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Properties

Label 16.0.33005678220...4401.4
Degree $16$
Signature $[0, 8]$
Discriminant $61^{14}\cdot 109^{14}$
Root discriminant $2212.65$
Ramified primes $61, 109$
Class number $2209102096$ (GRH)
Class group $[2, 2, 552275524]$ (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![7082452187739053435, 792523107994436141, 11150815222856179, -20341549562763644, 301979753790703, 32537334329978, 24334833783094, 245536064601, -16715660403, 1065246568, 141652975, 4998536, 323884, -12027, 607, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 607*x^14 - 12027*x^13 + 323884*x^12 + 4998536*x^11 + 141652975*x^10 + 1065246568*x^9 - 16715660403*x^8 + 245536064601*x^7 + 24334833783094*x^6 + 32537334329978*x^5 + 301979753790703*x^4 - 20341549562763644*x^3 + 11150815222856179*x^2 + 792523107994436141*x + 7082452187739053435)
 
gp: K = bnfinit(x^16 - 3*x^15 + 607*x^14 - 12027*x^13 + 323884*x^12 + 4998536*x^11 + 141652975*x^10 + 1065246568*x^9 - 16715660403*x^8 + 245536064601*x^7 + 24334833783094*x^6 + 32537334329978*x^5 + 301979753790703*x^4 - 20341549562763644*x^3 + 11150815222856179*x^2 + 792523107994436141*x + 7082452187739053435, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 607 x^{14} - 12027 x^{13} + 323884 x^{12} + 4998536 x^{11} + 141652975 x^{10} + 1065246568 x^{9} - 16715660403 x^{8} + 245536064601 x^{7} + 24334833783094 x^{6} + 32537334329978 x^{5} + 301979753790703 x^{4} - 20341549562763644 x^{3} + 11150815222856179 x^{2} + 792523107994436141 x + 7082452187739053435 \)

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:  \(330056782202198660511172839698929216685213271046904401=61^{14}\cdot 109^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2212.65$
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:  $61, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{25} a^{13} - \frac{2}{25} a^{12} - \frac{2}{25} a^{11} + \frac{2}{25} a^{10} + \frac{1}{25} a^{9} - \frac{1}{25} a^{8} + \frac{11}{25} a^{7} + \frac{3}{25} a^{6} - \frac{9}{25} a^{5} + \frac{3}{25} a^{4} + \frac{6}{25} a^{3} - \frac{8}{25} a - \frac{1}{5}$, $\frac{1}{5975} a^{14} + \frac{108}{5975} a^{13} - \frac{552}{5975} a^{12} + \frac{97}{5975} a^{11} - \frac{279}{5975} a^{10} - \frac{481}{5975} a^{9} - \frac{2339}{5975} a^{8} - \frac{162}{5975} a^{7} - \frac{84}{5975} a^{6} - \frac{962}{5975} a^{5} + \frac{406}{5975} a^{4} - \frac{66}{1195} a^{3} + \frac{1047}{5975} a^{2} - \frac{284}{1195} a + \frac{118}{239}$, $\frac{1}{38651184345624361241733405473829833257960536726012869716630066349751226806409785143921095323019806960976238594125307782090075} a^{15} - \frac{106068440284932338764233607372122566534775181544176681308746003257111698645216580079154510367845907261527540966351503872}{1546047373824974449669336218953193330318421469040514788665202653990049072256391405756843812920792278439049543765012311283603} a^{14} - \frac{418162055098598870833831759389453117357550371659421951718123634400343144525393828445497776989024977386953865418223561409336}{38651184345624361241733405473829833257960536726012869716630066349751226806409785143921095323019806960976238594125307782090075} a^{13} + \frac{1485671594167294906499194912222319948177502488829915271278451671336807441262510976569015200350015183987926722407204347484543}{38651184345624361241733405473829833257960536726012869716630066349751226806409785143921095323019806960976238594125307782090075} a^{12} - \frac{398035197339273011702336632459125823389808501840920477368853204973831947363412801797682394359114191652811195733888017779918}{7730236869124872248346681094765966651592107345202573943326013269950245361281957028784219064603961392195247718825061556418015} a^{11} + \frac{2518647072501098233373351902536455669464400324526714770996896773281400114024349114972158771394309113033842022787829885783781}{38651184345624361241733405473829833257960536726012869716630066349751226806409785143921095323019806960976238594125307782090075} a^{10} - \frac{3826693837451900995069167091505989891585262109354653916187530787654056492399072155408845842708991073390942456177954503753066}{38651184345624361241733405473829833257960536726012869716630066349751226806409785143921095323019806960976238594125307782090075} a^{9} + \frac{22441240663239937552758943902013557903806146365047316496065079336175733395453467810263893784281946920177312604218954806024}{56425086635947972615669205071284428113811002519726817104569439926644126724685817728351963975211397023322976049817967565095} a^{8} + \frac{18362831421536740104569154631995069771896754198511316487346644941921216474420682776713604046250402960618973245931689700551422}{38651184345624361241733405473829833257960536726012869716630066349751226806409785143921095323019806960976238594125307782090075} a^{7} - \frac{1400375852847678798355578549103773453697500323617869103283956088697900153424067647862123418836134769317379868269863981050002}{7730236869124872248346681094765966651592107345202573943326013269950245361281957028784219064603961392195247718825061556418015} a^{6} + \frac{9712388904187096968279394844743621064807383674400249964894582890366660802401347724002401244389379891246746676040684293690217}{38651184345624361241733405473829833257960536726012869716630066349751226806409785143921095323019806960976238594125307782090075} a^{5} - \frac{17471858841014909042814692894798480998065053418678578960842784211832525276005355231837704289570253939895783434359934825141873}{38651184345624361241733405473829833257960536726012869716630066349751226806409785143921095323019806960976238594125307782090075} a^{4} + \frac{5285346668188873686468210403390741201873342707083765641310786006196996608566058604642131129005018492005323876005176719184787}{38651184345624361241733405473829833257960536726012869716630066349751226806409785143921095323019806960976238594125307782090075} a^{3} + \frac{17068493030899109286325036789780345285792851789534670567277539873804019792752324484585645848757331455668460730806644110011894}{38651184345624361241733405473829833257960536726012869716630066349751226806409785143921095323019806960976238594125307782090075} a^{2} - \frac{1591261648543106584254109006709543329163336973453489990274526912221023112510925882309968282912991984575739845668531205237182}{7730236869124872248346681094765966651592107345202573943326013269950245361281957028784219064603961392195247718825061556418015} a + \frac{366716796871402702267230051102034688992434553363010078058914672866234432000347820302341767673518289954073152217465393646336}{1546047373824974449669336218953193330318421469040514788665202653990049072256391405756843812920792278439049543765012311283603}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{552275524}$, which has order $2209102096$ (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:  \( 185489178554 \) (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

\(\Q(\sqrt{109}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{6649}) \), \(\Q(\sqrt{61}, \sqrt{109})\), 8.8.86404825551402914547601.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$61.8.7.2$x^{8} - 244$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
61.8.7.2$x^{8} - 244$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$109$109.8.7.2$x^{8} - 3924$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
109.8.7.2$x^{8} - 3924$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$