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Properties

Label 16.0.18875793996...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 3^{8}\cdot 5^{12}\cdot 7^{8}\cdot 41^{8}$
Root discriminant $138.75$
Ramified primes $2, 3, 5, 7, 41$
Class number $192$ (GRH)
Class group $[2, 2, 2, 2, 12]$ (GRH)
Galois group 16T813

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![862485775, -744073275, 474767975, -186256050, 66855015, -16653470, 3262910, -172835, -49309, 21415, -5461, 1005, -29, -25, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 9*x^14 - 25*x^13 - 29*x^12 + 1005*x^11 - 5461*x^10 + 21415*x^9 - 49309*x^8 - 172835*x^7 + 3262910*x^6 - 16653470*x^5 + 66855015*x^4 - 186256050*x^3 + 474767975*x^2 - 744073275*x + 862485775)
 
gp: K = bnfinit(x^16 + 9*x^14 - 25*x^13 - 29*x^12 + 1005*x^11 - 5461*x^10 + 21415*x^9 - 49309*x^8 - 172835*x^7 + 3262910*x^6 - 16653470*x^5 + 66855015*x^4 - 186256050*x^3 + 474767975*x^2 - 744073275*x + 862485775, 1)
 

Normalized defining polynomial

\( x^{16} + 9 x^{14} - 25 x^{13} - 29 x^{12} + 1005 x^{11} - 5461 x^{10} + 21415 x^{9} - 49309 x^{8} - 172835 x^{7} + 3262910 x^{6} - 16653470 x^{5} + 66855015 x^{4} - 186256050 x^{3} + 474767975 x^{2} - 744073275 x + 862485775 \)

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:  \(18875793996821517790730062500000000=2^{8}\cdot 3^{8}\cdot 5^{12}\cdot 7^{8}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $138.75$
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:  $2, 3, 5, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{10} + \frac{1}{5} a^{8} - \frac{1}{5} a^{6} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{11} + \frac{1}{5} a^{9} - \frac{1}{5} a^{7} + \frac{1}{5} a^{5}$, $\frac{1}{4305} a^{14} + \frac{307}{4305} a^{13} + \frac{103}{1435} a^{12} - \frac{437}{4305} a^{11} + \frac{278}{615} a^{10} + \frac{682}{4305} a^{9} + \frac{1564}{4305} a^{8} - \frac{206}{615} a^{7} + \frac{113}{615} a^{6} - \frac{1048}{4305} a^{5} + \frac{22}{861} a^{4} - \frac{184}{861} a^{3} + \frac{394}{861} a^{2} - \frac{169}{861} a - \frac{85}{861}$, $\frac{1}{7501023660047428563743665117986047787263035176546448725} a^{15} - \frac{33306297194394837249228335589198702638805950657500}{300040946401897142549746604719441911490521407061857949} a^{14} - \frac{548393760215457248559205186792204548398593932094580666}{7501023660047428563743665117986047787263035176546448725} a^{13} - \frac{18052142117406681242178524133819771536612058467440354}{1500204732009485712748733023597209557452607035309289745} a^{12} - \frac{290912816568990337945680346548170793920306355228839811}{833447073338603173749296124220671976362559464060716525} a^{11} - \frac{10519193368099960199913661973017537975490951484811469}{100013648800632380849915534906480637163507135687285983} a^{10} - \frac{382994325113204846864666739794742269225819708791842163}{1071574808578204080534809302569435398180433596649492675} a^{9} - \frac{94515772087193271996466149561565084449828381981262931}{1500204732009485712748733023597209557452607035309289745} a^{8} + \frac{131846185330434090470041803844544840469940771070572}{10823987965436404853886962652216519173539733299489825} a^{7} - \frac{527273288942950020050340237524057526627857848414391458}{1500204732009485712748733023597209557452607035309289745} a^{6} + \frac{28252320310827340326667688359115096180189879990173247}{115400364000729670211441001815169965957892848869945365} a^{5} - \frac{111683225000998897501400659342366056716650328514840566}{500068244003161904249577674532403185817535678436429915} a^{4} - \frac{15759127991829146001031902848911170114954773189441902}{45460749454832900386325243139309380528866879857857265} a^{3} + \frac{17984531292702913542805312283881105366675968072929647}{100013648800632380849915534906480637163507135687285983} a^{2} - \frac{927985744993851427796059289449810585274232309809167}{42862992343128163221392372102777415927217343865979707} a - \frac{116853791265783888945578244816692182203788167013081378}{300040946401897142549746604719441911490521407061857949}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{12}$, which has order $192$ (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:  \( -1 \) (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:  \( 374826594.216 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T813:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 38 conjugacy class representatives for t16n813
Character table for t16n813 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{105}) \), 4.0.46125.1, 4.0.251125.1, \(\Q(\sqrt{5}, \sqrt{21})\), 8.0.5108165015625.3

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 R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$2$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$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}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
41Data not computed