↑

Properties

Label 16.8.19361901958...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 13^{4}\cdot 19^{2}\cdot 29^{8}\cdot 31^{2}$
Root discriminant $50.75$
Ramified primes $5, 13, 19, 29, 31$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T608)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49069, 25835, -86453, 130898, -81600, 5559, -17831, -23444, 9218, 3098, 1767, -6, -309, -17, -9, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 9*x^14 - 17*x^13 - 309*x^12 - 6*x^11 + 1767*x^10 + 3098*x^9 + 9218*x^8 - 23444*x^7 - 17831*x^6 + 5559*x^5 - 81600*x^4 + 130898*x^3 - 86453*x^2 + 25835*x + 49069)
 
gp: K = bnfinit(x^16 - x^15 - 9*x^14 - 17*x^13 - 309*x^12 - 6*x^11 + 1767*x^10 + 3098*x^9 + 9218*x^8 - 23444*x^7 - 17831*x^6 + 5559*x^5 - 81600*x^4 + 130898*x^3 - 86453*x^2 + 25835*x + 49069, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 9 x^{14} - 17 x^{13} - 309 x^{12} - 6 x^{11} + 1767 x^{10} + 3098 x^{9} + 9218 x^{8} - 23444 x^{7} - 17831 x^{6} + 5559 x^{5} - 81600 x^{4} + 130898 x^{3} - 86453 x^{2} + 25835 x + 49069 \)

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:  \(1936190195826058295484765625=5^{8}\cdot 13^{4}\cdot 19^{2}\cdot 29^{8}\cdot 31^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.75$
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:  $5, 13, 19, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{35} a^{10} - \frac{4}{35} a^{9} + \frac{1}{35} a^{8} - \frac{2}{7} a^{7} - \frac{9}{35} a^{6} - \frac{2}{5} a^{5} + \frac{16}{35} a^{4} - \frac{16}{35} a^{3} - \frac{13}{35} a^{2} - \frac{13}{35} a + \frac{16}{35}$, $\frac{1}{35} a^{11} - \frac{3}{7} a^{9} - \frac{6}{35} a^{8} - \frac{2}{5} a^{7} - \frac{3}{7} a^{6} - \frac{1}{7} a^{5} + \frac{13}{35} a^{4} - \frac{1}{5} a^{3} + \frac{1}{7} a^{2} - \frac{1}{35} a - \frac{6}{35}$, $\frac{1}{35} a^{12} + \frac{4}{35} a^{9} + \frac{1}{35} a^{8} + \frac{2}{7} a^{7} + \frac{13}{35} a^{5} - \frac{12}{35} a^{4} + \frac{2}{7} a^{3} + \frac{2}{5} a^{2} + \frac{9}{35} a - \frac{1}{7}$, $\frac{1}{35} a^{13} + \frac{17}{35} a^{9} + \frac{6}{35} a^{8} + \frac{1}{7} a^{7} + \frac{2}{5} a^{6} + \frac{9}{35} a^{5} + \frac{16}{35} a^{4} + \frac{8}{35} a^{3} - \frac{9}{35} a^{2} + \frac{12}{35} a + \frac{6}{35}$, $\frac{1}{92365} a^{14} + \frac{131}{13195} a^{13} + \frac{772}{92365} a^{12} + \frac{223}{92365} a^{11} + \frac{433}{92365} a^{10} - \frac{9041}{92365} a^{9} - \frac{1153}{92365} a^{8} - \frac{38153}{92365} a^{7} - \frac{31377}{92365} a^{6} - \frac{12854}{92365} a^{5} + \frac{10141}{92365} a^{4} - \frac{3597}{18473} a^{3} + \frac{27744}{92365} a^{2} + \frac{673}{1885} a - \frac{393}{18473}$, $\frac{1}{970535736516116243295742700912495} a^{15} + \frac{1130127410442693213266512637}{970535736516116243295742700912495} a^{14} - \frac{2320016211605545191728111641653}{194107147303223248659148540182499} a^{13} - \frac{3381879256152790274065121043497}{970535736516116243295742700912495} a^{12} + \frac{7070385459989025696127055090233}{970535736516116243295742700912495} a^{11} - \frac{19385841653553784938369626003}{194107147303223248659148540182499} a^{10} - \frac{385397215692656870483764476691921}{970535736516116243295742700912495} a^{9} + \frac{253716296584549966899585383433511}{970535736516116243295742700912495} a^{8} - \frac{243793424180612982988412531785787}{970535736516116243295742700912495} a^{7} + \frac{11252174746942094109609156845088}{74656595116624326407364823147115} a^{6} + \frac{415639069276547562430011150468859}{970535736516116243295742700912495} a^{5} - \frac{462782559126389077914162370874696}{970535736516116243295742700912495} a^{4} + \frac{40735679667256508239661484781682}{970535736516116243295742700912495} a^{3} + \frac{10546749831373310445957369913124}{74656595116624326407364823147115} a^{2} - \frac{423228234171176932501753533171827}{970535736516116243295742700912495} a + \frac{91880643228278840470543902246544}{970535736516116243295742700912495}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 20924009.7493 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T608):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), 4.4.4205.1 x2, 4.4.725.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.442050625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$19$19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$29$29.4.2.1$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}$
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}$
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.2.2$x^{4} - 31 x^{2} + 11532$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$