↑

Properties

Label 16.8.65790008098...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 5^{12}\cdot 89^{4}$
Root discriminant $41.08$
Ramified primes $2, 5, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T210)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1296, 864, -48816, -81408, 167864, 67856, -96624, -15616, 20920, 2904, -2268, -528, 160, 36, 8, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 8*x^14 + 36*x^13 + 160*x^12 - 528*x^11 - 2268*x^10 + 2904*x^9 + 20920*x^8 - 15616*x^7 - 96624*x^6 + 67856*x^5 + 167864*x^4 - 81408*x^3 - 48816*x^2 + 864*x + 1296)
 
gp: K = bnfinit(x^16 - 8*x^15 + 8*x^14 + 36*x^13 + 160*x^12 - 528*x^11 - 2268*x^10 + 2904*x^9 + 20920*x^8 - 15616*x^7 - 96624*x^6 + 67856*x^5 + 167864*x^4 - 81408*x^3 - 48816*x^2 + 864*x + 1296, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 8 x^{14} + 36 x^{13} + 160 x^{12} - 528 x^{11} - 2268 x^{10} + 2904 x^{9} + 20920 x^{8} - 15616 x^{7} - 96624 x^{6} + 67856 x^{5} + 167864 x^{4} - 81408 x^{3} - 48816 x^{2} + 864 x + 1296 \)

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:  \(65790008098816000000000000=2^{32}\cdot 5^{12}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.08$
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:  $2, 5, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{24} a^{12} + \frac{1}{12} a^{11} - \frac{1}{12} a^{10} - \frac{1}{12} a^{9} + \frac{1}{12} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{24} a^{13} + \frac{1}{12} a^{10} - \frac{1}{12} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{2952} a^{14} - \frac{7}{369} a^{13} + \frac{19}{1476} a^{12} - \frac{2}{123} a^{11} - \frac{35}{738} a^{10} - \frac{2}{123} a^{9} - \frac{7}{492} a^{8} + \frac{9}{41} a^{7} + \frac{74}{369} a^{6} - \frac{17}{369} a^{5} - \frac{28}{123} a^{4} + \frac{136}{369} a^{3} - \frac{155}{369} a^{2} + \frac{34}{123} a + \frac{15}{41}$, $\frac{1}{106777271793707973509087762712} a^{15} - \frac{999516481358793985727503}{53388635896853986754543881356} a^{14} - \frac{75110998031433395680805899}{3813473992632427625324562954} a^{13} - \frac{7049252405818947209995391}{4449052991404498896211990113} a^{12} + \frac{1055523059668114506562896938}{13347158974213496688635970339} a^{11} + \frac{669486393526201703491044163}{17796211965617995584847960452} a^{10} + \frac{693828597144018597086684905}{5932070655205998528282653484} a^{9} + \frac{14869261307248847712888155}{434053950380926721581657572} a^{8} + \frac{29916565896335415328157590}{325540462785695041186243179} a^{7} + \frac{2790073012589867761267312132}{13347158974213496688635970339} a^{6} - \frac{939873993340014732469770677}{4449052991404498896211990113} a^{5} + \frac{88276572257573123517738589}{1906736996316213812662281477} a^{4} - \frac{2982923204700957134461319387}{13347158974213496688635970339} a^{3} + \frac{50354159648003124315016199}{635578998772071270887427159} a^{2} + \frac{366836013167383942560818455}{1483017663801499632070663371} a - \frac{41310910029275566164390936}{494339221267166544023554457}$

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:  \( 6569018.97425 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T210):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.4.178000.1, 4.4.712000.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.4.91136000000.1, 8.4.227840000.1, 8.8.8111104000000.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
5Data not computed
89Data not computed