↑

Properties

Label 16.0.89779938981...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{4}\cdot 5^{10}\cdot 19^{4}\cdot 89^{2}$
Root discriminant $74.49$
Ramified primes $2, 3, 5, 19, 89$
Class number $24$ (GRH)
Class group $[2, 2, 6]$ (GRH)
Galois group 16T799

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1148841, 4998924, 45297924, 28664052, 17162450, 2083248, 2974708, 532960, 268071, 17504, 22468, 720, 876, 4, 44, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 44*x^14 + 4*x^13 + 876*x^12 + 720*x^11 + 22468*x^10 + 17504*x^9 + 268071*x^8 + 532960*x^7 + 2974708*x^6 + 2083248*x^5 + 17162450*x^4 + 28664052*x^3 + 45297924*x^2 + 4998924*x + 1148841)
 
gp: K = bnfinit(x^16 - 4*x^15 + 44*x^14 + 4*x^13 + 876*x^12 + 720*x^11 + 22468*x^10 + 17504*x^9 + 268071*x^8 + 532960*x^7 + 2974708*x^6 + 2083248*x^5 + 17162450*x^4 + 28664052*x^3 + 45297924*x^2 + 4998924*x + 1148841, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 44 x^{14} + 4 x^{13} + 876 x^{12} + 720 x^{11} + 22468 x^{10} + 17504 x^{9} + 268071 x^{8} + 532960 x^{7} + 2974708 x^{6} + 2083248 x^{5} + 17162450 x^{4} + 28664052 x^{3} + 45297924 x^{2} + 4998924 x + 1148841 \)

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:  \(897799389814236119040000000000=2^{40}\cdot 3^{4}\cdot 5^{10}\cdot 19^{4}\cdot 89^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.49$
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:  $2, 3, 5, 19, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{9} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{7} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{465} a^{14} - \frac{7}{465} a^{13} - \frac{1}{15} a^{12} - \frac{68}{465} a^{11} - \frac{74}{155} a^{10} - \frac{28}{155} a^{9} + \frac{136}{465} a^{8} + \frac{143}{465} a^{7} + \frac{77}{155} a^{6} + \frac{157}{465} a^{5} + \frac{148}{465} a^{4} - \frac{2}{31} a^{3} + \frac{2}{15} a^{2} - \frac{13}{155} a - \frac{71}{155}$, $\frac{1}{5563192857703552017879357433125533189380665021351501975} a^{15} + \frac{117045659646440065229546172242837903700730197452}{7178313364778776797263687010484558954039567769485809} a^{14} - \frac{131999925403455318849948524350232454321623457688064761}{5563192857703552017879357433125533189380665021351501975} a^{13} - \frac{29202135763265561472531475752171945941428365197048953}{1112638571540710403575871486625106637876133004270300395} a^{12} + \frac{306285089724869904591152020245130619472794103385001634}{618132539744839113097706381458392576597851669039055775} a^{11} - \frac{203869773200054626246101585959949374432586399975949652}{1854397619234517339293119144375177729793555007117167325} a^{10} - \frac{168137130330117193461894294183497501660874732345325751}{5563192857703552017879357433125533189380665021351501975} a^{9} + \frac{235048933221213489397969278226816967807235471255660427}{1112638571540710403575871486625106637876133004270300395} a^{8} - \frac{122916282586809234630400059775246593065432125796891916}{618132539744839113097706381458392576597851669039055775} a^{7} - \frac{792746220054306259653734718195588350863169849218891616}{5563192857703552017879357433125533189380665021351501975} a^{6} + \frac{1501966551915849466841952353029812215668197449731559674}{5563192857703552017879357433125533189380665021351501975} a^{5} + \frac{351770047042548895985289610939505433936605063190932833}{1854397619234517339293119144375177729793555007117167325} a^{4} + \frac{2157012416240329795298849265812361462276849083588373431}{5563192857703552017879357433125533189380665021351501975} a^{3} + \frac{172883667347862521831660702963577333077794100536330217}{1854397619234517339293119144375177729793555007117167325} a^{2} - \frac{65270740424249075499534571219856100321866983217301389}{1854397619234517339293119144375177729793555007117167325} a - \frac{65798986580198340215425846231498508377657340521529726}{618132539744839113097706381458392576597851669039055775}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{6}$, which has order $24$ (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:  \( 11014507.1976 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T799:

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 t16n799 are not computed
Character table for t16n799 is not computed

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), 4.0.7600.1, 4.0.30400.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.14786560000.2

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$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}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$