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Properties

Label 16.8.29029740969...0000.4
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{8}\cdot 29^{8}\cdot 941^{2}$
Root discriminant $80.15$
Ramified primes $2, 5, 29, 941$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1439

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![74941, 1253132, 2368316, 1106608, -423938, -279840, 39972, -33988, -21049, 14704, 3508, -1740, -98, 116, -16, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 16*x^14 + 116*x^13 - 98*x^12 - 1740*x^11 + 3508*x^10 + 14704*x^9 - 21049*x^8 - 33988*x^7 + 39972*x^6 - 279840*x^5 - 423938*x^4 + 1106608*x^3 + 2368316*x^2 + 1253132*x + 74941)
 
gp: K = bnfinit(x^16 - 4*x^15 - 16*x^14 + 116*x^13 - 98*x^12 - 1740*x^11 + 3508*x^10 + 14704*x^9 - 21049*x^8 - 33988*x^7 + 39972*x^6 - 279840*x^5 - 423938*x^4 + 1106608*x^3 + 2368316*x^2 + 1253132*x + 74941, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 16 x^{14} + 116 x^{13} - 98 x^{12} - 1740 x^{11} + 3508 x^{10} + 14704 x^{9} - 21049 x^{8} - 33988 x^{7} + 39972 x^{6} - 279840 x^{5} - 423938 x^{4} + 1106608 x^{3} + 2368316 x^{2} + 1253132 x + 74941 \)

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:  \(2902974096966413457817600000000=2^{24}\cdot 5^{8}\cdot 29^{8}\cdot 941^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $80.15$
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:  $2, 5, 29, 941$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{20} a^{8} + \frac{1}{10} a^{7} - \frac{1}{10} a^{6} + \frac{2}{5} a^{5} + \frac{1}{4} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{3}{10} a + \frac{1}{20}$, $\frac{1}{20} a^{9} - \frac{3}{10} a^{7} - \frac{2}{5} a^{6} + \frac{9}{20} a^{5} - \frac{3}{10} a^{4} + \frac{1}{5} a^{3} - \frac{1}{2} a^{2} - \frac{7}{20} a - \frac{1}{10}$, $\frac{1}{20} a^{10} + \frac{1}{5} a^{7} - \frac{3}{20} a^{6} + \frac{1}{10} a^{5} - \frac{3}{10} a^{4} - \frac{3}{10} a^{3} + \frac{1}{4} a^{2} + \frac{1}{10} a + \frac{3}{10}$, $\frac{1}{300} a^{11} + \frac{7}{300} a^{10} - \frac{1}{300} a^{8} + \frac{11}{60} a^{7} + \frac{11}{300} a^{6} + \frac{7}{75} a^{5} - \frac{53}{300} a^{4} - \frac{37}{300} a^{3} - \frac{21}{100} a^{2} + \frac{11}{30} a + \frac{17}{300}$, $\frac{1}{900} a^{12} - \frac{1}{225} a^{10} + \frac{7}{450} a^{9} + \frac{17}{900} a^{8} + \frac{113}{450} a^{7} + \frac{43}{450} a^{6} - \frac{32}{75} a^{5} + \frac{349}{900} a^{4} + \frac{53}{450} a^{3} + \frac{43}{450} a^{2} - \frac{11}{50} a + \frac{94}{225}$, $\frac{1}{13500} a^{13} - \frac{7}{13500} a^{12} + \frac{1}{2700} a^{11} + \frac{4}{225} a^{10} + \frac{13}{750} a^{9} + \frac{139}{6750} a^{8} - \frac{6491}{13500} a^{7} + \frac{1657}{3375} a^{6} + \frac{1217}{6750} a^{5} + \frac{131}{2250} a^{4} - \frac{1619}{13500} a^{3} - \frac{1741}{6750} a^{2} - \frac{6563}{13500} a + \frac{4181}{13500}$, $\frac{1}{40500} a^{14} + \frac{1}{40500} a^{13} - \frac{1}{6750} a^{12} + \frac{1}{4050} a^{11} + \frac{7}{3375} a^{10} + \frac{151}{8100} a^{9} + \frac{409}{20250} a^{8} - \frac{541}{1350} a^{7} - \frac{919}{2250} a^{6} + \frac{9593}{40500} a^{5} + \frac{3271}{10125} a^{4} + \frac{123}{250} a^{3} + \frac{4937}{13500} a^{2} + \frac{5471}{20250} a + \frac{6689}{20250}$, $\frac{1}{286806292014242901550500} a^{15} - \frac{19668325886805802}{23900524334520241795875} a^{14} + \frac{5652837838045335221}{286806292014242901550500} a^{13} - \frac{361641612912934607}{2019762619818611982750} a^{12} + \frac{290925956972874913289}{286806292014242901550500} a^{11} + \frac{1250863171465292692147}{57361258402848580310100} a^{10} - \frac{2080158627886422442433}{95602097338080967183500} a^{9} + \frac{984016083209715515363}{143403146007121450775250} a^{8} + \frac{31862797546981701860437}{95602097338080967183500} a^{7} - \frac{83059091912833250286871}{286806292014242901550500} a^{6} - \frac{100359201576082026878053}{286806292014242901550500} a^{5} - \frac{2109615196346377021211}{143403146007121450775250} a^{4} + \frac{755556629081867716289}{23900524334520241795875} a^{3} + \frac{22015270244863163735353}{286806292014242901550500} a^{2} - \frac{10402175863620438034769}{23900524334520241795875} a - \frac{25240126333964129374769}{143403146007121450775250}$

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:  \( 3384293389.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1439:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 74 conjugacy class representatives for t16n1439 are not computed
Character table for t16n1439 is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.106488227360000.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.8.12.19$x^{8} + 12 x^{4} + 80$$4$$2$$12$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
2.8.12.20$x^{8} + 8 x^{6} + 12 x^{4} + 80$$4$$2$$12$$C_2^3: C_4$$[2, 2, 2]^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{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}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
941Data not computed