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Properties

Label 16.0.10359132067...4289.2
Degree $16$
Signature $[0, 8]$
Discriminant $37^{14}\cdot 101^{14}$
Root discriminant $1336.47$
Ramified primes $37, 101$
Class number $504409680$ (GRH)
Class group $[2, 5022, 50220]$ (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![157304641041896641, 65945974579414745, 14048409623804496, 1891810596789859, 288007893045025, 28925010527445, 2497035252601, 184403499414, 12919660196, 795562016, 59677021, 2243990, 118511, 8198, 369, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 369*x^14 + 8198*x^13 + 118511*x^12 + 2243990*x^11 + 59677021*x^10 + 795562016*x^9 + 12919660196*x^8 + 184403499414*x^7 + 2497035252601*x^6 + 28925010527445*x^5 + 288007893045025*x^4 + 1891810596789859*x^3 + 14048409623804496*x^2 + 65945974579414745*x + 157304641041896641)
 
gp: K = bnfinit(x^16 - 7*x^15 + 369*x^14 + 8198*x^13 + 118511*x^12 + 2243990*x^11 + 59677021*x^10 + 795562016*x^9 + 12919660196*x^8 + 184403499414*x^7 + 2497035252601*x^6 + 28925010527445*x^5 + 288007893045025*x^4 + 1891810596789859*x^3 + 14048409623804496*x^2 + 65945974579414745*x + 157304641041896641, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 369 x^{14} + 8198 x^{13} + 118511 x^{12} + 2243990 x^{11} + 59677021 x^{10} + 795562016 x^{9} + 12919660196 x^{8} + 184403499414 x^{7} + 2497035252601 x^{6} + 28925010527445 x^{5} + 288007893045025 x^{4} + 1891810596789859 x^{3} + 14048409623804496 x^{2} + 65945974579414745 x + 157304641041896641 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(103591320678630890586725126366901805607295040854289=37^{14}\cdot 101^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1336.47$
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:  $37, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{11} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{99938607481505541516964012171819662783835101700557435251826838367755195955943568651538362814453602929040384342884284} a^{15} + \frac{1624882640840837393324345416305737441403865748049328042767140357309918798696433244696274954213107211462570388842073}{49969303740752770758482006085909831391917550850278717625913419183877597977971784325769181407226801464520192171442142} a^{14} + \frac{12276312772845979008270156429891991120592478788605098667405988214328117757782959004128906016341781974825269051237053}{99938607481505541516964012171819662783835101700557435251826838367755195955943568651538362814453602929040384342884284} a^{13} - \frac{3526786678958238141455664506910876850892542657142841264263870987627476695710380778705734424817904891342897902031347}{49969303740752770758482006085909831391917550850278717625913419183877597977971784325769181407226801464520192171442142} a^{12} + \frac{836116784939667021545386965163799495869055095356231706496060028750153436061768799563076647293977426261680550626725}{99938607481505541516964012171819662783835101700557435251826838367755195955943568651538362814453602929040384342884284} a^{11} + \frac{6456115389571948333811525615546284583607950882293288120774922454127965584498959535014763401468660205797887408756005}{49969303740752770758482006085909831391917550850278717625913419183877597977971784325769181407226801464520192171442142} a^{10} - \frac{8431890996600900580038218402105855819855609201506442815641748428280136632501917285566786002921537889947386095676275}{49969303740752770758482006085909831391917550850278717625913419183877597977971784325769181407226801464520192171442142} a^{9} + \frac{17464459760446114453482715697155805511388206769090940347353257755769796288618634754822791061410400343120150623647081}{99938607481505541516964012171819662783835101700557435251826838367755195955943568651538362814453602929040384342884284} a^{8} + \frac{2479598292110276980810799153779551742915918040724966545384981825300149786157951386225603269522768055577616448318665}{24984651870376385379241003042954915695958775425139358812956709591938798988985892162884590703613400732260096085721071} a^{7} + \frac{31050574425882026442832521855347301100840107095544826506028251441058224833366330305208362364677084010628836186918673}{99938607481505541516964012171819662783835101700557435251826838367755195955943568651538362814453602929040384342884284} a^{6} + \frac{1799871604794046536427545270491618374969484541166045500702391065097471325472636014144206702656601370303205357713621}{24984651870376385379241003042954915695958775425139358812956709591938798988985892162884590703613400732260096085721071} a^{5} - \frac{1121902233334430377941973669519261777157649435650484297116606567116010770929020743509468236629518504884542998074011}{49969303740752770758482006085909831391917550850278717625913419183877597977971784325769181407226801464520192171442142} a^{4} - \frac{5411745870001774377885925600081196660458926225911484751241120638766320913452283361053177443687505118628601616113677}{99938607481505541516964012171819662783835101700557435251826838367755195955943568651538362814453602929040384342884284} a^{3} - \frac{5923680777821737771327585568768575685906082375658017916640826211476425308199070646805828123330175986042659685356615}{24984651870376385379241003042954915695958775425139358812956709591938798988985892162884590703613400732260096085721071} a^{2} + \frac{1332048864114260381618600673075102098528981520760462776037475735045166634126800037730973560510705694274049035011153}{99938607481505541516964012171819662783835101700557435251826838367755195955943568651538362814453602929040384342884284} a - \frac{34690947787028647121515073367101965488217333514456251173683501954479287476368178006603681507151947453595438131206131}{99938607481505541516964012171819662783835101700557435251826838367755195955943568651538362814453602929040384342884284}$

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

Class group and class number

$C_{2}\times C_{5022}\times C_{50220}$, which has order $504409680$ (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:  $7$
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:  \( 2450440555.59 \) (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

\(\Q(\sqrt{101}) \), \(\Q(\sqrt{3737}) \), \(\Q(\sqrt{37}) \), 4.4.52187836553.1, 4.4.52187836553.2, \(\Q(\sqrt{37}, \sqrt{101})\), 8.8.2723570284082642921809.3

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. 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
$37$37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$101$101.8.7.1$x^{8} - 101$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
101.8.7.1$x^{8} - 101$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$