↑

Properties

Label 16.0.10737418240...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{42}\cdot 5^{12}$
Root discriminant $20.63$
Ramified primes $2, 5$
Class number $4$
Class group $[4]$
Galois group $C_8:C_2^2$ (as 16T45)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 16, 96, 264, 394, 348, 212, 52, -36, -64, -64, -48, 18, 28, -4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 4*x^14 + 28*x^13 + 18*x^12 - 48*x^11 - 64*x^10 - 64*x^9 - 36*x^8 + 52*x^7 + 212*x^6 + 348*x^5 + 394*x^4 + 264*x^3 + 96*x^2 + 16*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 - 4*x^14 + 28*x^13 + 18*x^12 - 48*x^11 - 64*x^10 - 64*x^9 - 36*x^8 + 52*x^7 + 212*x^6 + 348*x^5 + 394*x^4 + 264*x^3 + 96*x^2 + 16*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 4 x^{14} + 28 x^{13} + 18 x^{12} - 48 x^{11} - 64 x^{10} - 64 x^{9} - 36 x^{8} + 52 x^{7} + 212 x^{6} + 348 x^{5} + 394 x^{4} + 264 x^{3} + 96 x^{2} + 16 x + 1 \)

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:  \(1073741824000000000000=2^{42}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.63$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{979} a^{14} - \frac{335}{979} a^{13} + \frac{476}{979} a^{12} + \frac{125}{979} a^{11} - \frac{299}{979} a^{10} + \frac{380}{979} a^{9} - \frac{338}{979} a^{8} + \frac{34}{89} a^{7} - \frac{129}{979} a^{6} + \frac{467}{979} a^{5} + \frac{70}{979} a^{4} - \frac{405}{979} a^{3} + \frac{202}{979} a^{2} + \frac{12}{89} a + \frac{269}{979}$, $\frac{1}{795411144341} a^{15} - \frac{163759324}{795411144341} a^{14} - \frac{26569349077}{795411144341} a^{13} + \frac{109342547599}{795411144341} a^{12} - \frac{216510611003}{795411144341} a^{11} + \frac{9116421885}{25658424011} a^{10} + \frac{331718017756}{795411144341} a^{9} - \frac{79620497577}{795411144341} a^{8} - \frac{36381718326}{795411144341} a^{7} - \frac{33586186580}{72310104031} a^{6} - \frac{193931336002}{795411144341} a^{5} + \frac{247012420016}{795411144341} a^{4} - \frac{129511966279}{795411144341} a^{3} - \frac{27442799508}{795411144341} a^{2} - \frac{135705866203}{795411144341} a + \frac{231755432600}{795411144341}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( -\frac{567417922}{564521749} a^{15} + \frac{2296378330}{564521749} a^{14} + \frac{2191276870}{564521749} a^{13} - \frac{16127816755}{564521749} a^{12} - \frac{9489149920}{564521749} a^{11} + \frac{920357388}{18210379} a^{10} + \frac{35002721360}{564521749} a^{9} + \frac{33233943350}{564521749} a^{8} + \frac{17749389390}{564521749} a^{7} - \frac{2876148910}{51320159} a^{6} - \frac{118991529606}{564521749} a^{5} - \frac{189948096905}{564521749} a^{4} - \frac{209219214200}{564521749} a^{3} - \frac{132938490340}{564521749} a^{2} - \frac{40818620340}{564521749} a - \frac{3978716187}{564521749} \) (order $4$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 10024.846831 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T45):

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 $C_8:C_2^2$
Character table for $C_8:C_2^2$

Intermediate fields

\(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{10}) \), 4.0.320.1, 4.0.1280.1, \(\Q(i, \sqrt{10})\), 8.0.163840000.2

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$