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Properties

Label 16.0.28273563576...000.19
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 5^{12}\cdot 37^{12}$
Root discriminant $337.45$
Ramified primes $2, 5, 37$
Class number $118340000$ (GRH)
Class group $[2, 10, 10, 10, 59170]$ (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![122963703210000, 0, 27325267380000, 0, 2334546117000, 0, 99381186000, 0, 2305396000, 0, 29844200, 0, 210530, 0, 740, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 740*x^14 + 210530*x^12 + 29844200*x^10 + 2305396000*x^8 + 99381186000*x^6 + 2334546117000*x^4 + 27325267380000*x^2 + 122963703210000)
 
gp: K = bnfinit(x^16 + 740*x^14 + 210530*x^12 + 29844200*x^10 + 2305396000*x^8 + 99381186000*x^6 + 2334546117000*x^4 + 27325267380000*x^2 + 122963703210000, 1)
 

Normalized defining polynomial

\( x^{16} + 740 x^{14} + 210530 x^{12} + 29844200 x^{10} + 2305396000 x^{8} + 99381186000 x^{6} + 2334546117000 x^{4} + 27325267380000 x^{2} + 122963703210000 \)

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:  \(28273563576220552539381170176000000000000=2^{44}\cdot 5^{12}\cdot 37^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $337.45$
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:  $2, 5, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2960=2^{4}\cdot 5\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{2960}(1,·)$, $\chi_{2960}(43,·)$, $\chi_{2960}(961,·)$, $\chi_{2960}(2441,·)$, $\chi_{2960}(1227,·)$, $\chi_{2960}(1363,·)$, $\chi_{2960}(2707,·)$, $\chi_{2960}(1849,·)$, $\chi_{2960}(2843,·)$, $\chi_{2960}(1067,·)$, $\chi_{2960}(2369,·)$, $\chi_{2960}(1523,·)$, $\chi_{2960}(369,·)$, $\chi_{2960}(2547,·)$, $\chi_{2960}(1481,·)$, $\chi_{2960}(889,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{370} a^{4}$, $\frac{1}{370} a^{5}$, $\frac{1}{370} a^{6}$, $\frac{1}{1110} a^{7} + \frac{1}{1110} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{410700} a^{8} + \frac{1}{1110} a^{6} + \frac{1}{1110} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{410700} a^{9} - \frac{1}{3} a$, $\frac{1}{1232100} a^{10} - \frac{1}{1232100} a^{8} - \frac{1}{3330} a^{6} - \frac{1}{3330} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{3696300} a^{11} + \frac{1}{1848150} a^{9} + \frac{1}{4995} a^{7} + \frac{11}{9990} a^{5} - \frac{8}{27} a^{3}$, $\frac{1}{2297620080000} a^{12} - \frac{17}{110889000} a^{10} - \frac{7}{22177800} a^{8} + \frac{4393}{8391600} a^{6} + \frac{19}{59940} a^{4} + \frac{17}{90} a^{2} - \frac{271}{560}$, $\frac{1}{6892860240000} a^{13} - \frac{17}{332667000} a^{11} + \frac{47}{66533400} a^{9} - \frac{10727}{25174800} a^{7} + \frac{47}{35964} a^{5} - \frac{133}{270} a^{3} - \frac{271}{1680} a$, $\frac{1}{9367397066160000} a^{14} - \frac{47}{1170924633270000} a^{12} - \frac{65213}{452094453000} a^{10} + \frac{153317}{253172893680} a^{8} - \frac{9311797}{8553138300} a^{6} - \frac{7577}{6788205} a^{4} - \frac{541679}{2283120} a^{2} + \frac{11993}{31710}$, $\frac{1}{28102191198480000} a^{15} - \frac{47}{3512773899810000} a^{13} - \frac{65213}{1356283359000} a^{11} + \frac{153317}{759518681040} a^{9} - \frac{9311797}{25659414900} a^{7} - \frac{7577}{20364615} a^{5} + \frac{1741441}{6849360} a^{3} + \frac{43703}{95130} a$

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

Class group and class number

$C_{2}\times C_{10}\times C_{10}\times C_{10}\times C_{59170}$, which has order $118340000$ (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:  \( 1116781.2621991208 \) (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

\(\Q(\sqrt{74}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{370}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{185}) \), \(\Q(\sqrt{2}, \sqrt{37})\), \(\Q(\sqrt{5}, \sqrt{74})\), \(\Q(\sqrt{10}, \sqrt{74})\), \(\Q(\sqrt{10}, \sqrt{37})\), \(\Q(\sqrt{5}, \sqrt{37})\), \(\Q(\sqrt{2}, \sqrt{185})\), \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.12967168000.8, 4.0.12967168000.5, 4.0.12967168000.7, 4.0.12967168000.3, 8.8.4797852160000.1, 8.0.168147445940224000000.4, 8.0.168147445940224000000.5, 8.0.168147445940224000000.2, 8.0.168147445940224000000.6, 8.0.168147445940224000000.3, 8.0.168147445940224000000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.22.7$x^{8} + 2 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.8.22.7$x^{8} + 2 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$37$37.8.6.1$x^{8} - 1147 x^{4} + 855625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
37.8.6.1$x^{8} - 1147 x^{4} + 855625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$