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Properties

Label 16.0.15364319033...0000.7
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 5^{8}\cdot 31^{8}$
Root discriminant $182.67$
Ramified primes $2, 5, 31$
Class number $14745600$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 480, 480]$ (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![7910928927679, -1403153880664, 1515042967660, -229015759264, 127759270042, -16345577288, 6197750976, -661072120, 189148283, -16360264, 3716400, -247800, 45850, -2128, 324, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 324*x^14 - 2128*x^13 + 45850*x^12 - 247800*x^11 + 3716400*x^10 - 16360264*x^9 + 189148283*x^8 - 661072120*x^7 + 6197750976*x^6 - 16345577288*x^5 + 127759270042*x^4 - 229015759264*x^3 + 1515042967660*x^2 - 1403153880664*x + 7910928927679)
 
gp: K = bnfinit(x^16 - 8*x^15 + 324*x^14 - 2128*x^13 + 45850*x^12 - 247800*x^11 + 3716400*x^10 - 16360264*x^9 + 189148283*x^8 - 661072120*x^7 + 6197750976*x^6 - 16345577288*x^5 + 127759270042*x^4 - 229015759264*x^3 + 1515042967660*x^2 - 1403153880664*x + 7910928927679, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 324 x^{14} - 2128 x^{13} + 45850 x^{12} - 247800 x^{11} + 3716400 x^{10} - 16360264 x^{9} + 189148283 x^{8} - 661072120 x^{7} + 6197750976 x^{6} - 16345577288 x^{5} + 127759270042 x^{4} - 229015759264 x^{3} + 1515042967660 x^{2} - 1403153880664 x + 7910928927679 \)

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:  \(1536431903362769340925896294400000000=2^{62}\cdot 5^{8}\cdot 31^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $182.67$
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:  $2, 5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4960=2^{5}\cdot 5\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{4960}(1,·)$, $\chi_{4960}(1861,·)$, $\chi_{4960}(3721,·)$, $\chi_{4960}(1549,·)$, $\chi_{4960}(3409,·)$, $\chi_{4960}(1241,·)$, $\chi_{4960}(3101,·)$, $\chi_{4960}(929,·)$, $\chi_{4960}(2789,·)$, $\chi_{4960}(4649,·)$, $\chi_{4960}(621,·)$, $\chi_{4960}(2481,·)$, $\chi_{4960}(4341,·)$, $\chi_{4960}(2169,·)$, $\chi_{4960}(4029,·)$, $\chi_{4960}(309,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{47} a^{13} + \frac{17}{47} a^{12} + \frac{6}{47} a^{11} + \frac{15}{47} a^{10} + \frac{9}{47} a^{9} + \frac{10}{47} a^{8} - \frac{18}{47} a^{7} - \frac{2}{47} a^{6} - \frac{22}{47} a^{5} - \frac{1}{47} a^{4} + \frac{17}{47} a^{2} + \frac{10}{47} a - \frac{21}{47}$, $\frac{1}{260101322894689053199758673} a^{14} - \frac{7}{260101322894689053199758673} a^{13} - \frac{5770097498513485582620752}{260101322894689053199758673} a^{12} + \frac{34620584991080913495724603}{260101322894689053199758673} a^{11} - \frac{4247805696782034629879754}{260101322894689053199758673} a^{10} - \frac{36015011039642480694984918}{260101322894689053199758673} a^{9} - \frac{32931961848224063374224777}{260101322894689053199758673} a^{8} - \frac{51659620151425973479496484}{260101322894689053199758673} a^{7} - \frac{42891687346138878193218125}{260101322894689053199758673} a^{6} + \frac{115525971063114843545670755}{260101322894689053199758673} a^{5} - \frac{72508590729094282022478746}{260101322894689053199758673} a^{4} - \frac{62103523372344036307249921}{260101322894689053199758673} a^{3} - \frac{40578736810833723981731557}{260101322894689053199758673} a^{2} - \frac{61540844455885851975268991}{260101322894689053199758673} a - \frac{68316179031966754183221300}{260101322894689053199758673}$, $\frac{1}{3897158204336714757263273341812463} a^{15} + \frac{7491608}{3897158204336714757263273341812463} a^{14} - \frac{22152326306533775570404945543639}{3897158204336714757263273341812463} a^{13} + \frac{100762769136094685657843648860406}{3897158204336714757263273341812463} a^{12} + \frac{1888120848049751445670592267061040}{3897158204336714757263273341812463} a^{11} - \frac{830980800087641368925754991711476}{3897158204336714757263273341812463} a^{10} - \frac{854457639632689283556451938437527}{3897158204336714757263273341812463} a^{9} + \frac{36533997916470166342716126873751}{3897158204336714757263273341812463} a^{8} + \frac{1497410371635884947732794498606965}{3897158204336714757263273341812463} a^{7} - \frac{1736653613087467150935932563821204}{3897158204336714757263273341812463} a^{6} - \frac{1869314613553475883477597413755442}{3897158204336714757263273341812463} a^{5} - \frac{997353290594722717254666388032767}{3897158204336714757263273341812463} a^{4} - \frac{1216226769822458580018335333296397}{3897158204336714757263273341812463} a^{3} - \frac{1863342027099302419329812731616515}{3897158204336714757263273341812463} a^{2} + \frac{1599905372388102021885562591440026}{3897158204336714757263273341812463} a + \frac{1622075314792336824766013051930435}{3897158204336714757263273341812463}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{480}\times C_{480}$, which has order $14745600$ (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

\(\Q(\sqrt{-155}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-310}) \), \(\Q(\sqrt{2}, \sqrt{-155})\), \(\Q(\zeta_{16})^+\), 4.0.49203200.5, 8.0.2420954890240000.110, \(\Q(\zeta_{32})^+\), 8.0.1239528903802880000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
5Data not computed
$31$31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$