Properties

Label 16.0.666...433.3
Degree $16$
Signature $[0, 8]$
Discriminant $6.669\times 10^{31}$
Root discriminant $97.50$
Ramified primes $13, 17$
Class number $3104$ (GRH)
Class group $[2, 2, 2, 388]$ (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 26*x^14 - 129*x^13 + 302*x^12 - 1025*x^11 + 4282*x^10 + 16326*x^9 + 95981*x^8 + 338337*x^7 + 1003331*x^6 + 2604733*x^5 + 6573615*x^4 + 13657839*x^3 + 25464387*x^2 + 29036657*x + 45417983)
 
gp: K = bnfinit(x^16 - 3*x^15 + 26*x^14 - 129*x^13 + 302*x^12 - 1025*x^11 + 4282*x^10 + 16326*x^9 + 95981*x^8 + 338337*x^7 + 1003331*x^6 + 2604733*x^5 + 6573615*x^4 + 13657839*x^3 + 25464387*x^2 + 29036657*x + 45417983, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![45417983, 29036657, 25464387, 13657839, 6573615, 2604733, 1003331, 338337, 95981, 16326, 4282, -1025, 302, -129, 26, -3, 1]);
 

\( x^{16} - 3 x^{15} + 26 x^{14} - 129 x^{13} + 302 x^{12} - 1025 x^{11} + 4282 x^{10} + 16326 x^{9} + 95981 x^{8} + 338337 x^{7} + 1003331 x^{6} + 2604733 x^{5} + 6573615 x^{4} + 13657839 x^{3} + 25464387 x^{2} + 29036657 x + 45417983 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(66688975910627504451630153142433\)\(\medspace = 13^{12}\cdot 17^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $97.50$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $13, 17$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} + \frac{6}{13} a^{11} - \frac{4}{13} a^{10} + \frac{3}{13} a^{9} + \frac{3}{13} a^{8} - \frac{5}{13} a^{7} - \frac{1}{13} a^{6} + \frac{4}{13} a^{5} - \frac{4}{13} a^{4}$, $\frac{1}{13} a^{13} - \frac{1}{13} a^{11} + \frac{1}{13} a^{10} - \frac{2}{13} a^{9} + \frac{3}{13} a^{8} + \frac{3}{13} a^{7} - \frac{3}{13} a^{6} - \frac{2}{13} a^{5} - \frac{2}{13} a^{4}$, $\frac{1}{13} a^{14} - \frac{6}{13} a^{11} - \frac{6}{13} a^{10} + \frac{6}{13} a^{9} + \frac{6}{13} a^{8} + \frac{5}{13} a^{7} - \frac{3}{13} a^{6} + \frac{2}{13} a^{5} - \frac{4}{13} a^{4}$, $\frac{1}{6390145980772722025343275212010336388135095353991913} a^{15} - \frac{56479902511564403777606531558310708271504673280143}{6390145980772722025343275212010336388135095353991913} a^{14} + \frac{179413787225065635050735023749175329720378102133639}{6390145980772722025343275212010336388135095353991913} a^{13} - \frac{49759378359578254339367385246314419390661822985721}{6390145980772722025343275212010336388135095353991913} a^{12} + \frac{2459276856342958744393999524535693014240294518326939}{6390145980772722025343275212010336388135095353991913} a^{11} - \frac{1503510415522994842904008218446201989160176480206942}{6390145980772722025343275212010336388135095353991913} a^{10} + \frac{3080040473029832734799400045663208198827898074991340}{6390145980772722025343275212010336388135095353991913} a^{9} + \frac{416287503837916329705495462025592766137662452590730}{6390145980772722025343275212010336388135095353991913} a^{8} + \frac{108185744556916953538992090863645637187861012016962}{6390145980772722025343275212010336388135095353991913} a^{7} - \frac{2526121402358469064168913861724917683948060037476648}{6390145980772722025343275212010336388135095353991913} a^{6} - \frac{3706465445833796417318634383198777141646288769022}{42886885776998134398277014845706955625067754053637} a^{5} - \frac{42674384931036570437057554480758632389199923440557}{491549690828670925026405785539256645241161181076301} a^{4} + \frac{15293801759874770867388919338703383912410718362742}{491549690828670925026405785539256645241161181076301} a^{3} + \frac{12929082218154644752952313163307793829890897798319}{491549690828670925026405785539256645241161181076301} a^{2} + \frac{180633217363136685515038433073984310144720440701581}{491549690828670925026405785539256645241161181076301} a + \frac{875988098350203749498989479873597516482690243767}{4866828622066048762637681044943135101397635456201}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{388}$, which has order $3104$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 464752.47625 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 464752.47625 \cdot 3104}{2\sqrt{66688975910627504451630153142433}}\approx 0.21454843552$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.8.11719682839553.1

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

Sibling fields

Galois closure: Deg 32

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$13$13.4.3.3$x^{4} + 26$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.4$x^{4} + 104$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.3$x^{4} + 26$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.4$x^{4} + 104$$4$$1$$3$$C_4$$[\ ]_{4}$
17Data not computed