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Properties

Label 16.0.73146118344...0000.6
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{12}\cdot 17^{8}$
Root discriminant $55.15$
Ramified primes $2, 5, 17$
Class number $1728$ (GRH)
Class group $[3, 24, 24]$ (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![4174025, 3039300, 4390350, 1962800, 1509355, 384060, 225310, 24320, 23426, 1920, 2992, 200, 194, -20, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 8*x^14 - 20*x^13 + 194*x^12 + 200*x^11 + 2992*x^10 + 1920*x^9 + 23426*x^8 + 24320*x^7 + 225310*x^6 + 384060*x^5 + 1509355*x^4 + 1962800*x^3 + 4390350*x^2 + 3039300*x + 4174025)
 
gp: K = bnfinit(x^16 + 8*x^14 - 20*x^13 + 194*x^12 + 200*x^11 + 2992*x^10 + 1920*x^9 + 23426*x^8 + 24320*x^7 + 225310*x^6 + 384060*x^5 + 1509355*x^4 + 1962800*x^3 + 4390350*x^2 + 3039300*x + 4174025, 1)
 

Normalized defining polynomial

\( x^{16} + 8 x^{14} - 20 x^{13} + 194 x^{12} + 200 x^{11} + 2992 x^{10} + 1920 x^{9} + 23426 x^{8} + 24320 x^{7} + 225310 x^{6} + 384060 x^{5} + 1509355 x^{4} + 1962800 x^{3} + 4390350 x^{2} + 3039300 x + 4174025 \)

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:  \(7314611834454016000000000000=2^{32}\cdot 5^{12}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.15$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(680=2^{3}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{680}(1,·)$, $\chi_{680}(69,·)$, $\chi_{680}(647,·)$, $\chi_{680}(271,·)$, $\chi_{680}(339,·)$, $\chi_{680}(341,·)$, $\chi_{680}(409,·)$, $\chi_{680}(33,·)$, $\chi_{680}(611,·)$, $\chi_{680}(679,·)$, $\chi_{680}(103,·)$, $\chi_{680}(237,·)$, $\chi_{680}(307,·)$, $\chi_{680}(373,·)$, $\chi_{680}(577,·)$, $\chi_{680}(443,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{15} a^{12} - \frac{7}{15} a^{10} - \frac{2}{5} a^{8} - \frac{1}{5} a^{6} - \frac{4}{15} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{15} a^{13} - \frac{7}{15} a^{11} - \frac{2}{5} a^{9} - \frac{1}{5} a^{7} - \frac{4}{15} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{615} a^{14} + \frac{2}{123} a^{13} + \frac{7}{615} a^{12} + \frac{13}{123} a^{11} + \frac{301}{615} a^{10} - \frac{5}{41} a^{9} + \frac{71}{205} a^{8} - \frac{4}{41} a^{7} - \frac{286}{615} a^{6} - \frac{31}{123} a^{5} + \frac{4}{615} a^{4} + \frac{14}{41} a^{3} + \frac{32}{123} a^{2} - \frac{14}{41} a + \frac{52}{123}$, $\frac{1}{8242396832460106396322761351286235} a^{15} + \frac{562488880820051912838088969003}{8242396832460106396322761351286235} a^{14} + \frac{189575431970967378133908079646}{86762071920632698908660645803013} a^{13} - \frac{146969223206344762749048646411021}{8242396832460106396322761351286235} a^{12} + \frac{666721859065994085954591123469583}{1648479366492021279264552270257247} a^{11} - \frac{1126832306430513441925294695144141}{2747465610820035465440920450428745} a^{10} + \frac{101966103637470358840504142629412}{549493122164007093088184090085749} a^{9} + \frac{16709787697795964339768979598308}{144603453201054498181101076338355} a^{8} + \frac{420978286035542178815990134975510}{1648479366492021279264552270257247} a^{7} - \frac{2364606825700835784099769256288882}{8242396832460106396322761351286235} a^{6} + \frac{3431966471173673000736532877437387}{8242396832460106396322761351286235} a^{5} - \frac{159877472906081475990728500177647}{549493122164007093088184090085749} a^{4} - \frac{543137629892857057278072145716448}{1648479366492021279264552270257247} a^{3} - \frac{48333697361214275260802811389673}{549493122164007093088184090085749} a^{2} + \frac{325226872043530014244321753157809}{1648479366492021279264552270257247} a + \frac{44169322816263344061308302322}{2761271970673402477830070804451}$

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

Class group and class number

$C_{3}\times C_{24}\times C_{24}$, which has order $1728$ (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:  \( 7114.135357253273 \) (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

\(\Q(\sqrt{-170}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-85}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{2}, \sqrt{-85})\), \(\Q(\sqrt{5}, \sqrt{-34})\), \(\Q(\sqrt{10}, \sqrt{-17})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-17})\), \(\Q(\sqrt{5}, \sqrt{-17})\), \(\Q(\sqrt{10}, \sqrt{-34})\), 4.0.2312000.1, \(\Q(\zeta_{20})^+\), 4.0.36125.1, 4.4.8000.1, 8.0.3421020160000.8, 8.0.85525504000000.22, 8.0.5345344000000.4, 8.0.5345344000000.5, \(\Q(\zeta_{40})^+\), 8.0.85525504000000.30, 8.0.334084000000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$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}$
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$