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Properties

Label 16.0.65437948737...3125.4
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 101^{7}$
Root discriminant $30.80$
Ramified primes $5, 101$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 16T1281

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4061, 5700, -4487, 640, 7323, -5905, 2431, 1975, -2160, 1260, -124, -140, 148, -60, 23, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 23*x^14 - 60*x^13 + 148*x^12 - 140*x^11 - 124*x^10 + 1260*x^9 - 2160*x^8 + 1975*x^7 + 2431*x^6 - 5905*x^5 + 7323*x^4 + 640*x^3 - 4487*x^2 + 5700*x + 4061)
 
gp: K = bnfinit(x^16 - 5*x^15 + 23*x^14 - 60*x^13 + 148*x^12 - 140*x^11 - 124*x^10 + 1260*x^9 - 2160*x^8 + 1975*x^7 + 2431*x^6 - 5905*x^5 + 7323*x^4 + 640*x^3 - 4487*x^2 + 5700*x + 4061, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 23 x^{14} - 60 x^{13} + 148 x^{12} - 140 x^{11} - 124 x^{10} + 1260 x^{9} - 2160 x^{8} + 1975 x^{7} + 2431 x^{6} - 5905 x^{5} + 7323 x^{4} + 640 x^{3} - 4487 x^{2} + 5700 x + 4061 \)

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:  \(654379487370001220703125=5^{14}\cdot 101^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.80$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{61} a^{13} - \frac{1}{61} a^{12} - \frac{27}{61} a^{11} - \frac{27}{61} a^{10} + \frac{28}{61} a^{9} - \frac{9}{61} a^{8} + \frac{13}{61} a^{7} - \frac{6}{61} a^{6} + \frac{23}{61} a^{5} - \frac{21}{61} a^{4} - \frac{13}{61} a^{3} + \frac{9}{61} a - \frac{10}{61}$, $\frac{1}{7442} a^{14} + \frac{9}{3721} a^{13} - \frac{1243}{3721} a^{12} + \frac{889}{3721} a^{11} - \frac{212}{3721} a^{10} + \frac{1207}{3721} a^{9} - \frac{79}{3721} a^{8} - \frac{1740}{3721} a^{7} + \frac{46}{3721} a^{6} + \frac{2917}{7442} a^{5} + \frac{770}{3721} a^{4} + \frac{944}{3721} a^{3} - \frac{357}{7442} a^{2} + \frac{3211}{7442} a - \frac{3423}{7442}$, $\frac{1}{50392324488317080258} a^{15} - \frac{321291165700321}{25196162244158540129} a^{14} - \frac{204464160435337145}{25196162244158540129} a^{13} - \frac{367951459233663402}{1326113802324133691} a^{12} - \frac{11798113508735928228}{25196162244158540129} a^{11} - \frac{431526046476339129}{25196162244158540129} a^{10} - \frac{32529476975600235}{2290560204014412739} a^{9} + \frac{4331138135293489578}{25196162244158540129} a^{8} - \frac{8702072773718720082}{25196162244158540129} a^{7} - \frac{572395681772275491}{2652227604648267382} a^{6} - \frac{154048114184098714}{25196162244158540129} a^{5} + \frac{4233096731566042789}{25196162244158540129} a^{4} - \frac{17613737819400402899}{50392324488317080258} a^{3} - \frac{19437208704996859149}{50392324488317080258} a^{2} - \frac{1854145324492163981}{4581120408028825478} a - \frac{546574498440826690}{25196162244158540129}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( \frac{44569710751021387}{50392324488317080258} a^{15} - \frac{131656711918119643}{25196162244158540129} a^{14} + \frac{545867941976646658}{25196162244158540129} a^{13} - \frac{84013903592922540}{1326113802324133691} a^{12} + \frac{3634186056863628411}{25196162244158540129} a^{11} - \frac{4676804072823601797}{25196162244158540129} a^{10} - \frac{263090171171962626}{2290560204014412739} a^{9} + \frac{25752337185017199132}{25196162244158540129} a^{8} - \frac{52985074036064823297}{25196162244158540129} a^{7} + \frac{4243545259081657347}{2652227604648267382} a^{6} + \frac{27667551915643283129}{25196162244158540129} a^{5} - \frac{113167945587267231270}{25196162244158540129} a^{4} + \frac{242952679043774988087}{50392324488317080258} a^{3} - \frac{63688884853552565401}{50392324488317080258} a^{2} - \frac{9291403389072836475}{4581120408028825478} a + \frac{18081370936949947965}{25196162244158540129} \) (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:  \( 208333.790853 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1281:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1281
Character table for t16n1281 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.159390625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$101$101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.3.4$x^{4} + 808$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.0.1$x^{4} - x + 12$$1$$4$$0$$C_4$$[\ ]^{4}$