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Properties

Label 16.0.13174026235...000.12
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 5^{12}\cdot 61^{8}$
Root discriminant $208.92$
Ramified primes $2, 5, 61$
Class number $24336000$ (GRH)
Class group $[2, 390, 31200]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2592668017521, 187061526120, 453261043160, 21278260880, 36125621744, 988227240, 1746492120, 23128160, 56878606, 230360, 1289160, -1200, 20064, -40, 200, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 200*x^14 - 40*x^13 + 20064*x^12 - 1200*x^11 + 1289160*x^10 + 230360*x^9 + 56878606*x^8 + 23128160*x^7 + 1746492120*x^6 + 988227240*x^5 + 36125621744*x^4 + 21278260880*x^3 + 453261043160*x^2 + 187061526120*x + 2592668017521)
 
gp: K = bnfinit(x^16 + 200*x^14 - 40*x^13 + 20064*x^12 - 1200*x^11 + 1289160*x^10 + 230360*x^9 + 56878606*x^8 + 23128160*x^7 + 1746492120*x^6 + 988227240*x^5 + 36125621744*x^4 + 21278260880*x^3 + 453261043160*x^2 + 187061526120*x + 2592668017521, 1)
 

Normalized defining polynomial

\( x^{16} + 200 x^{14} - 40 x^{13} + 20064 x^{12} - 1200 x^{11} + 1289160 x^{10} + 230360 x^{9} + 56878606 x^{8} + 23128160 x^{7} + 1746492120 x^{6} + 988227240 x^{5} + 36125621744 x^{4} + 21278260880 x^{3} + 453261043160 x^{2} + 187061526120 x + 2592668017521 \)

