↑

Properties

Label 16.0.20723941878...0625.4
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 13^{8}\cdot 101^{4}$
Root discriminant $38.22$
Ramified primes $5, 13, 101$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T268)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![30416, -78712, 94988, -55758, -2717, 26358, -13816, -2124, 5883, -2752, 31, 560, -247, 16, 22, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 22*x^14 + 16*x^13 - 247*x^12 + 560*x^11 + 31*x^10 - 2752*x^9 + 5883*x^8 - 2124*x^7 - 13816*x^6 + 26358*x^5 - 2717*x^4 - 55758*x^3 + 94988*x^2 - 78712*x + 30416)
 
gp: K = bnfinit(x^16 - 8*x^15 + 22*x^14 + 16*x^13 - 247*x^12 + 560*x^11 + 31*x^10 - 2752*x^9 + 5883*x^8 - 2124*x^7 - 13816*x^6 + 26358*x^5 - 2717*x^4 - 55758*x^3 + 94988*x^2 - 78712*x + 30416, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 22 x^{14} + 16 x^{13} - 247 x^{12} + 560 x^{11} + 31 x^{10} - 2752 x^{9} + 5883 x^{8} - 2124 x^{7} - 13816 x^{6} + 26358 x^{5} - 2717 x^{4} - 55758 x^{3} + 94988 x^{2} - 78712 x + 30416 \)

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:  \(20723941878730254150390625=5^{12}\cdot 13^{8}\cdot 101^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.22$
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:  $5, 13, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} + \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} + \frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{86069701172806012576662968} a^{15} - \frac{1214417313341009644498605}{21517425293201503144165742} a^{14} - \frac{211851283714886541397745}{10758712646600751572082871} a^{13} + \frac{3129129505102123692498521}{43034850586403006288331484} a^{12} - \frac{1070305670447686177313537}{7824518288436910234242088} a^{11} - \frac{10310928280317639758458241}{43034850586403006288331484} a^{10} - \frac{2491416437745158211208181}{86069701172806012576662968} a^{9} + \frac{7299344559726866763580417}{43034850586403006288331484} a^{8} + \frac{2854592048953969888158095}{86069701172806012576662968} a^{7} - \frac{11844431651564636393250035}{43034850586403006288331484} a^{6} + \frac{3890738206803030355910001}{10758712646600751572082871} a^{5} + \frac{8969614301086017840552711}{21517425293201503144165742} a^{4} + \frac{12262495334033080100333285}{86069701172806012576662968} a^{3} + \frac{16458929230980961304683121}{43034850586403006288331484} a^{2} + \frac{2510016146776755387316792}{10758712646600751572082871} a + \frac{2756033621661158397616835}{10758712646600751572082871}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( \frac{1683757683586639528755}{3912259144218455117121044} a^{15} - \frac{21750305240284588821215}{7824518288436910234242088} a^{14} + \frac{41999554812265047931483}{7824518288436910234242088} a^{13} + \frac{27614117957621432082663}{1956129572109227558560522} a^{12} - \frac{651466902725506420947589}{7824518288436910234242088} a^{11} + \frac{470577913352969674535057}{3912259144218455117121044} a^{10} + \frac{1331580622252624406882733}{7824518288436910234242088} a^{9} - \frac{884126509071557328378929}{978064786054613779280261} a^{8} + \frac{9686432984766970987817053}{7824518288436910234242088} a^{7} + \frac{2847395751222325821861699}{3912259144218455117121044} a^{6} - \frac{36569804148382758239152515}{7824518288436910234242088} a^{5} + \frac{36705613952619293009503499}{7824518288436910234242088} a^{4} + \frac{18887718111008369742746467}{3912259144218455117121044} a^{3} - \frac{16222879798626179633244761}{978064786054613779280261} a^{2} + \frac{35606669293308618151941469}{1956129572109227558560522} a - \frac{9124794854493681845340002}{978064786054613779280261} \) (order $10$)
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:  \( 1041596.08737 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T268):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 29 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{65}) \), \(\Q(\zeta_{5})\), 4.0.21125.1, \(\Q(\sqrt{5}, \sqrt{13})\), 8.0.446265625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
101Data not computed