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Properties

Label 16.16.1184608536...5625.1
Degree $16$
Signature $[16, 0]$
Discriminant $5^{12}\cdot 13^{8}\cdot 29^{6}$
Root discriminant $42.62$
Ramified primes $5, 13, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2^2\times C_4):C_2$ (as 16T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11600, -29000, 25800, 91600, 1845, -96075, -28530, 45250, 20071, -10250, -5823, 1045, 789, -35, -47, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 47*x^14 - 35*x^13 + 789*x^12 + 1045*x^11 - 5823*x^10 - 10250*x^9 + 20071*x^8 + 45250*x^7 - 28530*x^6 - 96075*x^5 + 1845*x^4 + 91600*x^3 + 25800*x^2 - 29000*x - 11600)
 
gp: K = bnfinit(x^16 - 47*x^14 - 35*x^13 + 789*x^12 + 1045*x^11 - 5823*x^10 - 10250*x^9 + 20071*x^8 + 45250*x^7 - 28530*x^6 - 96075*x^5 + 1845*x^4 + 91600*x^3 + 25800*x^2 - 29000*x - 11600, 1)
 

Normalized defining polynomial

\( x^{16} - 47 x^{14} - 35 x^{13} + 789 x^{12} + 1045 x^{11} - 5823 x^{10} - 10250 x^{9} + 20071 x^{8} + 45250 x^{7} - 28530 x^{6} - 96075 x^{5} + 1845 x^{4} + 91600 x^{3} + 25800 x^{2} - 29000 x - 11600 \)

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:  \(118460853639390732666015625=5^{12}\cdot 13^{8}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.62$
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:  $5, 13, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{10} a^{12} - \frac{1}{5} a^{10} + \frac{2}{5} a^{8} - \frac{1}{2} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{13} - \frac{1}{5} a^{11} - \frac{1}{10} a^{9} + \frac{1}{5} a^{7} - \frac{1}{2} a^{6} + \frac{1}{10} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{20} a^{14} - \frac{1}{20} a^{12} - \frac{1}{4} a^{11} - \frac{3}{20} a^{10} - \frac{1}{4} a^{9} - \frac{9}{20} a^{8} + \frac{3}{20} a^{6} - \frac{1}{5} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{1688188760446360} a^{15} - \frac{9813842928107}{844094380223180} a^{14} - \frac{4403865185047}{337637752089272} a^{13} - \frac{1282444234849}{337637752089272} a^{12} - \frac{48333062948129}{337637752089272} a^{11} + \frac{33620767636719}{337637752089272} a^{10} + \frac{83827619970335}{337637752089272} a^{9} - \frac{3109435138174}{42204719011159} a^{8} + \frac{2269372393377}{4755461297032} a^{7} - \frac{5576204433355}{84409438022318} a^{6} - \frac{342318217765599}{844094380223180} a^{5} - \frac{177832098714683}{1688188760446360} a^{4} + \frac{96126068565339}{337637752089272} a^{3} - \frac{41467651977169}{168818876044636} a^{2} + \frac{12751495603183}{42204719011159} a - \frac{2633381696757}{42204719011159}$

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:  \( 71853302.9688 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times C_4):C_2$ (as 16T37):

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 $(C_2^2\times C_4):C_2$
Character table for $(C_2^2\times C_4):C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{5}, \sqrt{13})\), 4.4.725.1, 4.4.122525.1, 8.8.15012375625.1, 8.8.375309390625.1, 8.8.375309390625.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$