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Properties

Label 16.0.89432107247...0000.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{12}\cdot 31^{8}$
Root discriminant $74.47$
Ramified primes $2, 5, 31$
Class number $38400$ (GRH)
Class group $[2, 4, 20, 240]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![84589601, -47911596, 60832030, -28026260, 19732124, -7599516, 3826142, -1242420, 487381, -132356, 41682, -9212, 2324, -392, 76, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 76*x^14 - 392*x^13 + 2324*x^12 - 9212*x^11 + 41682*x^10 - 132356*x^9 + 487381*x^8 - 1242420*x^7 + 3826142*x^6 - 7599516*x^5 + 19732124*x^4 - 28026260*x^3 + 60832030*x^2 - 47911596*x + 84589601)
 
gp: K = bnfinit(x^16 - 8*x^15 + 76*x^14 - 392*x^13 + 2324*x^12 - 9212*x^11 + 41682*x^10 - 132356*x^9 + 487381*x^8 - 1242420*x^7 + 3826142*x^6 - 7599516*x^5 + 19732124*x^4 - 28026260*x^3 + 60832030*x^2 - 47911596*x + 84589601, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 76 x^{14} - 392 x^{13} + 2324 x^{12} - 9212 x^{11} + 41682 x^{10} - 132356 x^{9} + 487381 x^{8} - 1242420 x^{7} + 3826142 x^{6} - 7599516 x^{5} + 19732124 x^{4} - 28026260 x^{3} + 60832030 x^{2} - 47911596 x + 84589601 \)

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:  \(894321072475734016000000000000=2^{32}\cdot 5^{12}\cdot 31^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.47$
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:  \(1240=2^{3}\cdot 5\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{1240}(1,·)$, $\chi_{1240}(187,·)$, $\chi_{1240}(929,·)$, $\chi_{1240}(867,·)$, $\chi_{1240}(869,·)$, $\chi_{1240}(743,·)$, $\chi_{1240}(681,·)$, $\chi_{1240}(683,·)$, $\chi_{1240}(621,·)$, $\chi_{1240}(807,·)$, $\chi_{1240}(309,·)$, $\chi_{1240}(247,·)$, $\chi_{1240}(249,·)$, $\chi_{1240}(123,·)$, $\chi_{1240}(61,·)$, $\chi_{1240}(63,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{392857309738930922} a^{14} - \frac{7}{392857309738930922} a^{13} + \frac{13569631729425296}{196428654869465461} a^{12} + \frac{16796537058181000}{196428654869465461} a^{11} - \frac{73065837942294999}{392857309738930922} a^{10} + \frac{90130786123067405}{392857309738930922} a^{9} - \frac{9803115235779973}{196428654869465461} a^{8} - \frac{46417989786091180}{196428654869465461} a^{7} + \frac{705542805714553}{392857309738930922} a^{6} + \frac{194334728113769579}{392857309738930922} a^{5} + \frac{97536607024585843}{196428654869465461} a^{4} - \frac{16065664791893619}{196428654869465461} a^{3} - \frac{7143037141138854}{196428654869465461} a^{2} + \frac{41903076462048682}{196428654869465461} a - \frac{68767143337333582}{196428654869465461}$, $\frac{1}{861552187407174808113802} a^{15} + \frac{1096513}{861552187407174808113802} a^{14} - \frac{113464431773569969320705}{861552187407174808113802} a^{13} + \frac{60556712355177112586399}{430776093703587404056901} a^{12} + \frac{81339852040700368512849}{861552187407174808113802} a^{11} + \frac{70228822804699413544247}{861552187407174808113802} a^{10} + \frac{129949399641130465029579}{861552187407174808113802} a^{9} + \frac{55260294703901756618449}{430776093703587404056901} a^{8} + \frac{307852309256845711795477}{861552187407174808113802} a^{7} + \frac{260258053501162804060749}{861552187407174808113802} a^{6} - \frac{351932753872289166509663}{861552187407174808113802} a^{5} - \frac{38284005282486687485056}{430776093703587404056901} a^{4} - \frac{140126524798871506684902}{430776093703587404056901} a^{3} - \frac{78727078441102849392596}{430776093703587404056901} a^{2} + \frac{59769194934912220616323}{430776093703587404056901} a - \frac{22992668605362767168776}{430776093703587404056901}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{20}\times C_{240}$, which has order $38400$ (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:  \( 7114.135357253273 \) (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

\(\Q(\sqrt{-62}) \), \(\Q(\sqrt{-31}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-310}) \), \(\Q(\sqrt{-155}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{-31})\), \(\Q(\sqrt{5}, \sqrt{-62})\), \(\Q(\sqrt{10}, \sqrt{-62})\), \(\Q(\sqrt{5}, \sqrt{-31})\), \(\Q(\sqrt{10}, \sqrt{-31})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-155})\), 4.4.8000.1, 4.0.1922000.2, 4.0.7688000.5, \(\Q(\zeta_{20})^+\), 8.0.2364213760000.27, 8.0.945685504000000.40, 8.0.945685504000000.49, 8.0.59105344000000.25, 8.0.3694084000000.1, \(\Q(\zeta_{40})^+\), 8.0.945685504000000.55

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$31$31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$