{3} + 31$$3$$1$$2$$C_3$$$[\ ]_{3}$$
31.1.3.2a1.1$x^{3} + 31$$3$$1$$2$$C_3$$$[\ ]_{3}$$
31.1.3.2a1.1$x^{3} + 31$$3$$1$$2$$C_3$$$[\ ]_{3}$$
31.1.3.2a1.1$x^{3} + 31$$3$$1$$2$$C_3$$$[\ ]_{3}$$

Spectrum of ring of integers

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