↑

Properties

Label 16.0.11684193984...000.24
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 5^{12}\cdot 19^{8}$
Root discriminant $100.98$
Ramified primes $2, 3, 5, 19$
Class number $163840$ (GRH)
Class group $[2, 2, 2, 4, 4, 16, 80]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![79747180, -27037000, 31551500, -9495240, 5955986, -1570952, 729070, -185100, 60301, -13560, 6826, -2140, 296, 0, 20, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 20*x^14 + 296*x^12 - 2140*x^11 + 6826*x^10 - 13560*x^9 + 60301*x^8 - 185100*x^7 + 729070*x^6 - 1570952*x^5 + 5955986*x^4 - 9495240*x^3 + 31551500*x^2 - 27037000*x + 79747180)
 
gp: K = bnfinit(x^16 - 8*x^15 + 20*x^14 + 296*x^12 - 2140*x^11 + 6826*x^10 - 13560*x^9 + 60301*x^8 - 185100*x^7 + 729070*x^6 - 1570952*x^5 + 5955986*x^4 - 9495240*x^3 + 31551500*x^2 - 27037000*x + 79747180, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 20 x^{14} + 296 x^{12} - 2140 x^{11} + 6826 x^{10} - 13560 x^{9} + 60301 x^{8} - 185100 x^{7} + 729070 x^{6} - 1570952 x^{5} + 5955986 x^{4} - 9495240 x^{3} + 31551500 x^{2} - 27037000 x + 79747180 \)

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:  \(116841939847873560576000000000000=2^{32}\cdot 3^{8}\cdot 5^{12}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $100.98$
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:  $2, 3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2280=2^{3}\cdot 3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{2280}(1,·)$, $\chi_{2280}(2051,·)$, $\chi_{2280}(77,·)$, $\chi_{2280}(721,·)$, $\chi_{2280}(533,·)$, $\chi_{2280}(343,·)$, $\chi_{2280}(1369,·)$, $\chi_{2280}(797,·)$, $\chi_{2280}(607,·)$, $\chi_{2280}(419,·)$, $\chi_{2280}(1253,·)$, $\chi_{2280}(1063,·)$, $\chi_{2280}(2089,·)$, $\chi_{2280}(1139,·)$, $\chi_{2280}(1331,·)$, $\chi_{2280}(2167,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{18} a^{8} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{1}{9} a^{5} + \frac{1}{18} a^{4} - \frac{1}{9} a^{3} + \frac{4}{9} a^{2} - \frac{4}{9} a - \frac{4}{9}$, $\frac{1}{18} a^{9} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{6} a^{5} + \frac{1}{9} a^{4} + \frac{1}{3} a^{2} + \frac{1}{9} a + \frac{2}{9}$, $\frac{1}{18} a^{10} + \frac{1}{9} a^{7} + \frac{1}{18} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{3} - \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{18} a^{11} - \frac{1}{6} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{2}{9} a^{3} + \frac{4}{9} a^{2} - \frac{1}{9}$, $\frac{1}{54} a^{12} + \frac{1}{54} a^{9} - \frac{2}{27} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{5}{27} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{5}{27}$, $\frac{1}{3186} a^{13} + \frac{23}{3186} a^{12} + \frac{4}{531} a^{11} + \frac{14}{1593} a^{10} - \frac{49}{3186} a^{9} + \frac{2}{177} a^{8} + \frac{223}{1593} a^{7} + \frac{131}{1593} a^{6} - \frac{56}{531} a^{5} + \frac{125}{3186} a^{4} + \frac{239}{1593} a^{3} + \frac{239}{531} a^{2} - \frac{382}{1593} a + \frac{163}{1593}$, $\frac{1}{1235054911551800526} a^{14} - \frac{7}{1235054911551800526} a^{13} + \frac{528501108810898}{205842485258633421} a^{12} - \frac{19026039917192237}{1235054911551800526} a^{11} + \frac{14828825883706981}{617527455775900263} a^{10} + \frac{8705702356841843}{411684970517266842} a^{9} - \frac{11409367606142665}{1235054911551800526} a^{8} + \frac{91333359429570007}{1235054911551800526} a^{7} - \frac{57879626723211367}{411684970517266842} a^{6} - \frac{48649642242585728}{617527455775900263} a^{5} + \frac{29670415998332809}{1235054911551800526} a^{4} - \frac{31390258961947592}{68614161752877807} a^{3} + \frac{265962742340370188}{617527455775900263} a^{2} + \frac{1309518710963975}{10466567047049157} a - \frac{18024427282348525}{205842485258633421}$, $\frac{1}{45868704360122319735114} a^{15} + \frac{9281}{22934352180061159867557} a^{14} - \frac{350852716758161429}{4169882214556574521374} a^{13} + \frac{83969977288591519841}{45868704360122319735114} a^{12} - \frac{48337497298570131230}{2084941107278287260687} a^{11} + \frac{358054980548844616211}{45868704360122319735114} a^{10} - \frac{344444251761917537953}{15289568120040773245038} a^{9} - \frac{439401928617729225679}{45868704360122319735114} a^{8} + \frac{201693953333753441297}{4169882214556574521374} a^{7} + \frac{5999566137248856377923}{45868704360122319735114} a^{6} + \frac{661307087926950303130}{22934352180061159867557} a^{5} - \frac{3700577026162419593296}{22934352180061159867557} a^{4} - \frac{972417638034294998873}{7644784060020386622519} a^{3} - \frac{473315390469685439209}{22934352180061159867557} a^{2} + \frac{5458509672430439888305}{22934352180061159867557} a + \frac{3048021790260040891106}{22934352180061159867557}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{16}\times C_{80}$, which has order $163840$ (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:  \( 58630.77118534252 \) (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

\(\Q(\sqrt{-95}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-570}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-114}) \), \(\Q(\sqrt{6}, \sqrt{-95})\), \(\Q(\sqrt{5}, \sqrt{-19})\), \(\Q(\sqrt{30}, \sqrt{-95})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{-19})\), \(\Q(\sqrt{5}, \sqrt{-114})\), \(\Q(\sqrt{-19}, \sqrt{30})\), \(\Q(\zeta_{20})^+\), 4.0.722000.3, 4.4.72000.1, 4.0.25992000.5, 8.0.27023362560000.101, 8.0.521284000000.1, 8.0.675584064000000.148, 8.8.82944000000.2, 8.0.10809345024000000.191, 8.0.10809345024000000.246, 8.0.10809345024000000.162

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.16.1$x^{8} + 6 x^{6} + 2 x^{4} + 4 x^{2} + 8 x + 12$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.1$x^{8} + 6 x^{6} + 2 x^{4} + 4 x^{2} + 8 x + 12$$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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$