↑

Properties

Label 16.0.34984505969...9921.2
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 37^{12}$
Root discriminant $34.20$
Ramified primes $3, 37$
Class number $16$ (GRH)
Class group $[2, 8]$ (GRH)
Galois group $C_2^2.D_4$ (as 16T33)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4288, 3008, 2064, -6392, -3372, 360, 11425, -8108, 198, 662, 301, -96, -75, 14, 18, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 18*x^14 + 14*x^13 - 75*x^12 - 96*x^11 + 301*x^10 + 662*x^9 + 198*x^8 - 8108*x^7 + 11425*x^6 + 360*x^5 - 3372*x^4 - 6392*x^3 + 2064*x^2 + 3008*x + 4288)
 
gp: K = bnfinit(x^16 - 8*x^15 + 18*x^14 + 14*x^13 - 75*x^12 - 96*x^11 + 301*x^10 + 662*x^9 + 198*x^8 - 8108*x^7 + 11425*x^6 + 360*x^5 - 3372*x^4 - 6392*x^3 + 2064*x^2 + 3008*x + 4288, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 18 x^{14} + 14 x^{13} - 75 x^{12} - 96 x^{11} + 301 x^{10} + 662 x^{9} + 198 x^{8} - 8108 x^{7} + 11425 x^{6} + 360 x^{5} - 3372 x^{4} - 6392 x^{3} + 2064 x^{2} + 3008 x + 4288 \)

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:  \(3498450596935634189769921=3^{12}\cdot 37^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.20$
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:  $3, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{592} a^{12} - \frac{3}{296} a^{11} + \frac{31}{592} a^{10} + \frac{3}{37} a^{9} - \frac{17}{148} a^{8} + \frac{33}{296} a^{7} + \frac{53}{592} a^{6} + \frac{35}{296} a^{5} + \frac{43}{592} a^{4} + \frac{107}{296} a^{3} + \frac{29}{148} a^{2} + \frac{3}{74} a + \frac{11}{37}$, $\frac{1}{592} a^{13} - \frac{5}{592} a^{11} + \frac{3}{148} a^{10} - \frac{1}{296} a^{9} - \frac{23}{296} a^{8} + \frac{5}{592} a^{7} - \frac{7}{74} a^{6} - \frac{129}{592} a^{5} - \frac{23}{296} a^{4} - \frac{3}{296} a^{3} - \frac{21}{74} a^{2} + \frac{3}{74} a - \frac{8}{37}$, $\frac{1}{3981580064} a^{14} - \frac{1}{568797152} a^{13} - \frac{1420145}{3981580064} a^{12} + \frac{501233}{234210592} a^{11} - \frac{377787}{16729328} a^{10} + \frac{185728777}{1990790032} a^{9} + \frac{69364931}{3981580064} a^{8} - \frac{421684159}{3981580064} a^{7} + \frac{280468899}{3981580064} a^{6} - \frac{885306333}{3981580064} a^{5} - \frac{194623371}{995395016} a^{4} - \frac{2098717}{7319081} a^{3} - \frac{38996310}{124424377} a^{2} - \frac{4530969}{124424377} a - \frac{8898283}{124424377}$, $\frac{1}{75088618426976} a^{15} + \frac{673}{5363472744784} a^{14} - \frac{18075905949}{37544309213488} a^{13} + \frac{15668383707}{37544309213488} a^{12} + \frac{680118033}{90142399072} a^{11} - \frac{516588305559}{37544309213488} a^{10} - \frac{8329650657175}{75088618426976} a^{9} + \frac{1386412617945}{18772154606744} a^{8} - \frac{602047659309}{37544309213488} a^{7} + \frac{81306565077}{18772154606744} a^{6} + \frac{11129820065785}{75088618426976} a^{5} - \frac{8231224363819}{37544309213488} a^{4} - \frac{4857675686}{2346519325843} a^{3} + \frac{1641390361855}{9386077303372} a^{2} + \frac{1250891990043}{4693038651686} a + \frac{121235049132}{335217046549}$

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

Class group and class number

$C_{2}\times C_{8}$, which has order $16$ (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:  \( \frac{4530209}{46123229992} a^{15} - \frac{67953135}{92246459984} a^{14} + \frac{108394345}{92246459984} a^{13} + \frac{326059305}{92246459984} a^{12} - \frac{45858481}{5426262352} a^{11} - \frac{783126993}{46123229992} a^{10} + \frac{1399553513}{46123229992} a^{9} + \frac{9389550963}{92246459984} a^{8} + \frac{3984310193}{92246459984} a^{7} - \frac{82702181595}{92246459984} a^{6} + \frac{42016992569}{92246459984} a^{5} + \frac{35496571503}{23061614996} a^{4} - \frac{55397886083}{46123229992} a^{3} - \frac{7237070979}{23061614996} a^{2} - \frac{517958041}{5765403749} a + \frac{3899189587}{5765403749} \) (order $6$)
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:  \( 292944.962933 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^2.D_4$
Character table for $C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{-111}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{37}) \), 4.2.151959.1 x2, \(\Q(\sqrt{-3}, \sqrt{37})\), 4.0.455877.1 x2, 8.0.207823839129.2, 8.0.50551744653.1 x2, 8.0.1870414552161.1 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$37$37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$