↑

Properties

Label 16.16.3543087250...0304.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{52}\cdot 3^{12}\cdot 23^{6}$
Root discriminant $70.28$
Ramified primes $2, 3, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4^2:C_2$ (as 16T27)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-31079, -621736, -1252012, 634320, 2949200, 1276792, -1122272, -696072, 193408, 135984, -20876, -12448, 1532, 528, -64, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 64*x^14 + 528*x^13 + 1532*x^12 - 12448*x^11 - 20876*x^10 + 135984*x^9 + 193408*x^8 - 696072*x^7 - 1122272*x^6 + 1276792*x^5 + 2949200*x^4 + 634320*x^3 - 1252012*x^2 - 621736*x - 31079)
 
gp: K = bnfinit(x^16 - 8*x^15 - 64*x^14 + 528*x^13 + 1532*x^12 - 12448*x^11 - 20876*x^10 + 135984*x^9 + 193408*x^8 - 696072*x^7 - 1122272*x^6 + 1276792*x^5 + 2949200*x^4 + 634320*x^3 - 1252012*x^2 - 621736*x - 31079, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 64 x^{14} + 528 x^{13} + 1532 x^{12} - 12448 x^{11} - 20876 x^{10} + 135984 x^{9} + 193408 x^{8} - 696072 x^{7} - 1122272 x^{6} + 1276792 x^{5} + 2949200 x^{4} + 634320 x^{3} - 1252012 x^{2} - 621736 x - 31079 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(354308725098774913512640610304=2^{52}\cdot 3^{12}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.28$
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:  $2, 3, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{69} a^{14} + \frac{5}{69} a^{13} - \frac{1}{69} a^{12} - \frac{1}{69} a^{11} + \frac{1}{23} a^{10} - \frac{10}{69} a^{9} + \frac{10}{69} a^{8} - \frac{26}{69} a^{7} + \frac{11}{69} a^{6} + \frac{34}{69} a^{5} - \frac{4}{69} a^{4} - \frac{4}{23} a^{3} - \frac{8}{69} a^{2} - \frac{8}{69} a - \frac{26}{69}$, $\frac{1}{357845970033614235208133134371017589} a^{15} - \frac{798527430004240926765483328269770}{119281990011204745069377711457005863} a^{14} + \frac{48517236382746322375447780625552995}{357845970033614235208133134371017589} a^{13} + \frac{54151053225499532721668481125562364}{357845970033614235208133134371017589} a^{12} - \frac{8499042097942644055068039854837134}{119281990011204745069377711457005863} a^{11} - \frac{43233189088139314329115576899106865}{357845970033614235208133134371017589} a^{10} - \frac{7125177462789200155520393856882541}{119281990011204745069377711457005863} a^{9} + \frac{45091754349575055540595305439333862}{357845970033614235208133134371017589} a^{8} + \frac{55884259294478395564837659621034249}{357845970033614235208133134371017589} a^{7} + \frac{122843496920758609456176394407630469}{357845970033614235208133134371017589} a^{6} + \frac{176195055614748810215158027498487182}{357845970033614235208133134371017589} a^{5} + \frac{49509024839606869240502313430610617}{119281990011204745069377711457005863} a^{4} + \frac{623458255729216153183816576232461}{5186173478748032394320770063348081} a^{3} + \frac{43452957518712365361381335375871598}{357845970033614235208133134371017589} a^{2} - \frac{30354700698676250369018819451528730}{357845970033614235208133134371017589} a + \frac{24589197937537074653539602289715875}{119281990011204745069377711457005863}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 5742846751.42 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2:C_2$ (as 16T27):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4^2:C_2$
Character table for $C_4^2:C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.6469982355456.1, 8.8.6469982355456.2, 8.8.44930433024.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
3Data not computed
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$