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Properties

Label 16.8.17328553034...8849.5
Degree $16$
Signature $[8, 4]$
Discriminant $61^{12}\cdot 97^{14}$
Root discriminant $1195.15$
Ramified primes $61, 97$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group $OD_{16}.C_2$ (as 16T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![100142470449872159, 51111767470296455, -7553763870360839, -1639799050910109, 28489887648616, 118985616572265, -12312300080542, -902801827044, 130023633221, 3540601088, -621467375, -8736888, 1582461, 12208, -2022, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 - 2022*x^14 + 12208*x^13 + 1582461*x^12 - 8736888*x^11 - 621467375*x^10 + 3540601088*x^9 + 130023633221*x^8 - 902801827044*x^7 - 12312300080542*x^6 + 118985616572265*x^5 + 28489887648616*x^4 - 1639799050910109*x^3 - 7553763870360839*x^2 + 51111767470296455*x + 100142470449872159)
 
gp: K = bnfinit(x^16 - 7*x^15 - 2022*x^14 + 12208*x^13 + 1582461*x^12 - 8736888*x^11 - 621467375*x^10 + 3540601088*x^9 + 130023633221*x^8 - 902801827044*x^7 - 12312300080542*x^6 + 118985616572265*x^5 + 28489887648616*x^4 - 1639799050910109*x^3 - 7553763870360839*x^2 + 51111767470296455*x + 100142470449872159, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} - 2022 x^{14} + 12208 x^{13} + 1582461 x^{12} - 8736888 x^{11} - 621467375 x^{10} + 3540601088 x^{9} + 130023633221 x^{8} - 902801827044 x^{7} - 12312300080542 x^{6} + 118985616572265 x^{5} + 28489887648616 x^{4} - 1639799050910109 x^{3} - 7553763870360839 x^{2} + 51111767470296455 x + 100142470449872159 \)

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:  \(17328553034618181867273005889005225191371184588849=61^{12}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1195.15$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{12} + \frac{1}{6} a^{9} - \frac{1}{2} a^{8} + \frac{1}{6} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{36} a^{14} - \frac{1}{12} a^{13} + \frac{5}{36} a^{12} - \frac{1}{4} a^{11} - \frac{1}{18} a^{10} + \frac{7}{36} a^{9} - \frac{5}{36} a^{8} - \frac{13}{36} a^{7} - \frac{1}{6} a^{6} - \frac{17}{36} a^{5} + \frac{1}{6} a^{4} - \frac{4}{9} a^{3} - \frac{1}{3} a^{2} - \frac{17}{36} a + \frac{17}{36}$, $\frac{1}{5954258058002335164041321541354387672589074505345477482643430047597337641799061433216431130420715145776036642089264056} a^{15} + \frac{26042432692458582643842795514207132905872887369188229478686926849277024627821557433624091971253685150391460958660291}{2977129029001167582020660770677193836294537252672738741321715023798668820899530716608215565210357572888018321044632028} a^{14} + \frac{16962872838338627236688024604129732428913228820243797804166952788930649969128836394967951532187834321793987294681423}{1488564514500583791010330385338596918147268626336369370660857511899334410449765358304107782605178786444009160522316014} a^{13} - \frac{274822147651836819441585915827551218591315269591122239878897597044811139757679225281818740663301832681080468130380823}{1488564514500583791010330385338596918147268626336369370660857511899334410449765358304107782605178786444009160522316014} a^{12} + \frac{4657852833756279902440776994495211880638239915350810234916189549901784548244185373322744689871810294144008411524321}{52692549185861373133109040188976882058310393852614844979145398651303872936274879939968417083369160582088819841497912} a^{11} + \frac{892175972113684056173320077605277925287326836831006504145316056439691057637906752588638384098224881317638757874511853}{5954258058002335164041321541354387672589074505345477482643430047597337641799061433216431130420715145776036642089264056} a^{10} - \frac{365728447633116153635851054971430254406417943159042022739410391295038013632961714499868783957129182238454905583580647}{2977129029001167582020660770677193836294537252672738741321715023798668820899530716608215565210357572888018321044632028} a^{9} + \frac{241367562876449898173579111243289462887378142505479362830228291948534971902270264818755292634623514028726625358055103}{992376343000389194006886923559064612098179084224246247107238341266222940299843572202738521736785857629339440348210676} a^{8} + \frac{2895293290532190477340902988672764033973804399720940269762566496505194489064130912872453077028173957683781157998223843}{5954258058002335164041321541354387672589074505345477482643430047597337641799061433216431130420715145776036642089264056} a^{7} - \frac{2786928750420260679520652922575441784505775349723768258904986485795116719855683675149645610303508104276761240907531129}{5954258058002335164041321541354387672589074505345477482643430047597337641799061433216431130420715145776036642089264056} a^{6} - \frac{2293991827429973954834497875074388710888934550692292778120169145712187481455566219927201657433171661222010737258369907}{5954258058002335164041321541354387672589074505345477482643430047597337641799061433216431130420715145776036642089264056} a^{5} - \frac{872849206901521967975901815879946461132307139261612712047218746213429441876388885245114897972630120062515705193589279}{2977129029001167582020660770677193836294537252672738741321715023798668820899530716608215565210357572888018321044632028} a^{4} + \frac{478037805084817871067036347103042578271976593147302513770919564330790586433446653338213792304912506805971033665159267}{2977129029001167582020660770677193836294537252672738741321715023798668820899530716608215565210357572888018321044632028} a^{3} + \frac{1147778782897180236056362493821783891940305422710861854906933908729379574325787672470444225932228679912505536987134893}{5954258058002335164041321541354387672589074505345477482643430047597337641799061433216431130420715145776036642089264056} a^{2} + \frac{108560406315143222020240564647617052244247768560939511918365614004259693530631312277014051406182615484166924140925765}{992376343000389194006886923559064612098179084224246247107238341266222940299843572202738521736785857629339440348210676} a - \frac{10567619689439540511730576751462774819427389358339530311094068213909054894665542339370398472004208371611659493409957}{57808330660216846252828364479168812355233733061606577501392524733954734386398654691421661460395292677437248952322952}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( 713145714456000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}.C_2$ (as 16T40):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $OD_{16}.C_2$
Character table for $OD_{16}.C_2$

Intermediate fields

\(\Q(\sqrt{97}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{5917}) \), 4.4.3396056233.1, 4.4.912673.1, \(\Q(\sqrt{61}, \sqrt{97})\), 8.8.11533197937698150289.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
61Data not computed
$97$97.8.7.3$x^{8} - 60625$$8$$1$$7$$C_8$$[\ ]_{8}$
97.8.7.3$x^{8} - 60625$$8$$1$$7$$C_8$$[\ ]_{8}$