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Properties

Label 16.0.47991168245...000.49
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 5^{12}\cdot 17^{8}$
Root discriminant $95.52$
Ramified primes $2, 3, 5, 17$
Class number $261120$ (GRH)
Class group $[2, 4, 4, 8160]$ (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![2376274876, -995805576, 1190242220, -412056200, 265837154, -77057656, 34806462, -8448140, 2927421, -586696, 161722, -25852, 5704, -672, 116, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 116*x^14 - 672*x^13 + 5704*x^12 - 25852*x^11 + 161722*x^10 - 586696*x^9 + 2927421*x^8 - 8448140*x^7 + 34806462*x^6 - 77057656*x^5 + 265837154*x^4 - 412056200*x^3 + 1190242220*x^2 - 995805576*x + 2376274876)
 
gp: K = bnfinit(x^16 - 8*x^15 + 116*x^14 - 672*x^13 + 5704*x^12 - 25852*x^11 + 161722*x^10 - 586696*x^9 + 2927421*x^8 - 8448140*x^7 + 34806462*x^6 - 77057656*x^5 + 265837154*x^4 - 412056200*x^3 + 1190242220*x^2 - 995805576*x + 2376274876, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 116 x^{14} - 672 x^{13} + 5704 x^{12} - 25852 x^{11} + 161722 x^{10} - 586696 x^{9} + 2927421 x^{8} - 8448140 x^{7} + 34806462 x^{6} - 77057656 x^{5} + 265837154 x^{4} - 412056200 x^{3} + 1190242220 x^{2} - 995805576 x + 2376274876 \)

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:  \(47991168245852798976000000000000=2^{32}\cdot 3^{8}\cdot 5^{12}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.52$
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:  $2, 3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2040=2^{3}\cdot 3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{2040}(1,·)$, $\chi_{2040}(1223,·)$, $\chi_{2040}(203,·)$, $\chi_{2040}(1427,·)$, $\chi_{2040}(1429,·)$, $\chi_{2040}(407,·)$, $\chi_{2040}(409,·)$, $\chi_{2040}(1121,·)$, $\chi_{2040}(1123,·)$, $\chi_{2040}(101,·)$, $\chi_{2040}(103,·)$, $\chi_{2040}(1327,·)$, $\chi_{2040}(307,·)$, $\chi_{2040}(1021,·)$, $\chi_{2040}(1529,·)$, $\chi_{2040}(509,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{92937983149830856522} a^{14} - \frac{7}{92937983149830856522} a^{13} + \frac{13915088121536291779}{92937983149830856522} a^{12} + \frac{9447454420613105939}{92937983149830856522} a^{11} - \frac{8187700869392609181}{46468991574915428261} a^{10} + \frac{5382503514972214978}{46468991574915428261} a^{9} - \frac{8912673495892213645}{92937983149830856522} a^{8} - \frac{17955332719184416063}{92937983149830856522} a^{7} + \frac{33362536778499990933}{92937983149830856522} a^{6} - \frac{21602528399467391175}{92937983149830856522} a^{5} + \frac{3491339663542516519}{46468991574915428261} a^{4} - \frac{7473519574435885774}{46468991574915428261} a^{3} + \frac{9199872151164322404}{46468991574915428261} a^{2} - \frac{6539767238903242827}{46468991574915428261} a + \frac{6195710676670226937}{46468991574915428261}$, $\frac{1}{1100400683509284463988380042} a^{15} + \frac{5920073}{1100400683509284463988380042} a^{14} + \frac{7340039756255614676105995}{550200341754642231994190021} a^{13} + \frac{131029047945043377970760868}{550200341754642231994190021} a^{12} - \frac{268186760337921398409227359}{1100400683509284463988380042} a^{11} + \frac{259420931389389539836427805}{1100400683509284463988380042} a^{10} - \frac{249883117513106962002686147}{1100400683509284463988380042} a^{9} + \frac{137486349622838831091922438}{550200341754642231994190021} a^{8} + \frac{40515272713917423465167375}{550200341754642231994190021} a^{7} + \frac{156864022701477593756572885}{550200341754642231994190021} a^{6} + \frac{120879107187586291321600353}{1100400683509284463988380042} a^{5} - \frac{57810293186445439006958887}{550200341754642231994190021} a^{4} - \frac{120863988975695676799628061}{550200341754642231994190021} a^{3} - \frac{60092872453314268168492059}{550200341754642231994190021} a^{2} - \frac{29929953864923223272875431}{550200341754642231994190021} a + \frac{243952924526854346873107196}{550200341754642231994190021}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{8160}$, which has order $261120$ (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

\(\Q(\sqrt{-510}) \), \(\Q(\sqrt{-255}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-102}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{-255})\), \(\Q(\sqrt{5}, \sqrt{-102})\), \(\Q(\sqrt{10}, \sqrt{-51})\), \(\Q(\sqrt{5}, \sqrt{-51})\), \(\Q(\sqrt{10}, \sqrt{-102})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-51})\), 4.4.8000.1, 4.0.5202000.2, 4.0.20808000.2, \(\Q(\zeta_{20})^+\), 8.0.17318914560000.86, 8.0.6927565824000000.86, 8.0.6927565824000000.105, 8.0.432972864000000.105, 8.0.27060804000000.38, \(\Q(\zeta_{40})^+\), 8.0.6927565824000000.151

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 R 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 ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$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}$