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Properties

Label 16.8.26997420280...5529.1
Degree $16$
Signature $[8, 4]$
Discriminant $53^{14}\cdot 89^{14}$
Root discriminant $1638.54$
Ramified primes $53, 89$
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![-90570842522372189, 247314848830313952, 119340689913643006, 16479084532307579, 1367334826234034, 185632816258606, 20752520925940, 1043907848018, -17719251843, -8455181166, -714832698, -18245116, 471345, 5961, -1215, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 1215*x^14 + 5961*x^13 + 471345*x^12 - 18245116*x^11 - 714832698*x^10 - 8455181166*x^9 - 17719251843*x^8 + 1043907848018*x^7 + 20752520925940*x^6 + 185632816258606*x^5 + 1367334826234034*x^4 + 16479084532307579*x^3 + 119340689913643006*x^2 + 247314848830313952*x - 90570842522372189)
 
gp: K = bnfinit(x^16 - 2*x^15 - 1215*x^14 + 5961*x^13 + 471345*x^12 - 18245116*x^11 - 714832698*x^10 - 8455181166*x^9 - 17719251843*x^8 + 1043907848018*x^7 + 20752520925940*x^6 + 185632816258606*x^5 + 1367334826234034*x^4 + 16479084532307579*x^3 + 119340689913643006*x^2 + 247314848830313952*x - 90570842522372189, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 1215 x^{14} + 5961 x^{13} + 471345 x^{12} - 18245116 x^{11} - 714832698 x^{10} - 8455181166 x^{9} - 17719251843 x^{8} + 1043907848018 x^{7} + 20752520925940 x^{6} + 185632816258606 x^{5} + 1367334826234034 x^{4} + 16479084532307579 x^{3} + 119340689913643006 x^{2} + 247314848830313952 x - 90570842522372189 \)

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:  \(2699742028062005762663832012742665627137111010645529=53^{14}\cdot 89^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1638.54$
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:  $53, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{106} a^{12} - \frac{25}{106} a^{11} + \frac{3}{106} a^{10} - \frac{21}{53} a^{9} - \frac{12}{53} a^{8} - \frac{11}{53} a^{7} + \frac{3}{106} a^{6} + \frac{41}{106} a^{5} - \frac{3}{106} a^{4} - \frac{45}{106} a^{3} + \frac{1}{53} a^{2} + \frac{17}{106} a - \frac{11}{106}$, $\frac{1}{106} a^{13} + \frac{7}{53} a^{11} + \frac{33}{106} a^{10} - \frac{7}{53} a^{9} + \frac{7}{53} a^{8} - \frac{17}{106} a^{7} + \frac{5}{53} a^{6} - \frac{19}{53} a^{5} - \frac{7}{53} a^{4} + \frac{43}{106} a^{3} - \frac{39}{106} a^{2} - \frac{5}{53} a + \frac{43}{106}$, $\frac{1}{106} a^{14} - \frac{41}{106} a^{11} + \frac{25}{53} a^{10} - \frac{17}{53} a^{9} + \frac{1}{106} a^{8} + \frac{13}{53} a^{6} + \frac{24}{53} a^{5} - \frac{21}{106} a^{4} - \frac{45}{106} a^{3} - \frac{19}{53} a^{2} + \frac{17}{106} a + \frac{24}{53}$, $\frac{1}{2787164533618557575001164039436693033675662715664315393598294661051056952255640608408584164718763794958007723500930558317878} a^{15} - \frac{9334689771342081390169505311374627266518114576701214923013970752898088261849184692581437778869719733469645786625252536181}{2787164533618557575001164039436693033675662715664315393598294661051056952255640608408584164718763794958007723500930558317878} a^{14} - \frac{6493685571621831811362799291084193386899600338508267231293763580188957019371141597012999889052856363245592669197666278756}{1393582266809278787500582019718346516837831357832157696799147330525528476127820304204292082359381897479003861750465279158939} a^{13} - \frac{5464652289910305928871022186070227324523629877383476086227146121344863278038887065236488338766780928096289657399728941}{26294005034137335613218528673931066355430780336455805599983911896708084455241892532156454384139281084509506825480476965263} a^{12} - \frac{538989189459221822005212768488300121781905079832122550302385715300337719622941505206505422778082357885189977678615190809960}{1393582266809278787500582019718346516837831357832157696799147330525528476127820304204292082359381897479003861750465279158939} a^{11} - \frac{668733988873080162978948734353014259076031862508799294539355603589469812625755981683604766594405530243027651630794575136877}{2787164533618557575001164039436693033675662715664315393598294661051056952255640608408584164718763794958007723500930558317878} a^{10} + \frac{7336748475338482214072180186725946960844848967522747148449580809638783620575934461460612222545046427418802285632875541705}{2787164533618557575001164039436693033675662715664315393598294661051056952255640608408584164718763794958007723500930558317878} a^{9} + \frac{1038962138665729433266751678991910928798778116662976249055879525519664030818729868453480931637118683706550995656344931747283}{2787164533618557575001164039436693033675662715664315393598294661051056952255640608408584164718763794958007723500930558317878} a^{8} + \frac{215722647520156511312318808117972091130262191154698371368912540992817207924898715734032799220478064698261079366950008484009}{1393582266809278787500582019718346516837831357832157696799147330525528476127820304204292082359381897479003861750465279158939} a^{7} - \frac{200714292315324485454976604305270436175826675983347352633822382120395916363179351744848550749984510008193690653985135288513}{2787164533618557575001164039436693033675662715664315393598294661051056952255640608408584164718763794958007723500930558317878} a^{6} - \frac{562819182778751573007874299920424149176249090547170190642084869408458164531966638984031481089478656201990828462003199140567}{1393582266809278787500582019718346516837831357832157696799147330525528476127820304204292082359381897479003861750465279158939} a^{5} - \frac{823687859526317306628786577680547781432088957629693088859100437686391781135381668188965845268089502965940107060221004442691}{2787164533618557575001164039436693033675662715664315393598294661051056952255640608408584164718763794958007723500930558317878} a^{4} - \frac{5506727598698609392024939801906105977572128568315167604775691772919367810763438773575392839221319619750132962604710964301}{14366827492879162757737958966168520792142591317857295843290178665211633774513611383549402910921462860608287234540879166587} a^{3} + \frac{416905363199612654986549765801222028351445455212352940060541283763895551911526295367372537686224304089580407068398246097119}{2787164533618557575001164039436693033675662715664315393598294661051056952255640608408584164718763794958007723500930558317878} a^{2} + \frac{81887548180256040084251473565772574742072638842470296100186833009153528345977979010006742641231551151837469855651110974228}{1393582266809278787500582019718346516837831357832157696799147330525528476127820304204292082359381897479003861750465279158939} a + \frac{5688145212539012704567435425962837949591877418057395959664315992903340026131296885090928341658573307852262461084412476069}{28733654985758325515475917932337041584285182635714591686580357330423267549027222767098805821842925721216574469081758333174}$

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:  \( 9578247592160000000 \) (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

\(\Q(\sqrt{4717}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{89}) \), \(\Q(\sqrt{53}, \sqrt{89})\), 8.8.11015272807216227454969.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$53$53.8.7.1$x^{8} - 53$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
53.8.7.1$x^{8} - 53$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$89$89.8.7.2$x^{8} - 801$$8$$1$$7$$C_8$$[\ ]_{8}$
89.8.7.2$x^{8} - 801$$8$$1$$7$$C_8$$[\ ]_{8}$