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Properties

Label 16.8.21431487973...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{8}\cdot 5^{12}\cdot 7^{8}\cdot 29^{6}$
Root discriminant $44.23$
Ramified primes $2, 5, 7, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2^2\times C_4).C_2^4$ (as 16T471)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1331, -6897, 8173, -2524, -6699, 14737, 4443, -6587, -2568, -95, 497, 234, -43, -23, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 9*x^14 - 23*x^13 - 43*x^12 + 234*x^11 + 497*x^10 - 95*x^9 - 2568*x^8 - 6587*x^7 + 4443*x^6 + 14737*x^5 - 6699*x^4 - 2524*x^3 + 8173*x^2 - 6897*x + 1331)
 
gp: K = bnfinit(x^16 - 9*x^14 - 23*x^13 - 43*x^12 + 234*x^11 + 497*x^10 - 95*x^9 - 2568*x^8 - 6587*x^7 + 4443*x^6 + 14737*x^5 - 6699*x^4 - 2524*x^3 + 8173*x^2 - 6897*x + 1331, 1)
 

Normalized defining polynomial

\( x^{16} - 9 x^{14} - 23 x^{13} - 43 x^{12} + 234 x^{11} + 497 x^{10} - 95 x^{9} - 2568 x^{8} - 6587 x^{7} + 4443 x^{6} + 14737 x^{5} - 6699 x^{4} - 2524 x^{3} + 8173 x^{2} - 6897 x + 1331 \)

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:  \(214314879732757562500000000=2^{8}\cdot 5^{12}\cdot 7^{8}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.23$
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:  $2, 5, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{55} a^{14} + \frac{2}{55} a^{12} - \frac{23}{55} a^{11} + \frac{23}{55} a^{10} + \frac{14}{55} a^{9} + \frac{24}{55} a^{8} + \frac{3}{11} a^{7} - \frac{1}{11} a^{6} + \frac{2}{55} a^{5} - \frac{23}{55} a^{4} - \frac{3}{55} a^{3} - \frac{1}{5} a^{2} - \frac{27}{55} a$, $\frac{1}{417762322712121299576111813255} a^{15} + \frac{14594496405417969119709672}{3452581179439019004761254655} a^{14} + \frac{24317872869397658765802445288}{417762322712121299576111813255} a^{13} + \frac{28057936765067554781218085918}{417762322712121299576111813255} a^{12} + \frac{80969057364447988183079408668}{417762322712121299576111813255} a^{11} - \frac{155815941960381921174913871186}{417762322712121299576111813255} a^{10} - \frac{48847317298981025961262808152}{417762322712121299576111813255} a^{9} - \frac{133803986234749122927927061546}{417762322712121299576111813255} a^{8} - \frac{27238033525618364539905542029}{417762322712121299576111813255} a^{7} + \frac{33373201005997435640612404104}{417762322712121299576111813255} a^{6} - \frac{15509338965632114605482954241}{83552464542424259915222362651} a^{5} - \frac{6515277601788056957581105449}{417762322712121299576111813255} a^{4} - \frac{2011603360587883899015988768}{7595678594765841810474760241} a^{3} - \frac{150688532691690800755210065138}{417762322712121299576111813255} a^{2} - \frac{11612220608418346128443994108}{37978392973829209052373801205} a - \frac{288451264807477099432765993}{690516235887803800952250931}$

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:  \( 14469337.1322 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times C_4).C_2^4$ (as 16T471):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $(C_2^2\times C_4).C_2^4$
Character table for $(C_2^2\times C_4).C_2^4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.6125.1, 4.4.725.1, 4.4.177625.1, 8.8.31550640625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$