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Properties

Label 16.8.64479545841...7129.2
Degree $16$
Signature $[8, 4]$
Discriminant $61^{14}\cdot 97^{14}$
Root discriminant $1997.97$
Ramified primes $61, 97$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![59045036290459, 23287671874118, -23777866905041, -14946713792227, -1689580081024, 250305707770, 37856782679, 2903909665, 498030477, -26801036, -2417749, -252809, -57670, 161, -41, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 41*x^14 + 161*x^13 - 57670*x^12 - 252809*x^11 - 2417749*x^10 - 26801036*x^9 + 498030477*x^8 + 2903909665*x^7 + 37856782679*x^6 + 250305707770*x^5 - 1689580081024*x^4 - 14946713792227*x^3 - 23777866905041*x^2 + 23287671874118*x + 59045036290459)
 
gp: K = bnfinit(x^16 - 4*x^15 - 41*x^14 + 161*x^13 - 57670*x^12 - 252809*x^11 - 2417749*x^10 - 26801036*x^9 + 498030477*x^8 + 2903909665*x^7 + 37856782679*x^6 + 250305707770*x^5 - 1689580081024*x^4 - 14946713792227*x^3 - 23777866905041*x^2 + 23287671874118*x + 59045036290459, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 41 x^{14} + 161 x^{13} - 57670 x^{12} - 252809 x^{11} - 2417749 x^{10} - 26801036 x^{9} + 498030477 x^{8} + 2903909665 x^{7} + 37856782679 x^{6} + 250305707770 x^{5} - 1689580081024 x^{4} - 14946713792227 x^{3} - 23777866905041 x^{2} + 23287671874118 x + 59045036290459 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(64479545841814254728122854912988442937092177855107129=61^{14}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1997.97$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $61, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} - \frac{1}{3}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{1}{9} a^{5} + \frac{1}{9} a^{4} - \frac{1}{9} a^{3} - \frac{1}{9} a^{2} - \frac{1}{9} a - \frac{1}{9}$, $\frac{1}{2619} a^{8} + \frac{95}{2619} a^{7} + \frac{317}{2619} a^{6} - \frac{304}{2619} a^{5} + \frac{64}{873} a^{4} + \frac{175}{2619} a^{3} - \frac{713}{2619} a^{2} + \frac{529}{2619} a + \frac{617}{2619}$, $\frac{1}{2619} a^{9} + \frac{22}{2619} a^{7} + \frac{136}{2619} a^{6} + \frac{263}{2619} a^{5} + \frac{268}{2619} a^{4} + \frac{122}{2619} a^{3} + \frac{1043}{2619} a^{2} + \frac{41}{873} a - \frac{997}{2619}$, $\frac{1}{7857} a^{10} - \frac{1}{7857} a^{9} + \frac{1}{7857} a^{8} - \frac{5}{291} a^{7} + \frac{454}{7857} a^{6} + \frac{278}{7857} a^{5} - \frac{686}{7857} a^{4} + \frac{373}{873} a^{3} - \frac{788}{7857} a^{2} + \frac{1739}{7857} a + \frac{262}{7857}$, $\frac{1}{70713} a^{11} - \frac{1}{23571} a^{10} - \frac{4}{23571} a^{9} - \frac{2}{70713} a^{8} + \frac{124}{70713} a^{7} + \frac{3772}{23571} a^{6} - \frac{2314}{23571} a^{5} + \frac{439}{70713} a^{4} - \frac{27611}{70713} a^{3} - \frac{1007}{7857} a^{2} + \frac{6404}{23571} a - \frac{33224}{70713}$, $\frac{1}{424278} a^{12} - \frac{1}{212139} a^{11} - \frac{5}{141426} a^{10} + \frac{20}{212139} a^{9} + \frac{41}{424278} a^{8} - \frac{9319}{212139} a^{7} - \frac{695}{7857} a^{6} - \frac{14819}{424278} a^{5} + \frac{9445}{212139} a^{4} - \frac{91403}{424278} a^{3} + \frac{12847}{70713} a^{2} + \frac{138349}{424278} a - \frac{42755}{424278}$, $\frac{1}{3818502} a^{13} - \frac{2}{1909251} a^{12} - \frac{11}{3818502} a^{11} + \frac{35}{1909251} a^{10} - \frac{175}{1272834} a^{9} - \frac{14}{212139} a^{8} + \frac{93914}{1909251} a^{7} - \frac{374243}{3818502} a^{6} + \frac{29399}{212139} a^{5} + \frac{199777}{1272834} a^{4} + \frac{584840}{1909251} a^{3} + \frac{1017907}{3818502} a^{2} - \frac{650905}{3818502} a + \frac{537503}{1909251}$, $\frac{1}{1666466824338} a^{14} - \frac{42295}{555488941446} a^{13} + \frac{378079}{555488941446} a^{12} - \frac{8233897}{1666466824338} a^{11} + \frac{57691867}{1666466824338} a^{10} + \frac{86461103}{555488941446} a^{9} - \frac{81828208}{833233412169} a^{8} + \frac{26573036483}{1666466824338} a^{7} + \frac{181346367811}{1666466824338} a^{6} - \frac{23753408339}{555488941446} a^{5} + \frac{22507595791}{1666466824338} a^{4} - \frac{761897687101}{1666466824338} a^{3} + \frac{32294032451}{92581490241} a^{2} + \frac{21972093271}{555488941446} a + \frac{246069198401}{833233412169}$, $\frac{1}{658393180148743101792719622401415733062389738835977214546} a^{15} + \frac{2119534113813978678244710199631867093354660}{329196590074371550896359811200707866531194869417988607273} a^{14} + \frac{3927654264197783613005474367422336490037838600369}{73154797794304789088079958044601748118043304315108579394} a^{13} + \frac{300569789216426054651207971037645178596731537671553}{329196590074371550896359811200707866531194869417988607273} a^{12} + \frac{617637498109506974931518201138614167767872398328765}{219464393382914367264239874133805244354129912945325738182} a^{11} + \frac{10520483291125847504590163536099064729621259796338332}{329196590074371550896359811200707866531194869417988607273} a^{10} + \frac{38104847551567552839280952058386118447030396290778203}{329196590074371550896359811200707866531194869417988607273} a^{9} - \frac{18281626355937244295224983520882567718550585359187529}{219464393382914367264239874133805244354129912945325738182} a^{8} - \frac{2024484956648795068245719040954955525483849552713315005}{36577398897152394544039979022300874059021652157554289697} a^{7} - \frac{105242118086638153439724466055554391787486332841164617601}{658393180148743101792719622401415733062389738835977214546} a^{6} + \frac{44463372685252348520081551283653905582627572470721874371}{329196590074371550896359811200707866531194869417988607273} a^{5} - \frac{541320880870046705566656457804027692937704661537099225}{219464393382914367264239874133805244354129912945325738182} a^{4} + \frac{92062082733597027376742345667248776474089091304068619921}{658393180148743101792719622401415733062389738835977214546} a^{3} + \frac{46691360532601707854191716631029301603269042830925293078}{109732196691457183632119937066902622177064956472662869091} a^{2} - \frac{10554520672899217730578841642682936580409534367232976527}{329196590074371550896359811200707866531194869417988607273} a + \frac{31980197612870948423387952935799829887974742730851082964}{329196590074371550896359811200707866531194869417988607273}$

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

Class group and class number

$C_{4}$, which has order $4$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1718626291000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{97}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{5917}) \), \(\Q(\sqrt{61}, \sqrt{97})\), 8.8.42915029526174817225369.1

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

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$61$61.8.7.1$x^{8} - 61$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
61.8.7.1$x^{8} - 61$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$97$97.8.7.4$x^{8} - 1515625$$8$$1$$7$$C_8$$[\ ]_{8}$
97.8.7.2$x^{8} - 2425$$8$$1$$7$$C_8$$[\ ]_{8}$