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Properties

Label 16.0.10981017777...8944.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 47^{8}$
Root discriminant $100.59$
Ramified primes $2, 47$
Class number $289280$ (GRH)
Class group $[8, 8, 4520]$ (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![1365814462, -595699024, 718504936, -261201616, 170908336, -52180400, 23985408, -6139600, 2169449, -459208, 129180, -21864, 4918, -616, 108, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 108*x^14 - 616*x^13 + 4918*x^12 - 21864*x^11 + 129180*x^10 - 459208*x^9 + 2169449*x^8 - 6139600*x^7 + 23985408*x^6 - 52180400*x^5 + 170908336*x^4 - 261201616*x^3 + 718504936*x^2 - 595699024*x + 1365814462)
 
gp: K = bnfinit(x^16 - 8*x^15 + 108*x^14 - 616*x^13 + 4918*x^12 - 21864*x^11 + 129180*x^10 - 459208*x^9 + 2169449*x^8 - 6139600*x^7 + 23985408*x^6 - 52180400*x^5 + 170908336*x^4 - 261201616*x^3 + 718504936*x^2 - 595699024*x + 1365814462, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 108 x^{14} - 616 x^{13} + 4918 x^{12} - 21864 x^{11} + 129180 x^{10} - 459208 x^{9} + 2169449 x^{8} - 6139600 x^{7} + 23985408 x^{6} - 52180400 x^{5} + 170908336 x^{4} - 261201616 x^{3} + 718504936 x^{2} - 595699024 x + 1365814462 \)

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:  \(109810177778809754855611190738944=2^{62}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $100.59$
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:  $2, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1504=2^{5}\cdot 47\)
Dirichlet character group:    $\lbrace$$\chi_{1504}(1,·)$, $\chi_{1504}(1221,·)$, $\chi_{1504}(1409,·)$, $\chi_{1504}(1033,·)$, $\chi_{1504}(845,·)$, $\chi_{1504}(657,·)$, $\chi_{1504}(469,·)$, $\chi_{1504}(281,·)$, $\chi_{1504}(93,·)$, $\chi_{1504}(1317,·)$, $\chi_{1504}(1129,·)$, $\chi_{1504}(941,·)$, $\chi_{1504}(753,·)$, $\chi_{1504}(565,·)$, $\chi_{1504}(377,·)$, $\chi_{1504}(189,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{298899191120051914369} a^{14} - \frac{7}{298899191120051914369} a^{13} + \frac{56279077350892773498}{298899191120051914369} a^{12} - \frac{38775272985304726528}{298899191120051914369} a^{11} - \frac{24768429471381685692}{298899191120051914369} a^{10} - \frac{68699700664560088210}{298899191120051914369} a^{9} + \frac{48527802397041457420}{298899191120051914369} a^{8} + \frac{90458182680894622427}{298899191120051914369} a^{7} - \frac{68872497857907545557}{298899191120051914369} a^{6} + \frac{87782952461801462076}{298899191120051914369} a^{5} - \frac{52280449931772938155}{298899191120051914369} a^{4} + \frac{137375521024890636867}{298899191120051914369} a^{3} + \frac{42197612724209288743}{298899191120051914369} a^{2} + \frac{89674393391248657486}{298899191120051914369} a - \frac{104589981486423370601}{298899191120051914369}$, $\frac{1}{2677400563727936464881149153} a^{15} + \frac{4478761}{2677400563727936464881149153} a^{14} - \frac{809679294469488798467245277}{2677400563727936464881149153} a^{13} - \frac{1178525036419921971549085222}{2677400563727936464881149153} a^{12} - \frac{963696987925757407593578749}{2677400563727936464881149153} a^{11} + \frac{559533141062873375676343769}{2677400563727936464881149153} a^{10} + \frac{1284711011031806756389726828}{2677400563727936464881149153} a^{9} - \frac{106060479057227475663604585}{2677400563727936464881149153} a^{8} - \frac{1106854540397301901745020}{2229309378624426698485553} a^{7} - \frac{1185776318023177176398370205}{2677400563727936464881149153} a^{6} - \frac{342055468013031057027568585}{2677400563727936464881149153} a^{5} - \frac{264069286642094029996978121}{2677400563727936464881149153} a^{4} - \frac{1052244807221557191095468217}{2677400563727936464881149153} a^{3} + \frac{169277119876122794546043763}{2677400563727936464881149153} a^{2} + \frac{66138475458039090730347453}{2677400563727936464881149153} a + \frac{768755520319341934868204258}{2677400563727936464881149153}$

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

Class group and class number

$C_{8}\times C_{8}\times C_{4520}$, which has order $289280$ (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:  \( 15753.94986242651 \) (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

\(\Q(\sqrt{-47}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-94}) \), \(\Q(\sqrt{2}, \sqrt{-47})\), \(\Q(\zeta_{16})^+\), 4.0.4524032.5, 8.0.20466865537024.21, \(\Q(\zeta_{32})^+\), 8.0.10479035154956288.53

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.8.31.6$x^{8} + 16 x^{7} + 28 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
2.8.31.6$x^{8} + 16 x^{7} + 28 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
$47$47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$