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Properties

Label 16.0.58479365965...2064.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{34}\cdot 17^{8}\cdot 47^{4}$
Root discriminant $47.09$
Ramified primes $2, 17, 47$
Class number $40$ (GRH)
Class group $[2, 20]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T373)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3008, -19968, 62944, -119808, 149968, -125696, 70840, -28496, 11558, -6040, 2437, -634, 181, -44, 7, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 7*x^14 - 44*x^13 + 181*x^12 - 634*x^11 + 2437*x^10 - 6040*x^9 + 11558*x^8 - 28496*x^7 + 70840*x^6 - 125696*x^5 + 149968*x^4 - 119808*x^3 + 62944*x^2 - 19968*x + 3008)
 
gp: K = bnfinit(x^16 - 2*x^15 + 7*x^14 - 44*x^13 + 181*x^12 - 634*x^11 + 2437*x^10 - 6040*x^9 + 11558*x^8 - 28496*x^7 + 70840*x^6 - 125696*x^5 + 149968*x^4 - 119808*x^3 + 62944*x^2 - 19968*x + 3008, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 7 x^{14} - 44 x^{13} + 181 x^{12} - 634 x^{11} + 2437 x^{10} - 6040 x^{9} + 11558 x^{8} - 28496 x^{7} + 70840 x^{6} - 125696 x^{5} + 149968 x^{4} - 119808 x^{3} + 62944 x^{2} - 19968 x + 3008 \)

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:  \(584793659653495312365912064=2^{34}\cdot 17^{8}\cdot 47^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.09$
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:  $2, 17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{8} a^{9} - \frac{1}{16} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{1504} a^{14} - \frac{35}{1504} a^{13} - \frac{23}{752} a^{12} + \frac{11}{376} a^{11} + \frac{59}{1504} a^{10} - \frac{85}{1504} a^{9} - \frac{21}{752} a^{8} - \frac{1}{94} a^{7} + \frac{31}{376} a^{6} + \frac{69}{376} a^{5} - \frac{33}{188} a^{4} - \frac{10}{47} a^{3} - \frac{21}{94} a^{2} + \frac{11}{47} a$, $\frac{1}{1276699730043967593024} a^{15} + \frac{43766158880372663}{425566576681322531008} a^{14} + \frac{602187656986925620}{19948433281936993641} a^{13} + \frac{436436330026210857}{212783288340661265504} a^{12} - \frac{19797630628149132281}{1276699730043967593024} a^{11} - \frac{42708796912217130929}{1276699730043967593024} a^{10} + \frac{1942180783097326639}{26597911042582658188} a^{9} + \frac{48480864034216506739}{638349865021983796512} a^{8} - \frac{4169161580856691185}{53195822085165316376} a^{7} - \frac{2393744145706001035}{159587466255495949128} a^{6} - \frac{1749367013286264105}{53195822085165316376} a^{5} - \frac{2631932441829493211}{19948433281936993641} a^{4} + \frac{2116300837206753471}{6649477760645664547} a^{3} - \frac{5078375950621883665}{13298955521291329094} a^{2} - \frac{4999363571794314752}{19948433281936993641} a - \frac{180572074909750298}{424434750679510503}$

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

Class group and class number

$C_{2}\times C_{20}$, which has order $40$ (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:  \( 4627492.67606 \) (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

\(\Q(\sqrt{34}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), 4.0.434656.1, \(\Q(\sqrt{2}, \sqrt{17})\), 4.0.108664.1, 8.0.3022813413376.6, 8.4.257260716032.5, 8.4.257260716032.4

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.8.22.87$x^{8} + 4 x^{7} + 6 x^{4} + 12 x^{2} + 2$$8$$1$$22$$D_4$$[2, 3, 7/2]$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$