↑

Properties

Label 16.0.14710825712...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{4}\cdot 5^{14}\cdot 29^{14}$
Root discriminant $102.44$
Ramified primes $3, 5, 29$
Class number $2304$ (GRH)
Class group $[2, 2, 2, 2, 12, 12]$ (GRH)
Galois group $OD_{16}.C_2$ (as 16T40)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![40054095, 82809270, 92969235, 74177940, 39264590, 12847940, 3491765, 537290, 106831, 52088, -8597, 496, 1000, -174, 23, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 23*x^14 - 174*x^13 + 1000*x^12 + 496*x^11 - 8597*x^10 + 52088*x^9 + 106831*x^8 + 537290*x^7 + 3491765*x^6 + 12847940*x^5 + 39264590*x^4 + 74177940*x^3 + 92969235*x^2 + 82809270*x + 40054095)
 
gp: K = bnfinit(x^16 - 2*x^15 + 23*x^14 - 174*x^13 + 1000*x^12 + 496*x^11 - 8597*x^10 + 52088*x^9 + 106831*x^8 + 537290*x^7 + 3491765*x^6 + 12847940*x^5 + 39264590*x^4 + 74177940*x^3 + 92969235*x^2 + 82809270*x + 40054095, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 23 x^{14} - 174 x^{13} + 1000 x^{12} + 496 x^{11} - 8597 x^{10} + 52088 x^{9} + 106831 x^{8} + 537290 x^{7} + 3491765 x^{6} + 12847940 x^{5} + 39264590 x^{4} + 74177940 x^{3} + 92969235 x^{2} + 82809270 x + 40054095 \)

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:  \(147108257121214334362554931640625=3^{4}\cdot 5^{14}\cdot 29^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $102.44$
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:  $3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{10} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{4320221922} a^{14} + \frac{41190980}{2160110961} a^{13} - \frac{186946711}{4320221922} a^{12} - \frac{42725839}{1440073974} a^{11} - \frac{1051627325}{4320221922} a^{10} - \frac{101116013}{4320221922} a^{9} + \frac{670608523}{4320221922} a^{8} + \frac{799154494}{2160110961} a^{7} + \frac{392483617}{4320221922} a^{6} + \frac{712303585}{2160110961} a^{5} + \frac{1010828998}{2160110961} a^{4} - \frac{1914570463}{4320221922} a^{3} - \frac{1728354199}{4320221922} a^{2} + \frac{323479522}{720036987} a - \frac{15319796}{240012329}$, $\frac{1}{5361843573980573481085351909035320947024225868034} a^{15} - \frac{66044097542967604001895204975708316193}{5361843573980573481085351909035320947024225868034} a^{14} - \frac{374935833027577858151762477665724891751321859855}{5361843573980573481085351909035320947024225868034} a^{13} + \frac{2416880235674166664512258604514699249270847475}{595760397108952609009483545448368994113802874226} a^{12} + \frac{141357392506319567529679358318474158347765441890}{2680921786990286740542675954517660473512112934017} a^{11} - \frac{1075918041342549453846189786592130659756609482179}{5361843573980573481085351909035320947024225868034} a^{10} - \frac{27397838972329815779589244309963434749301751573}{2680921786990286740542675954517660473512112934017} a^{9} + \frac{185990092587158047264562830157547107235808099}{4695134478091570473805036697929352843278656627} a^{8} - \frac{1201861860691768742575873176931334479276519112417}{5361843573980573481085351909035320947024225868034} a^{7} - \frac{2538955409886355513002737272603669608517297027327}{5361843573980573481085351909035320947024225868034} a^{6} + \frac{24195518460451301431074842582289676422703781983}{2680921786990286740542675954517660473512112934017} a^{5} + \frac{1123020768186864731925455617909488552017202451108}{2680921786990286740542675954517660473512112934017} a^{4} - \frac{1086083355593484250356986792885849974163604720301}{5361843573980573481085351909035320947024225868034} a^{3} - \frac{29962041731869259867197338774517255592382230279}{893640595663428913514225318172553491170704311339} a^{2} - \frac{230428344582179793672743429474536859017026103591}{595760397108952609009483545448368994113802874226} a + \frac{23063086297691814974066979386478651904340252051}{99293399518158768168247257574728165685633812371}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{12}\times C_{12}$, which has order $2304$ (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:  \( 1611809.57536 \) (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

\(\Q(\sqrt{145}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{29}) \), 4.4.3048625.1, 4.4.3048625.2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.9294114390625.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$5$5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$29$29.8.7.1$x^{8} - 29$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.7.1$x^{8} - 29$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$