↑

Properties

Label 16.0.10055599390...8125.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 11^{8}\cdot 229^{3}$
Root discriminant $20.54$
Ramified primes $5, 11, 229$
Class number $2$
Class group $[2]$
Galois group 16T1741

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -90, 319, -462, 399, -423, 484, -435, 414, -402, 345, -267, 163, -75, 27, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 27*x^14 - 75*x^13 + 163*x^12 - 267*x^11 + 345*x^10 - 402*x^9 + 414*x^8 - 435*x^7 + 484*x^6 - 423*x^5 + 399*x^4 - 462*x^3 + 319*x^2 - 90*x + 9)
 
gp: K = bnfinit(x^16 - 6*x^15 + 27*x^14 - 75*x^13 + 163*x^12 - 267*x^11 + 345*x^10 - 402*x^9 + 414*x^8 - 435*x^7 + 484*x^6 - 423*x^5 + 399*x^4 - 462*x^3 + 319*x^2 - 90*x + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 27 x^{14} - 75 x^{13} + 163 x^{12} - 267 x^{11} + 345 x^{10} - 402 x^{9} + 414 x^{8} - 435 x^{7} + 484 x^{6} - 423 x^{5} + 399 x^{4} - 462 x^{3} + 319 x^{2} - 90 x + 9 \)

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:  \(1005559939055198828125=5^{8}\cdot 11^{8}\cdot 229^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.54$
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:  $5, 11, 229$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{41} a^{13} - \frac{17}{41} a^{12} + \frac{9}{41} a^{10} + \frac{7}{41} a^{9} - \frac{15}{41} a^{8} - \frac{10}{41} a^{7} + \frac{16}{41} a^{6} + \frac{10}{41} a^{5} + \frac{1}{41} a^{4} + \frac{17}{41} a^{3} + \frac{3}{41} a^{2} - \frac{20}{41} a - \frac{7}{41}$, $\frac{1}{451} a^{14} + \frac{2}{451} a^{13} + \frac{5}{451} a^{12} - \frac{155}{451} a^{11} - \frac{68}{451} a^{10} + \frac{118}{451} a^{9} - \frac{131}{451} a^{8} + \frac{72}{451} a^{7} - \frac{137}{451} a^{6} - \frac{219}{451} a^{5} + \frac{7}{41} a^{4} - \frac{43}{451} a^{3} - \frac{19}{41} a^{2} + \frac{105}{451} a - \frac{51}{451}$, $\frac{1}{114166290183} a^{15} + \frac{26227913}{38055430061} a^{14} - \frac{198567327}{38055430061} a^{13} - \frac{14018375551}{38055430061} a^{12} + \frac{13068927496}{114166290183} a^{11} + \frac{18935944917}{38055430061} a^{10} - \frac{11541530634}{38055430061} a^{9} - \frac{6530107945}{38055430061} a^{8} + \frac{768269020}{38055430061} a^{7} + \frac{12259506340}{38055430061} a^{6} + \frac{3283159901}{10378753653} a^{5} - \frac{13143666246}{38055430061} a^{4} - \frac{430028511}{3459584551} a^{3} + \frac{393192165}{928181221} a^{2} - \frac{12780312860}{114166290183} a + \frac{975347374}{3459584551}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4487.11850935 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1741:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 95 conjugacy class representatives for t16n1741 are not computed
Character table for t16n1741 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 8.4.2095493125.1

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 $16$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R $16$ R $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
229Data not computed