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Properties

Label 16.8.60210204307...3637.1
Degree $16$
Signature $[8, 4]$
Discriminant $3^{10}\cdot 13^{9}\cdot 1327^{8}$
Root discriminant $306.36$
Ramified primes $3, 13, 1327$
Class number $96$ (GRH)
Class group $[2, 48]$ (GRH)
Galois group 16T1675

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4653510391, -3995172348, -1983832568, 2074151994, -301240263, -41505565, 75430919, -15197529, 1234105, 477846, -202403, 26877, -2660, -220, 72, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 72*x^14 - 220*x^13 - 2660*x^12 + 26877*x^11 - 202403*x^10 + 477846*x^9 + 1234105*x^8 - 15197529*x^7 + 75430919*x^6 - 41505565*x^5 - 301240263*x^4 + 2074151994*x^3 - 1983832568*x^2 - 3995172348*x + 4653510391)
 
gp: K = bnfinit(x^16 - 8*x^15 + 72*x^14 - 220*x^13 - 2660*x^12 + 26877*x^11 - 202403*x^10 + 477846*x^9 + 1234105*x^8 - 15197529*x^7 + 75430919*x^6 - 41505565*x^5 - 301240263*x^4 + 2074151994*x^3 - 1983832568*x^2 - 3995172348*x + 4653510391, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 72 x^{14} - 220 x^{13} - 2660 x^{12} + 26877 x^{11} - 202403 x^{10} + 477846 x^{9} + 1234105 x^{8} - 15197529 x^{7} + 75430919 x^{6} - 41505565 x^{5} - 301240263 x^{4} + 2074151994 x^{3} - 1983832568 x^{2} - 3995172348 x + 4653510391 \)

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:  \(6021020430747726555546975715999071923637=3^{10}\cdot 13^{9}\cdot 1327^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $306.36$
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:  $3, 13, 1327$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{4766644528159188334871730969980170147954809851782313026982238751802679} a^{15} - \frac{273184637158300799201809656494064408851111377371743082303470900932072}{4766644528159188334871730969980170147954809851782313026982238751802679} a^{14} - \frac{50246594630597181851575125616467305146094903909846109843474943976380}{4766644528159188334871730969980170147954809851782313026982238751802679} a^{13} - \frac{88262538072060294353905540045074604387737182332999080172077685400453}{1588881509386396111623910323326723382651603283927437675660746250600893} a^{12} - \frac{2211467980968998448824807993760090566789573839699998063703805601997715}{4766644528159188334871730969980170147954809851782313026982238751802679} a^{11} - \frac{1132991995338671574882343887931446766825498491096400433670091788829325}{4766644528159188334871730969980170147954809851782313026982238751802679} a^{10} - \frac{254173179068974828659298476336309450438553848493006560635858897543066}{4766644528159188334871730969980170147954809851782313026982238751802679} a^{9} + \frac{1433505641464947476628963276385293500618255877199608553252557598409}{10857960200818196662577974874670091453200022441417569537544962988161} a^{8} + \frac{1339178332883934333437691857024893719666385682695019728418760488565635}{4766644528159188334871730969980170147954809851782313026982238751802679} a^{7} - \frac{542793613658454761228222773402798343763977423944024071132109723746487}{1588881509386396111623910323326723382651603283927437675660746250600893} a^{6} - \frac{2118522305695925521336390478757690665018230302722856464665383798904391}{4766644528159188334871730969980170147954809851782313026982238751802679} a^{5} + \frac{787671338359524401433851033196275069809384691461677545839922501118419}{1588881509386396111623910323326723382651603283927437675660746250600893} a^{4} + \frac{722154967771247898873447208738144056267020965035780616696574081023936}{1588881509386396111623910323326723382651603283927437675660746250600893} a^{3} + \frac{671438123983990065789945158552257121546542376209634507967798784081915}{4766644528159188334871730969980170147954809851782313026982238751802679} a^{2} + \frac{322128094727233012733772086174041303698470611219726406352837999595135}{1588881509386396111623910323326723382651603283927437675660746250600893} a + \frac{1072481254616696422090359649693493366128621224977593102076448039700937}{4766644528159188334871730969980170147954809851782313026982238751802679}$

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

Class group and class number

$C_{2}\times C_{48}$, which has order $96$ (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:  \( 1594068728730 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1675:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6144
The 54 conjugacy class representatives for t16n1675 are not computed
Character table for t16n1675 is not computed

Intermediate fields

\(\Q(\sqrt{51753}) \), 4.4.3981.1, 8.8.7173681975339714081.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$13$13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.12.6.1$x^{12} + 338 x^{8} + 8788 x^{6} + 28561 x^{4} + 19307236$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
1327Data not computed