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Properties

Label 16.8.64479545841...7129.3
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![8634653323364421, 3402687025817982, 72859130170740, -528998767606638, -77387700251526, 3620444460103, 588540102377, 100014470272, 15836693522, -72342590, -7269097, -3901834, -840713, 98, -23, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 23*x^14 + 98*x^13 - 840713*x^12 - 3901834*x^11 - 7269097*x^10 - 72342590*x^9 + 15836693522*x^8 + 100014470272*x^7 + 588540102377*x^6 + 3620444460103*x^5 - 77387700251526*x^4 - 528998767606638*x^3 + 72859130170740*x^2 + 3402687025817982*x + 8634653323364421)
 
gp: K = bnfinit(x^16 - 4*x^15 - 23*x^14 + 98*x^13 - 840713*x^12 - 3901834*x^11 - 7269097*x^10 - 72342590*x^9 + 15836693522*x^8 + 100014470272*x^7 + 588540102377*x^6 + 3620444460103*x^5 - 77387700251526*x^4 - 528998767606638*x^3 + 72859130170740*x^2 + 3402687025817982*x + 8634653323364421, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 23 x^{14} + 98 x^{13} - 840713 x^{12} - 3901834 x^{11} - 7269097 x^{10} - 72342590 x^{9} + 15836693522 x^{8} + 100014470272 x^{7} + 588540102377 x^{6} + 3620444460103 x^{5} - 77387700251526 x^{4} - 528998767606638 x^{3} + 72859130170740 x^{2} + 3402687025817982 x + 8634653323364421 \)

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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{27} a^{7} - \frac{2}{27} a^{5} + \frac{1}{9} a^{4} + \frac{1}{27} a^{3} - \frac{1}{9} a^{2}$, $\frac{1}{1647} a^{8} - \frac{2}{1647} a^{7} - \frac{44}{1647} a^{6} + \frac{205}{1647} a^{5} + \frac{16}{1647} a^{4} - \frac{113}{1647} a^{3} - \frac{28}{61} a^{2} - \frac{55}{183} a + \frac{19}{61}$, $\frac{1}{4941} a^{9} - \frac{16}{1647} a^{7} + \frac{13}{549} a^{6} - \frac{41}{1647} a^{5} + \frac{52}{549} a^{4} - \frac{433}{4941} a^{3} - \frac{101}{549} a^{2} + \frac{130}{549} a - \frac{28}{61}$, $\frac{1}{4941} a^{10} + \frac{7}{1647} a^{7} - \frac{13}{1647} a^{6} + \frac{142}{1647} a^{5} - \frac{763}{4941} a^{4} + \frac{85}{1647} a^{3} - \frac{181}{549} a^{2} + \frac{73}{183} a - \frac{1}{61}$, $\frac{1}{4941} a^{11} + \frac{1}{1647} a^{7} + \frac{28}{549} a^{6} - \frac{127}{4941} a^{5} + \frac{52}{549} a^{4} + \frac{248}{1647} a^{3} - \frac{152}{549} a^{2} + \frac{16}{183} a - \frac{11}{61}$, $\frac{1}{44469} a^{12} + \frac{1}{14823} a^{11} - \frac{4}{44469} a^{10} + \frac{1}{14823} a^{9} - \frac{1}{14823} a^{8} + \frac{22}{1647} a^{7} + \frac{302}{44469} a^{6} + \frac{1307}{14823} a^{5} + \frac{5410}{44469} a^{4} - \frac{625}{14823} a^{3} - \frac{1984}{4941} a^{2} - \frac{98}{1647} a - \frac{91}{183}$, $\frac{1}{133407} a^{13} - \frac{4}{133407} a^{11} - \frac{4}{44469} a^{10} - \frac{4}{44469} a^{9} + \frac{4}{14823} a^{8} + \frac{2408}{133407} a^{7} + \frac{731}{14823} a^{6} - \frac{8414}{133407} a^{5} + \frac{4153}{44469} a^{4} - \frac{161}{4941} a^{3} - \frac{463}{4941} a^{2} - \frac{430}{1647} a - \frac{11}{183}$, $\frac{1}{8009776125106766991} a^{14} - \frac{6868604904011}{2669925375035588997} a^{13} + \frac{80131114523636}{8009776125106766991} a^{12} - \frac{3554772502307}{2669925375035588997} a^{11} + \frac{178300700087407}{2669925375035588997} a^{10} - \frac{37040905165706}{889975125011862999} a^{9} + \frac{1935717540443600}{8009776125106766991} a^{8} + \frac{49335130770669263}{2669925375035588997} a^{7} - \frac{126693904358908010}{8009776125106766991} a^{6} - \frac{233630026583449039}{2669925375035588997} a^{5} - \frac{372413878800458657}{2669925375035588997} a^{4} - \frac{108510904995611029}{889975125011862999} a^{3} - \frac{7442141362060402}{296658375003954333} a^{2} + \frac{9202942349687527}{98886125001318111} a - \frac{769003005790396}{10987347222368679}$, $\frac{1}{302412507660331677356422384005350490064878434795726834892490710628066464921} a^{15} - \frac{1618786305009152476867131035706224002905608998319341322}{33601389740036853039602487111705610007208714977302981654721190069785162769} a^{14} + \frac{942615906542234796150681656996810529109194419277453595138858254910312}{302412507660331677356422384005350490064878434795726834892490710628066464921} a^{13} - \frac{972541705529213845760956507360629665799731728267105085637773371353589}{100804169220110559118807461335116830021626144931908944964163570209355488307} a^{12} + \frac{27296780394330196418998730007778364254687363400243409646371821461946677}{302412507660331677356422384005350490064878434795726834892490710628066464921} a^{11} - \frac{7036705600706834027614214304601085587084923160227363226106136722382917}{100804169220110559118807461335116830021626144931908944964163570209355488307} a^{10} - \frac{28894693848645980221780858277548691341383351476461567707779106448590019}{302412507660331677356422384005350490064878434795726834892490710628066464921} a^{9} - \frac{31169581018082885403366033920982535179004152712324611744600450221084}{102756543547513312047714027864543149869139801153831748179575504800566247} a^{8} + \frac{3484755058558188394840875841623149561255191020798292370085805761900332822}{302412507660331677356422384005350490064878434795726834892490710628066464921} a^{7} - \frac{82356875670887570407124922786171617178619952609699793783676705372368131}{1652527364264107526537827235001915246256166310359163032199402790317303087} a^{6} + \frac{24302514867481946828929651465386007177135809030859375906183640543527289066}{302412507660331677356422384005350490064878434795726834892490710628066464921} a^{5} + \frac{15701432041806920486481250421060744725058601987836120623634311854185296358}{100804169220110559118807461335116830021626144931908944964163570209355488307} a^{4} - \frac{377545941232389445277359140403556949482115592506940485920489295615323}{8662384568197177891106596316500543956485876508714354641588344952251911} a^{3} - \frac{2024480923829002764071844542450543377405410617926186491615379288553248004}{11200463246678951013200829037235203335736238325767660551573730023261720923} a^{2} - \frac{348027095350829347032572344266791576912784411124313976884186727577210112}{3733487748892983671066943012411734445245412775255886850524576674420573641} a - \frac{183297669189837209442484455433939406644998176342317925015067445162654162}{414831972099220407896327001379081605027268086139542983391619630491174849}$

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:  \( 28026804113300000000000 \) (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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$97.8.7.2$x^{8} - 2425$$8$$1$$7$$C_8$$[\ ]_{8}$
97.8.7.2$x^{8} - 2425$$8$$1$$7$$C_8$$[\ ]_{8}$