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Properties

Label 16.0.26970872071...0625.4
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 17^{14}$
Root discriminant $69.09$
Ramified primes $3, 5, 17$
Class number $35040$ (GRH)
Class group $[2, 17520]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65536, -212992, 335616, -293824, 196928, -29264, -12352, 22124, -3164, -6331, 4971, -1999, 255, 47, -15, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 15*x^14 + 47*x^13 + 255*x^12 - 1999*x^11 + 4971*x^10 - 6331*x^9 - 3164*x^8 + 22124*x^7 - 12352*x^6 - 29264*x^5 + 196928*x^4 - 293824*x^3 + 335616*x^2 - 212992*x + 65536)
 
gp: K = bnfinit(x^16 - x^15 - 15*x^14 + 47*x^13 + 255*x^12 - 1999*x^11 + 4971*x^10 - 6331*x^9 - 3164*x^8 + 22124*x^7 - 12352*x^6 - 29264*x^5 + 196928*x^4 - 293824*x^3 + 335616*x^2 - 212992*x + 65536, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 15 x^{14} + 47 x^{13} + 255 x^{12} - 1999 x^{11} + 4971 x^{10} - 6331 x^{9} - 3164 x^{8} + 22124 x^{7} - 12352 x^{6} - 29264 x^{5} + 196928 x^{4} - 293824 x^{3} + 335616 x^{2} - 212992 x + 65536 \)

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:  \(269708720716852904093994140625=3^{8}\cdot 5^{12}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.09$
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:  $3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(255=3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{255}(64,·)$, $\chi_{255}(1,·)$, $\chi_{255}(2,·)$, $\chi_{255}(4,·)$, $\chi_{255}(8,·)$, $\chi_{255}(128,·)$, $\chi_{255}(16,·)$, $\chi_{255}(223,·)$, $\chi_{255}(32,·)$, $\chi_{255}(127,·)$, $\chi_{255}(239,·)$, $\chi_{255}(247,·)$, $\chi_{255}(251,·)$, $\chi_{255}(253,·)$, $\chi_{255}(254,·)$, $\chi_{255}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{16} a^{7} + \frac{3}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{128} a^{8} - \frac{1}{32} a^{7} + \frac{1}{64} a^{6} - \frac{1}{16} a^{5} + \frac{1}{128} a^{4} - \frac{1}{32} a^{3} - \frac{1}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{128} a^{9} + \frac{1}{64} a^{7} + \frac{1}{128} a^{5} - \frac{1}{32} a^{3}$, $\frac{1}{1024} a^{10} - \frac{3}{1024} a^{9} - \frac{1}{256} a^{8} - \frac{15}{512} a^{7} + \frac{21}{1024} a^{6} - \frac{19}{1024} a^{5} + \frac{11}{512} a^{4} - \frac{43}{256} a^{3} - \frac{21}{128} a^{2} - \frac{5}{32} a - \frac{1}{2}$, $\frac{1}{2048} a^{11} - \frac{1}{2048} a^{10} + \frac{3}{1024} a^{9} - \frac{3}{1024} a^{8} - \frac{7}{2048} a^{7} - \frac{41}{2048} a^{6} - \frac{1}{16} a^{5} + \frac{1}{64} a^{4} + \frac{1}{32} a^{3} + \frac{25}{128} a^{2} + \frac{11}{32} a - \frac{1}{2}$, $\frac{1}{2048} a^{12} - \frac{1}{2048} a^{10} + \frac{1}{1024} a^{9} - \frac{5}{2048} a^{8} + \frac{9}{512} a^{7} + \frac{57}{2048} a^{6} + \frac{1}{1024} a^{5} + \frac{19}{512} a^{4} - \frac{21}{256} a^{3} - \frac{3}{16} a^{2} + \frac{3}{16} a$, $\frac{1}{4096} a^{13} - \frac{1}{4096} a^{11} - \frac{1}{2048} a^{10} + \frac{7}{4096} a^{9} - \frac{3}{1024} a^{8} + \frac{49}{4096} a^{7} - \frac{41}{2048} a^{6} - \frac{29}{512} a^{5} + \frac{1}{64} a^{4} + \frac{51}{256} a^{3} + \frac{25}{128} a^{2} + \frac{5}{32} a - \frac{1}{2}$, $\frac{1}{1612693504} a^{14} + \frac{82403}{1612693504} a^{13} + \frac{160433}{1612693504} a^{12} - \frac{108489}{1612693504} a^{11} + \frac{106367}{1612693504} a^{10} + \frac{3205841}{1612693504} a^{9} - \frac{6060869}{1612693504} a^{8} - \frac{23516035}{1612693504} a^{7} - \frac{11028959}{403173376} a^{6} - \frac{2963847}{201586688} a^{5} - \frac{1159779}{100793344} a^{4} + \frac{16689133}{100793344} a^{3} - \frac{4696869}{25198336} a^{2} + \frac{625623}{1574896} a + \frac{15704}{98431}$, $\frac{1}{103212384256} a^{15} + \frac{21}{103212384256} a^{14} - \frac{2924097}{103212384256} a^{13} - \frac{8247143}{103212384256} a^{12} + \frac{19798565}{103212384256} a^{11} - \frac{34596513}{103212384256} a^{10} + \frac{119309717}{103212384256} a^{9} - \frac{174798829}{103212384256} a^{8} - \frac{1084955485}{51606192128} a^{7} - \frac{47282041}{6450774016} a^{6} + \frac{163332395}{3225387008} a^{5} - \frac{208610497}{6450774016} a^{4} + \frac{239072703}{3225387008} a^{3} + \frac{6424291}{806346752} a^{2} + \frac{5810793}{12599168} a - \frac{73363}{3149792}$

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

Class group and class number

$C_{2}\times C_{17520}$, which has order $35040$ (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:  \( 101041909.11303878 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

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 $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{-255}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-15}, \sqrt{17})\), 4.4.122825.1, 4.0.44217.1, 8.0.1221964430625.1, 8.8.519334883015625.2, 8.0.6411541765625.2

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{16}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
3Data not computed
5Data not computed
$17$17.8.7.2$x^{8} - 153$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.2$x^{8} - 153$$8$$1$$7$$C_8$$[\ ]_{8}$