↑

Properties

Label 16.0.96313927189...0625.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{8}\cdot 149^{4}$
Root discriminant $42.07$
Ramified primes $5, 29, 149$
Class number $96$ (GRH)
Class group $[2, 48]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T373)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![400, 2400, 22260, 4200, 8049, 4262, 2696, -534, 1127, -976, 961, -238, 193, -48, 26, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 26*x^14 - 48*x^13 + 193*x^12 - 238*x^11 + 961*x^10 - 976*x^9 + 1127*x^8 - 534*x^7 + 2696*x^6 + 4262*x^5 + 8049*x^4 + 4200*x^3 + 22260*x^2 + 2400*x + 400)
 
gp: K = bnfinit(x^16 - 4*x^15 + 26*x^14 - 48*x^13 + 193*x^12 - 238*x^11 + 961*x^10 - 976*x^9 + 1127*x^8 - 534*x^7 + 2696*x^6 + 4262*x^5 + 8049*x^4 + 4200*x^3 + 22260*x^2 + 2400*x + 400, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 26 x^{14} - 48 x^{13} + 193 x^{12} - 238 x^{11} + 961 x^{10} - 976 x^{9} + 1127 x^{8} - 534 x^{7} + 2696 x^{6} + 4262 x^{5} + 8049 x^{4} + 4200 x^{3} + 22260 x^{2} + 2400 x + 400 \)

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:  \(96313927189328563031640625=5^{8}\cdot 29^{8}\cdot 149^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.07$
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:  $5, 29, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{10} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{2} a^{4} + \frac{1}{10} a^{3} + \frac{1}{10} a^{2} - \frac{1}{2} a$, $\frac{1}{30} a^{11} + \frac{1}{5} a^{9} - \frac{7}{30} a^{8} + \frac{1}{6} a^{7} - \frac{11}{30} a^{6} - \frac{7}{30} a^{5} + \frac{1}{30} a^{4} + \frac{2}{15} a^{3} - \frac{1}{15} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{60} a^{12} - \frac{1}{60} a^{11} + \frac{1}{12} a^{9} + \frac{1}{5} a^{8} + \frac{11}{60} a^{7} + \frac{7}{15} a^{6} - \frac{13}{60} a^{5} + \frac{3}{10} a^{4} + \frac{1}{20} a^{3} - \frac{29}{60} a^{2} - \frac{1}{3}$, $\frac{1}{60} a^{13} - \frac{1}{60} a^{11} - \frac{1}{60} a^{10} + \frac{1}{12} a^{9} - \frac{7}{60} a^{8} - \frac{3}{20} a^{7} + \frac{3}{20} a^{6} - \frac{1}{60} a^{5} + \frac{7}{20} a^{4} + \frac{7}{15} a^{3} - \frac{1}{12} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{360} a^{14} + \frac{1}{180} a^{13} - \frac{1}{180} a^{12} + \frac{1}{90} a^{11} + \frac{1}{40} a^{10} + \frac{2}{45} a^{9} + \frac{13}{360} a^{8} - \frac{31}{180} a^{7} + \frac{19}{360} a^{6} + \frac{29}{90} a^{5} + \frac{37}{90} a^{4} + \frac{13}{60} a^{3} - \frac{67}{360} a^{2} - \frac{5}{12} a + \frac{5}{18}$, $\frac{1}{50900235184130486067138960} a^{15} + \frac{1833224717821304656963}{3635731084580749004795640} a^{14} - \frac{23675029882535269485829}{25450117592065243033569480} a^{13} + \frac{5846860771716455821711}{1817865542290374502397820} a^{12} - \frac{183591884802499714267609}{16966745061376828689046320} a^{11} - \frac{112042160938663666297477}{2545011759206524303356948} a^{10} - \frac{8710269963695166184389719}{50900235184130486067138960} a^{9} - \frac{2624470208148222887838223}{25450117592065243033569480} a^{8} + \frac{2106168074111714764903291}{50900235184130486067138960} a^{7} + \frac{566990537088261136020533}{1272505879603262151678474} a^{6} + \frac{1952638653425624459540177}{6362529398016310758392370} a^{5} + \frac{9404288822140823325871}{403970120508972111643960} a^{4} + \frac{25135200631093615713316133}{50900235184130486067138960} a^{3} + \frac{1875440328675837691021363}{8483372530688414344523160} a^{2} - \frac{106386419134262386093273}{2545011759206524303356948} a + \frac{23885959501094565730102}{70694771089070119537693}$

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

Class group and class number

$C_{2}\times C_{48}$, which has order $96$ (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:  \( 100117.930697 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T373):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), 4.0.626545.1, 4.0.626545.2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.4.65865543125.1, 8.0.9813965925625.2, 8.4.65865543125.2

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$149$149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.4.2.1$x^{4} + 745 x^{2} + 199809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
149.4.2.1$x^{4} + 745 x^{2} + 199809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$