↑

Properties

Label 16.0.80639537988...0704.7
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 11^{8}\cdot 13^{8}$
Root discriminant $175.45$
Ramified primes $2, 11, 13$
Class number $21593520$ (GRH)
Class group $[3, 3, 2399280]$ (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![4300996454206, -817050280912, 886977486184, -143374847440, 80602553392, -11019895472, 4216870272, -480701584, 138892457, -12851080, 2947164, -210600, 39286, -1960, 300, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 300*x^14 - 1960*x^13 + 39286*x^12 - 210600*x^11 + 2947164*x^10 - 12851080*x^9 + 138892457*x^8 - 480701584*x^7 + 4216870272*x^6 - 11019895472*x^5 + 80602553392*x^4 - 143374847440*x^3 + 886977486184*x^2 - 817050280912*x + 4300996454206)
 
gp: K = bnfinit(x^16 - 8*x^15 + 300*x^14 - 1960*x^13 + 39286*x^12 - 210600*x^11 + 2947164*x^10 - 12851080*x^9 + 138892457*x^8 - 480701584*x^7 + 4216870272*x^6 - 11019895472*x^5 + 80602553392*x^4 - 143374847440*x^3 + 886977486184*x^2 - 817050280912*x + 4300996454206, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 300 x^{14} - 1960 x^{13} + 39286 x^{12} - 210600 x^{11} + 2947164 x^{10} - 12851080 x^{9} + 138892457 x^{8} - 480701584 x^{7} + 4216870272 x^{6} - 11019895472 x^{5} + 80602553392 x^{4} - 143374847440 x^{3} + 886977486184 x^{2} - 817050280912 x + 4300996454206 \)

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:  \(806395379885761262339860530624200704=2^{62}\cdot 11^{8}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $175.45$
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:  $2, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4576=2^{5}\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{4576}(1,·)$, $\chi_{4576}(3717,·)$, $\chi_{4576}(4289,·)$, $\chi_{4576}(3145,·)$, $\chi_{4576}(2573,·)$, $\chi_{4576}(2001,·)$, $\chi_{4576}(1429,·)$, $\chi_{4576}(857,·)$, $\chi_{4576}(285,·)$, $\chi_{4576}(4005,·)$, $\chi_{4576}(3433,·)$, $\chi_{4576}(2861,·)$, $\chi_{4576}(2289,·)$, $\chi_{4576}(1717,·)$, $\chi_{4576}(1145,·)$, $\chi_{4576}(573,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{47} a^{12} - \frac{6}{47} a^{11} + \frac{20}{47} a^{10} + \frac{2}{47} a^{9} - \frac{22}{47} a^{8} + \frac{10}{47} a^{7} - \frac{5}{47} a^{6} - \frac{2}{47} a^{5} + \frac{21}{47} a^{4} - \frac{17}{47} a^{3} - \frac{12}{47} a^{2} + \frac{10}{47} a - \frac{22}{47}$, $\frac{1}{799} a^{13} + \frac{2}{799} a^{12} + \frac{19}{799} a^{11} + \frac{256}{799} a^{10} - \frac{335}{799} a^{9} - \frac{354}{799} a^{8} + \frac{75}{799} a^{7} + \frac{52}{799} a^{6} - \frac{324}{799} a^{5} - \frac{84}{799} a^{4} + \frac{40}{799} a^{3} + \frac{337}{799} a^{2} + \frac{199}{799} a - \frac{129}{799}$, $\frac{1}{2156819527391553759735503} a^{14} - \frac{7}{2156819527391553759735503} a^{13} - \frac{12070779545929971849638}{2156819527391553759735503} a^{12} + \frac{4260275133857637123407}{126871736905385515278559} a^{11} - \frac{812313006598557834617951}{2156819527391553759735503} a^{10} - \frac{915966896816466798112342}{2156819527391553759735503} a^{9} + \frac{540909909992023190828595}{2156819527391553759735503} a^{8} - \frac{184805863821021352033939}{2156819527391553759735503} a^{7} + \frac{19366995753445691329915}{45889777178543697015649} a^{6} + \frac{605528098608476343552236}{2156819527391553759735503} a^{5} - \frac{668088631042272735066302}{2156819527391553759735503} a^{4} - \frac{534132927927555692211332}{2156819527391553759735503} a^{3} + \frac{1049876992331890506627664}{2156819527391553759735503} a^{2} - \frac{51610372868112980720909}{2156819527391553759735503} a + \frac{202561557988265433714118}{2156819527391553759735503}$, $\frac{1}{65773714791016023816104699630591} a^{15} + \frac{15247841}{65773714791016023816104699630591} a^{14} + \frac{210454571739635565792570922}{3869042046530354342123805860623} a^{13} - \frac{207308647970468732393317867717}{65773714791016023816104699630591} a^{12} + \frac{31047279610989557356690608576642}{65773714791016023816104699630591} a^{11} - \frac{12382206222970401993749052146120}{65773714791016023816104699630591} a^{10} - \frac{16134215170388372769213422283143}{65773714791016023816104699630591} a^{9} - \frac{15296893970183711765181941966797}{65773714791016023816104699630591} a^{8} + \frac{553778808004125550104591833797}{65773714791016023816104699630591} a^{7} - \frac{8093474495690253107261510827653}{65773714791016023816104699630591} a^{6} + \frac{6691574696031137046483272894486}{65773714791016023816104699630591} a^{5} + \frac{6720739108314702488706442962260}{65773714791016023816104699630591} a^{4} - \frac{12944638948548513605184350915126}{65773714791016023816104699630591} a^{3} - \frac{27234892459785132201140136054754}{65773714791016023816104699630591} a^{2} + \frac{17777058865459594379808423706109}{65773714791016023816104699630591} a + \frac{17838438497638117884915615142373}{65773714791016023816104699630591}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{2399280}$, which has order $21593520$ (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

\(\Q(\sqrt{-286}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-143}) \), \(\Q(\sqrt{2}, \sqrt{-143})\), \(\Q(\zeta_{16})^+\), 4.0.41879552.2, 8.0.1753896875720704.79, 8.0.897995200369000448.1, \(\Q(\zeta_{32})^+\)

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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]$
11Data not computed
13Data not computed