↑

Properties

Label 16.0.23043668220...1744.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 13^{8}\cdot 17^{14}$
Root discriminant $121.66$
Ramified primes $2, 13, 17$
Class number $1536000$ (GRH)
Class group $[4, 4, 20, 40, 120]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![361048092497, -24069872832, 91622805660, -5412835164, 10530368990, -542563352, 715763506, -31424078, 31478439, -1136632, 917938, -25718, 17357, -338, 195, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 195*x^14 - 338*x^13 + 17357*x^12 - 25718*x^11 + 917938*x^10 - 1136632*x^9 + 31478439*x^8 - 31424078*x^7 + 715763506*x^6 - 542563352*x^5 + 10530368990*x^4 - 5412835164*x^3 + 91622805660*x^2 - 24069872832*x + 361048092497)
 
gp: K = bnfinit(x^16 - 2*x^15 + 195*x^14 - 338*x^13 + 17357*x^12 - 25718*x^11 + 917938*x^10 - 1136632*x^9 + 31478439*x^8 - 31424078*x^7 + 715763506*x^6 - 542563352*x^5 + 10530368990*x^4 - 5412835164*x^3 + 91622805660*x^2 - 24069872832*x + 361048092497, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 195 x^{14} - 338 x^{13} + 17357 x^{12} - 25718 x^{11} + 917938 x^{10} - 1136632 x^{9} + 31478439 x^{8} - 31424078 x^{7} + 715763506 x^{6} - 542563352 x^{5} + 10530368990 x^{4} - 5412835164 x^{3} + 91622805660 x^{2} - 24069872832 x + 361048092497 \)

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:  \(2304366822037031859005514783391744=2^{24}\cdot 13^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $121.66$
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:  $2, 13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1768=2^{3}\cdot 13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{1768}(1,·)$, $\chi_{1768}(259,·)$, $\chi_{1768}(1665,·)$, $\chi_{1768}(1611,·)$, $\chi_{1768}(1041,·)$, $\chi_{1768}(467,·)$, $\chi_{1768}(729,·)$, $\chi_{1768}(987,·)$, $\chi_{1768}(417,·)$, $\chi_{1768}(155,·)$, $\chi_{1768}(937,·)$, $\chi_{1768}(625,·)$, $\chi_{1768}(883,·)$, $\chi_{1768}(1715,·)$, $\chi_{1768}(1249,·)$, $\chi_{1768}(1403,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{901530475602006767981577568356438702128606021417142043073} a^{15} - \frac{387871368256704415224329752793505411063592042848763485969}{901530475602006767981577568356438702128606021417142043073} a^{14} - \frac{448055741588232678194320089428774802796152699126809072451}{901530475602006767981577568356438702128606021417142043073} a^{13} + \frac{199138303632906226589393735326419698290701957104248767193}{901530475602006767981577568356438702128606021417142043073} a^{12} + \frac{442745031911451319906326730935479178147593578440826091721}{901530475602006767981577568356438702128606021417142043073} a^{11} + \frac{115199877919779059044015386690281926658285867771117419083}{901530475602006767981577568356438702128606021417142043073} a^{10} + \frac{47850390634860997189178734644459641166361439401751868764}{901530475602006767981577568356438702128606021417142043073} a^{9} + \frac{93673830313185343854790968939479429397048776865726520487}{901530475602006767981577568356438702128606021417142043073} a^{8} - \frac{256836042743004354850558436701224239716431339782687603530}{901530475602006767981577568356438702128606021417142043073} a^{7} - \frac{385557118738587688868288029070337916334755219140076345731}{901530475602006767981577568356438702128606021417142043073} a^{6} - \frac{255776999583558471303445220351797311092890354220739920614}{901530475602006767981577568356438702128606021417142043073} a^{5} + \frac{116021084303024126300866136434553495103035983745583971428}{901530475602006767981577568356438702128606021417142043073} a^{4} + \frac{374883478174531301935466096505151989243770333384380743009}{901530475602006767981577568356438702128606021417142043073} a^{3} - \frac{19217129302496381497660728882629224625392578705467875254}{901530475602006767981577568356438702128606021417142043073} a^{2} - \frac{358795519873290957914362769697606679408347629880762054291}{901530475602006767981577568356438702128606021417142043073} a + \frac{307864143123720428582942173939851605089623452334240482685}{901530475602006767981577568356438702128606021417142043073}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{20}\times C_{40}\times C_{120}$, which has order $1536000$ (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:  \( 3640.012213375973 \) (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

\(\Q(\sqrt{-442}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-26}) \), \(\Q(\sqrt{17}, \sqrt{-26})\), 4.4.4913.1, 4.0.53139008.6, 8.0.2823754171224064.89, 8.0.48003820910809088.30, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$