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Properties

Label 16.0.35687701699...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{15}\cdot 61^{9}$
Root discriminant $45.66$
Ramified primes $5, 61$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group 16T1192

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![75581, -317102, 704900, -1067545, 1147520, -819991, 309122, 23670, -81165, 27060, 3952, -4234, 485, 205, -45, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 45*x^14 + 205*x^13 + 485*x^12 - 4234*x^11 + 3952*x^10 + 27060*x^9 - 81165*x^8 + 23670*x^7 + 309122*x^6 - 819991*x^5 + 1147520*x^4 - 1067545*x^3 + 704900*x^2 - 317102*x + 75581)
 
gp: K = bnfinit(x^16 - 3*x^15 - 45*x^14 + 205*x^13 + 485*x^12 - 4234*x^11 + 3952*x^10 + 27060*x^9 - 81165*x^8 + 23670*x^7 + 309122*x^6 - 819991*x^5 + 1147520*x^4 - 1067545*x^3 + 704900*x^2 - 317102*x + 75581, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 45 x^{14} + 205 x^{13} + 485 x^{12} - 4234 x^{11} + 3952 x^{10} + 27060 x^{9} - 81165 x^{8} + 23670 x^{7} + 309122 x^{6} - 819991 x^{5} + 1147520 x^{4} - 1067545 x^{3} + 704900 x^{2} - 317102 x + 75581 \)

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:  \(356877016993229400634765625=5^{15}\cdot 61^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.66$
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:  $5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{1350448249928246629516842003143618} a^{15} - \frac{63591320723241051791520914583463}{1350448249928246629516842003143618} a^{14} - \frac{113192010324192382923925089123533}{1350448249928246629516842003143618} a^{13} - \frac{16175841617989758640393353362045}{675224124964123314758421001571809} a^{12} - \frac{254770855839043482925513742263986}{675224124964123314758421001571809} a^{11} + \frac{102930895329641142621692519350469}{675224124964123314758421001571809} a^{10} + \frac{251947051390684601165947681597379}{675224124964123314758421001571809} a^{9} + \frac{117498730530709730537541603401494}{675224124964123314758421001571809} a^{8} - \frac{422655056298370107851462398812761}{1350448249928246629516842003143618} a^{7} - \frac{140293174667551708286520626887558}{675224124964123314758421001571809} a^{6} + \frac{183880039832254615548341982611549}{675224124964123314758421001571809} a^{5} - \frac{239698565301028209947272186895805}{675224124964123314758421001571809} a^{4} + \frac{318392234075930529776230402432777}{675224124964123314758421001571809} a^{3} + \frac{272435348816172445293328114187541}{675224124964123314758421001571809} a^{2} + \frac{368168316893196528014360847342907}{1350448249928246629516842003143618} a - \frac{220072481675033585404410398092073}{675224124964123314758421001571809}$

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

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{10292306656452088317028331703}{675224124964123314758421001571809} a^{15} + \frac{91460796055445523077885334761}{675224124964123314758421001571809} a^{14} - \frac{532113080401908416303797888399}{675224124964123314758421001571809} a^{13} - \frac{3609816020886711508633656140694}{675224124964123314758421001571809} a^{12} + \frac{16195922061069743494156237653028}{675224124964123314758421001571809} a^{11} + \frac{45058360738799750890103411003329}{675224124964123314758421001571809} a^{10} - \frac{267824652356020230713526531455129}{675224124964123314758421001571809} a^{9} - \frac{14607230631577281399737167919136}{675224124964123314758421001571809} a^{8} + \frac{1919815716019992549392540355415987}{675224124964123314758421001571809} a^{7} - \frac{3038632058580416582660393375408032}{675224124964123314758421001571809} a^{6} - \frac{3195676136371535028799930168833798}{675224124964123314758421001571809} a^{5} + \frac{16590641528661228723680838916625002}{675224124964123314758421001571809} a^{4} - \frac{25723163140533543749130923482224794}{675224124964123314758421001571809} a^{3} + \frac{23751275087679619224679287503500166}{675224124964123314758421001571809} a^{2} - \frac{15202805160587009997390772804243493}{675224124964123314758421001571809} a + \frac{6172814585800023473376244299745257}{675224124964123314758421001571809} \) (order $10$)
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:  \( 1650674.25582 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1192:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 88 conjugacy class representatives for t16n1192 are not computed
Character table for t16n1192 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.17732890625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.4.3.4$x^{4} + 488$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.3$x^{4} + 122$$4$$1$$3$$C_4$$[\ ]_{4}$