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Properties

Label 16.4.27728016466...0000.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{28}\cdot 5^{8}\cdot 71^{4}\cdot 101^{4}$
Root discriminant $69.21$
Ramified primes $2, 5, 71, 101$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1275

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2500, -175000, -284600, -367600, -340336, -230312, -118016, -45012, -7660, -52, 2124, -44, -100, 88, -8, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 8*x^14 + 88*x^13 - 100*x^12 - 44*x^11 + 2124*x^10 - 52*x^9 - 7660*x^8 - 45012*x^7 - 118016*x^6 - 230312*x^5 - 340336*x^4 - 367600*x^3 - 284600*x^2 - 175000*x - 2500)
 
gp: K = bnfinit(x^16 - 2*x^15 - 8*x^14 + 88*x^13 - 100*x^12 - 44*x^11 + 2124*x^10 - 52*x^9 - 7660*x^8 - 45012*x^7 - 118016*x^6 - 230312*x^5 - 340336*x^4 - 367600*x^3 - 284600*x^2 - 175000*x - 2500, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 8 x^{14} + 88 x^{13} - 100 x^{12} - 44 x^{11} + 2124 x^{10} - 52 x^{9} - 7660 x^{8} - 45012 x^{7} - 118016 x^{6} - 230312 x^{5} - 340336 x^{4} - 367600 x^{3} - 284600 x^{2} - 175000 x - 2500 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(277280164669720467865600000000=2^{28}\cdot 5^{8}\cdot 71^{4}\cdot 101^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.21$
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:  $2, 5, 71, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{20} a^{9} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{10} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{3}{10} a$, $\frac{1}{20} a^{10} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{10} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{3}{10} a^{2}$, $\frac{1}{20} a^{11} - \frac{1}{10} a^{8} + \frac{1}{10} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{10} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{20} a^{12} + \frac{1}{10} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{2} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{20} a^{13} + \frac{1}{10} a^{8} - \frac{2}{5} a^{6} - \frac{3}{10} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{100} a^{14} - \frac{1}{50} a^{13} + \frac{1}{50} a^{12} - \frac{1}{50} a^{11} + \frac{1}{100} a^{9} + \frac{6}{25} a^{8} - \frac{8}{25} a^{7} - \frac{3}{10} a^{6} - \frac{1}{50} a^{5} + \frac{6}{25} a^{4} - \frac{3}{25} a^{3} + \frac{1}{25} a^{2} - \frac{1}{2} a$, $\frac{1}{37731964575322717264056482406146500} a^{15} - \frac{49723855262629659727339336630627}{37731964575322717264056482406146500} a^{14} + \frac{5706699794194239574642067839837}{325275556683816528138417951777125} a^{13} - \frac{897764639621550751964436277657187}{37731964575322717264056482406146500} a^{12} + \frac{18115044011030013116499245418}{10780561307235062075444709258899} a^{11} - \frac{114310663656291779814239344885169}{37731964575322717264056482406146500} a^{10} - \frac{25279551950592884730070963076213}{18865982287661358632028241203073250} a^{9} + \frac{1105483407415230814153221758545787}{9432991143830679316014120601536625} a^{8} + \frac{15421367000997073241412751230201}{130110222673526611255367180710850} a^{7} + \frac{3870184695822643726640397671673247}{9432991143830679316014120601536625} a^{6} - \frac{976232260718392111667258401865129}{9432991143830679316014120601536625} a^{5} - \frac{993622010367479077153445403480533}{2695140326808765518861177314724750} a^{4} - \frac{95412403014613549358887302430996}{325275556683816528138417951777125} a^{3} + \frac{364062522878529997755510130921223}{754639291506454345281129648122930} a^{2} + \frac{6582470104291296293247408327901}{34301785977566106603687711278315} a + \frac{30006901538455467413805260070742}{75463929150645434528112964812293}$

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:  $9$
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:  \( 238052362.797 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1275:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1275
Character table for t16n1275 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.28400.1, 8.4.1303400960000.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$71$71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
101Data not computed