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Properties

Label 16.0.27510376712...625.39
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 101^{12}$
Root discriminant $106.53$
Ramified primes $5, 101$
Class number $41472$ (GRH)
Class group $[2, 2, 2, 6, 6, 12, 12]$ (GRH)
Galois group $Q_{16}$ (as 16T14)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![730770125, 212747625, 495143825, 367664500, 101899505, 33449130, 20731015, 1410010, 2024471, -30422, 130737, -4435, 4761, -90, 98, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 98*x^14 - 90*x^13 + 4761*x^12 - 4435*x^11 + 130737*x^10 - 30422*x^9 + 2024471*x^8 + 1410010*x^7 + 20731015*x^6 + 33449130*x^5 + 101899505*x^4 + 367664500*x^3 + 495143825*x^2 + 212747625*x + 730770125)
 
gp: K = bnfinit(x^16 - x^15 + 98*x^14 - 90*x^13 + 4761*x^12 - 4435*x^11 + 130737*x^10 - 30422*x^9 + 2024471*x^8 + 1410010*x^7 + 20731015*x^6 + 33449130*x^5 + 101899505*x^4 + 367664500*x^3 + 495143825*x^2 + 212747625*x + 730770125, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 98 x^{14} - 90 x^{13} + 4761 x^{12} - 4435 x^{11} + 130737 x^{10} - 30422 x^{9} + 2024471 x^{8} + 1410010 x^{7} + 20731015 x^{6} + 33449130 x^{5} + 101899505 x^{4} + 367664500 x^{3} + 495143825 x^{2} + 212747625 x + 730770125 \)

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:  \(275103767122062920083301025390625=5^{12}\cdot 101^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $106.53$
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:  $5, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{8} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{25} a^{14} - \frac{1}{25} a^{13} - \frac{2}{25} a^{12} + \frac{2}{5} a^{11} + \frac{11}{25} a^{10} - \frac{2}{5} a^{9} + \frac{12}{25} a^{8} + \frac{3}{25} a^{7} - \frac{4}{25} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{5652105160071573431180470987933000887356906145958495910936613275} a^{15} + \frac{21286288161323872104418119782819159386058056853269518281782611}{1130421032014314686236094197586600177471381229191699182187322655} a^{14} + \frac{49853157442328039291311460103457032818695623925206260623381757}{5652105160071573431180470987933000887356906145958495910936613275} a^{13} - \frac{203274454241875894428574174277000511237777131382138810552866602}{5652105160071573431180470987933000887356906145958495910936613275} a^{12} + \frac{2398359773065931064103133281240261154818275986693959162568069451}{5652105160071573431180470987933000887356906145958495910936613275} a^{11} - \frac{1132582147905889870376999736167117365780490870637863600444989999}{5652105160071573431180470987933000887356906145958495910936613275} a^{10} - \frac{1171828363693487468165610482666614746782963126704518752044454833}{5652105160071573431180470987933000887356906145958495910936613275} a^{9} - \frac{384235095202508789642917297682162331352443664207526683859581882}{1130421032014314686236094197586600177471381229191699182187322655} a^{8} + \frac{22470665999017583822927681137004446616123336275601218337038564}{79607114930585541284231985745535223765590227407866139590656525} a^{7} + \frac{2226134551442776375653115080078309636965917011992920034297827136}{5652105160071573431180470987933000887356906145958495910936613275} a^{6} + \frac{67376768010575635237728731258628363218154532609982007543110537}{1130421032014314686236094197586600177471381229191699182187322655} a^{5} + \frac{326461113687500842059850436403968072402197463248990047869989417}{1130421032014314686236094197586600177471381229191699182187322655} a^{4} + \frac{121709862721088551849059783995418687859825040709281189199619842}{1130421032014314686236094197586600177471381229191699182187322655} a^{3} + \frac{303899343058556705725604931989826555760187226092196301664729681}{1130421032014314686236094197586600177471381229191699182187322655} a^{2} + \frac{20739460698859312106830422036956147863362749082263307249858875}{226084206402862937247218839517320035494276245838339836437464531} a + \frac{34733554007981274214807583627308801131891081671422976928724799}{226084206402862937247218839517320035494276245838339836437464531}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{6}\times C_{12}\times C_{12}$, which has order $41472$ (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:  \( 103071.243274 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_{16}$ (as 16T14):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $Q_{16}$
Character table for $Q_{16}$

Intermediate fields

\(\Q(\sqrt{505}) \), \(\Q(\sqrt{101}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{101})\), 4.4.51005.1 x2, 4.4.2525.1 x2, 8.8.65037750625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$101$101.4.3.1$x^{4} - 101$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.1$x^{4} - 101$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.1$x^{4} - 101$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.1$x^{4} - 101$$4$$1$$3$$C_4$$[\ ]_{4}$