↑

Properties

Label 16.16.8195019227...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{36}\cdot 5^{12}\cdot 29^{2}\cdot 241^{2}$
Root discriminant $48.09$
Ramified primes $2, 5, 29, 241$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T797

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![241, 2476, 5922, -9136, -30844, 25240, 36258, -32820, -10478, 14300, -62, -2560, 356, 184, -38, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 38*x^14 + 184*x^13 + 356*x^12 - 2560*x^11 - 62*x^10 + 14300*x^9 - 10478*x^8 - 32820*x^7 + 36258*x^6 + 25240*x^5 - 30844*x^4 - 9136*x^3 + 5922*x^2 + 2476*x + 241)
 
gp: K = bnfinit(x^16 - 4*x^15 - 38*x^14 + 184*x^13 + 356*x^12 - 2560*x^11 - 62*x^10 + 14300*x^9 - 10478*x^8 - 32820*x^7 + 36258*x^6 + 25240*x^5 - 30844*x^4 - 9136*x^3 + 5922*x^2 + 2476*x + 241, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 38 x^{14} + 184 x^{13} + 356 x^{12} - 2560 x^{11} - 62 x^{10} + 14300 x^{9} - 10478 x^{8} - 32820 x^{7} + 36258 x^{6} + 25240 x^{5} - 30844 x^{4} - 9136 x^{3} + 5922 x^{2} + 2476 x + 241 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(819501922779136000000000000=2^{36}\cdot 5^{12}\cdot 29^{2}\cdot 241^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.09$
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:  $2, 5, 29, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} + \frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} + \frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} + \frac{1}{4} a^{6} - \frac{1}{4} a^{2}$, $\frac{1}{6673641990932759044} a^{15} - \frac{587466802933533145}{6673641990932759044} a^{14} + \frac{198072855067214049}{3336820995466379522} a^{13} + \frac{276226597999103223}{6673641990932759044} a^{12} + \frac{1251143654076076843}{6673641990932759044} a^{11} + \frac{218047510438462299}{6673641990932759044} a^{10} + \frac{473598748369141335}{3336820995466379522} a^{9} - \frac{21858829395335005}{6673641990932759044} a^{8} + \frac{38424837102095527}{351244315312250476} a^{7} + \frac{958200830063266927}{6673641990932759044} a^{6} - \frac{297473707614237867}{3336820995466379522} a^{5} + \frac{2595482024414069871}{6673641990932759044} a^{4} + \frac{23730539176497139}{74984741471154596} a^{3} + \frac{1202628634587576927}{6673641990932759044} a^{2} + \frac{423537087436679361}{3336820995466379522} a - \frac{1909969879373526973}{6673641990932759044}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 170247755.755 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T797:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n797 are not computed
Character table for t16n797 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), 4.4.8000.1, \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{40})^+\)

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
5Data not computed
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
241Data not computed