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Properties

Label 16.0.25576314038...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{8}\cdot 29^{6}$
Root discriminant $38.72$
Ramified primes $2, 3, 5, 29$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_2^2:D_4$ (as 16T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![529, 414, 11617, -13132, 40752, -59650, 62980, -30706, 6773, 590, 590, -494, 2, 80, -15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 15*x^14 + 80*x^13 + 2*x^12 - 494*x^11 + 590*x^10 + 590*x^9 + 6773*x^8 - 30706*x^7 + 62980*x^6 - 59650*x^5 + 40752*x^4 - 13132*x^3 + 11617*x^2 + 414*x + 529)
 
gp: K = bnfinit(x^16 - 2*x^15 - 15*x^14 + 80*x^13 + 2*x^12 - 494*x^11 + 590*x^10 + 590*x^9 + 6773*x^8 - 30706*x^7 + 62980*x^6 - 59650*x^5 + 40752*x^4 - 13132*x^3 + 11617*x^2 + 414*x + 529, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 15 x^{14} + 80 x^{13} + 2 x^{12} - 494 x^{11} + 590 x^{10} + 590 x^{9} + 6773 x^{8} - 30706 x^{7} + 62980 x^{6} - 59650 x^{5} + 40752 x^{4} - 13132 x^{3} + 11617 x^{2} + 414 x + 529 \)

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:  \(25576314038393241600000000=2^{24}\cdot 3^{8}\cdot 5^{8}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.72$
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:  $2, 3, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{29} a^{11} + \frac{10}{29} a^{10} - \frac{13}{29} a^{9} + \frac{1}{29} a^{8} - \frac{6}{29} a^{7} - \frac{4}{29} a^{6} - \frac{10}{29} a^{5} - \frac{6}{29} a^{4} - \frac{4}{29} a^{3} - \frac{1}{29} a^{2} - \frac{7}{29} a + \frac{10}{29}$, $\frac{1}{58} a^{12} - \frac{13}{29} a^{10} - \frac{7}{29} a^{9} + \frac{13}{58} a^{8} - \frac{1}{29} a^{7} + \frac{1}{58} a^{6} - \frac{11}{29} a^{5} + \frac{27}{58} a^{4} + \frac{5}{29} a^{3} - \frac{13}{29} a^{2} + \frac{11}{29} a - \frac{13}{58}$, $\frac{1}{58} a^{13} + \frac{7}{29} a^{10} + \frac{23}{58} a^{9} + \frac{12}{29} a^{8} + \frac{19}{58} a^{7} - \frac{5}{29} a^{6} - \frac{1}{58} a^{5} + \frac{14}{29} a^{4} - \frac{7}{29} a^{3} - \frac{2}{29} a^{2} - \frac{21}{58} a + \frac{14}{29}$, $\frac{1}{1682} a^{14} + \frac{5}{1682} a^{13} - \frac{7}{841} a^{12} - \frac{7}{841} a^{11} + \frac{61}{1682} a^{10} + \frac{293}{1682} a^{9} + \frac{45}{1682} a^{8} - \frac{763}{1682} a^{7} - \frac{185}{1682} a^{6} + \frac{263}{1682} a^{5} + \frac{277}{841} a^{4} + \frac{239}{841} a^{3} + \frac{351}{1682} a^{2} - \frac{131}{1682} a - \frac{124}{841}$, $\frac{1}{410944782862578631551061016314} a^{15} + \frac{92269415219434728556603029}{410944782862578631551061016314} a^{14} + \frac{1225180315915409979823720980}{205472391431289315775530508157} a^{13} - \frac{717253598765022766914749216}{205472391431289315775530508157} a^{12} - \frac{3075058389936118172772369457}{410944782862578631551061016314} a^{11} + \frac{113240021244685927464319095985}{410944782862578631551061016314} a^{10} - \frac{80384959206066991308888574523}{410944782862578631551061016314} a^{9} - \frac{5079638574778530329189355417}{14170509753882021777622793666} a^{8} + \frac{157204281913887856504350567797}{410944782862578631551061016314} a^{7} + \frac{152240194078873723369676177}{366587674275270857761874234} a^{6} - \frac{69046713200298059313543604156}{205472391431289315775530508157} a^{5} - \frac{98469297143119362584552549838}{205472391431289315775530508157} a^{4} - \frac{3093394176855093070282934311}{14170509753882021777622793666} a^{3} - \frac{1020388658146703268366275409}{21628672782240980607950579806} a^{2} - \frac{34033740618813032362976277314}{205472391431289315775530508157} a + \frac{2937522416141299729456448201}{8933582236143013729370891659}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 181341.233613 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:D_4$ (as 16T34):

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 $C_2^2:D_4$
Character table for $C_2^2:D_4$

Intermediate fields

\(\Q(\sqrt{30}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{6}) \), 4.0.6525.1, 4.0.46400.1, 4.4.83520.1, 4.4.83520.2, \(\Q(\sqrt{5}, \sqrt{6})\), 8.8.174389760000.2, 8.0.174389760000.2, 8.0.174389760000.15

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

Sibling fields

Galois closure: data not computed
Degree 16 siblings: 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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$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$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$