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:  \(13174026235637722110754816000000000000=2^{48}\cdot 5^{12}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $208.92$
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:  $2, 5, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4880=2^{4}\cdot 5\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{4880}(1,·)$, $\chi_{4880}(2563,·)$, $\chi_{4880}(2439,·)$, $\chi_{4880}(2441,·)$, $\chi_{4880}(2317,·)$, $\chi_{4880}(4879,·)$, $\chi_{4880}(4757,·)$, $\chi_{4880}(3293,·)$, $\chi_{4880}(1951,·)$, $\chi_{4880}(4027,·)$, $\chi_{4880}(4391,·)$, $\chi_{4880}(489,·)$, $\chi_{4880}(2929,·)$, $\chi_{4880}(1587,·)$, $\chi_{4880}(123,·)$, $\chi_{4880}(853,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{6} a - \frac{1}{4}$, $\frac{1}{24} a^{9} - \frac{1}{24} a^{8} + \frac{1}{12} a^{7} + \frac{1}{12} a^{6} + \frac{1}{12} a^{5} + \frac{1}{12} a^{4} + \frac{1}{12} a^{3} + \frac{1}{12} a^{2} - \frac{11}{24} a - \frac{3}{8}$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{7}{24} a^{2} - \frac{1}{6} a + \frac{3}{8}$, $\frac{1}{24} a^{11} - \frac{1}{24} a^{8} + \frac{1}{12} a^{7} + \frac{1}{12} a^{6} + \frac{1}{12} a^{5} + \frac{1}{12} a^{4} - \frac{11}{24} a^{3} + \frac{1}{12} a^{2} + \frac{1}{12} a - \frac{3}{8}$, $\frac{1}{936} a^{12} - \frac{1}{936} a^{11} + \frac{11}{936} a^{10} + \frac{1}{312} a^{9} + \frac{1}{36} a^{8} - \frac{37}{234} a^{7} - \frac{7}{234} a^{6} - \frac{20}{117} a^{5} - \frac{101}{936} a^{4} - \frac{15}{104} a^{3} + \frac{103}{312} a^{2} + \frac{17}{936} a + \frac{49}{156}$, $\frac{1}{1872} a^{13} - \frac{1}{1872} a^{12} - \frac{7}{468} a^{11} - \frac{1}{52} a^{10} - \frac{1}{144} a^{9} - \frac{31}{1872} a^{8} + \frac{55}{234} a^{7} - \frac{10}{117} a^{6} - \frac{101}{1872} a^{5} + \frac{37}{208} a^{4} + \frac{29}{156} a^{3} + \frac{131}{468} a^{2} - \frac{15}{208} a - \frac{1}{16}$, $\frac{1}{418512906864} a^{14} - \frac{49701059}{209256453432} a^{13} + \frac{41367529}{139504302288} a^{12} + \frac{551338639}{52314113358} a^{11} + \frac{7935360697}{418512906864} a^{10} + \frac{1882330793}{104628226716} a^{9} - \frac{8965126595}{418512906864} a^{8} + \frac{16122477883}{104628226716} a^{7} + \frac{19234261949}{139504302288} a^{6} - \frac{30662391875}{209256453432} a^{5} - \frac{48593208779}{418512906864} a^{4} - \frac{45592327411}{104628226716} a^{3} + \frac{15297431575}{46501434096} a^{2} - \frac{21637773983}{52314113358} a + \frac{939966751}{1964849328}$, $\frac{1}{116944639444271868980053560869768707973116656} a^{15} + \frac{13502499385306058298728758519337}{58472319722135934490026780434884353986558328} a^{14} + \frac{23936841708519675748641615402975618290635}{116944639444271868980053560869768707973116656} a^{13} + \frac{6499003647953036751752814729915287735203}{58472319722135934490026780434884353986558328} a^{12} + \frac{629990971390277669202606420169367001924353}{116944639444271868980053560869768707973116656} a^{11} + \frac{46271446573575895096110748984465514684359}{29236159861067967245013390217442176993279164} a^{10} + \frac{1601355231677738635580630764659358728204329}{116944639444271868980053560869768707973116656} a^{9} - \frac{47220084908077466749784482586187952134687}{2165641471190219795926917793884605703205864} a^{8} + \frac{7367054006363555093517916220566749753015623}{116944639444271868980053560869768707973116656} a^{7} + \frac{5381704692662005178949081127040349883580533}{58472319722135934490026780434884353986558328} a^{6} - \frac{13702386781680032052934848951371038554152059}{116944639444271868980053560869768707973116656} a^{5} + \frac{8407004562092190845809044267545841716914883}{58472319722135934490026780434884353986558328} a^{4} - \frac{14196917010889839715580930259160321059170153}{116944639444271868980053560869768707973116656} a^{3} - \frac{2772501703151962087651230141884402831445641}{14618079930533983622506695108721088496639582} a^{2} + \frac{20335019993771459503183124553730531073582815}{116944639444271868980053560869768707973116656} a + \frac{123764984740883909522770581337509820071127}{274517932967774340328764227393823258152856}$

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

Class group and class number

$C_{2}\times C_{390}\times C_{31200}$, which has order $24336000$ (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:  \( 85299.42553126559 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-61}) \), \(\Q(\sqrt{-122}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-610}) \), \(\Q(\sqrt{-305}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}, \sqrt{-61})\), \(\Q(\sqrt{10}, \sqrt{-61})\), \(\Q(\sqrt{5}, \sqrt{-61})\), \(\Q(\sqrt{10}, \sqrt{-122})\), \(\Q(\sqrt{5}, \sqrt{-122})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-305})\), 4.0.952576000.2, 4.4.256000.2, 4.4.256000.1, 4.0.952576000.1, 8.0.567125647360000.45, 8.0.3629604143104000000.8, 8.0.3629604143104000000.7, 8.0.3629604143104000000.3, 8.0.3629604143104000000.4, 8.0.907401035776000000.5, 8.8.65536000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$61$61.8.4.1$x^{8} + 14884 x^{4} - 226981 x^{2} + 55383364$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
61.8.4.1$x^{8} + 14884 x^{4} - 226981 x^{2} + 55383364$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$