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Properties

Label 16.0.23712129003...569.11
Degree $16$
Signature $[0, 8]$
Discriminant $13^{6}\cdot 53^{12}$
Root discriminant $51.40$
Ramified primes $13, 53$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $D_4:C_4$ (as 16T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5077, -27292, 78283, -147037, 206961, -223525, 196253, -138670, 82520, -39950, 16658, -5534, 1620, -350, 70, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 70*x^14 - 350*x^13 + 1620*x^12 - 5534*x^11 + 16658*x^10 - 39950*x^9 + 82520*x^8 - 138670*x^7 + 196253*x^6 - 223525*x^5 + 206961*x^4 - 147037*x^3 + 78283*x^2 - 27292*x + 5077)
 
gp: K = bnfinit(x^16 - 8*x^15 + 70*x^14 - 350*x^13 + 1620*x^12 - 5534*x^11 + 16658*x^10 - 39950*x^9 + 82520*x^8 - 138670*x^7 + 196253*x^6 - 223525*x^5 + 206961*x^4 - 147037*x^3 + 78283*x^2 - 27292*x + 5077, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 70 x^{14} - 350 x^{13} + 1620 x^{12} - 5534 x^{11} + 16658 x^{10} - 39950 x^{9} + 82520 x^{8} - 138670 x^{7} + 196253 x^{6} - 223525 x^{5} + 206961 x^{4} - 147037 x^{3} + 78283 x^{2} - 27292 x + 5077 \)

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:  \(2371212900396504113756570569=13^{6}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.40$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3}$, $\frac{1}{21} a^{13} - \frac{1}{7} a^{12} - \frac{2}{21} a^{10} - \frac{1}{7} a^{9} + \frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{1}{3} a^{4} - \frac{5}{21} a^{3} - \frac{1}{3} a^{2} + \frac{8}{21} a - \frac{3}{7}$, $\frac{1}{112003563} a^{14} - \frac{1}{16000509} a^{13} - \frac{12420347}{112003563} a^{12} - \frac{146869}{112003563} a^{11} + \frac{9179273}{112003563} a^{10} + \frac{5891323}{37334521} a^{9} + \frac{2052424}{16000509} a^{8} - \frac{15788875}{112003563} a^{7} + \frac{9186}{59927} a^{6} + \frac{14088856}{37334521} a^{5} + \frac{24570884}{112003563} a^{4} - \frac{8274628}{37334521} a^{3} - \frac{1714306}{112003563} a^{2} + \frac{4337033}{112003563} a + \frac{25125703}{112003563}$, $\frac{1}{116819716209} a^{15} + \frac{514}{116819716209} a^{14} - \frac{348434683}{116819716209} a^{13} + \frac{15257543566}{116819716209} a^{12} - \frac{6107973415}{38939905403} a^{11} - \frac{17115288625}{116819716209} a^{10} + \frac{896321214}{38939905403} a^{9} + \frac{4866764988}{38939905403} a^{8} - \frac{3056671343}{116819716209} a^{7} + \frac{33590574221}{116819716209} a^{6} - \frac{44970012383}{116819716209} a^{5} - \frac{32547501814}{116819716209} a^{4} + \frac{31866467330}{116819716209} a^{3} + \frac{25021341181}{116819716209} a^{2} - \frac{44346096833}{116819716209} a - \frac{6995481035}{116819716209}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  \( 1019104.93477 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_4$ (as 16T26):

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 $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{53}) \), 4.4.36517.1, 4.0.148877.1, 4.0.1935401.1, 8.8.48695101400413.1, 8.0.17335386757.1, 8.0.3745777030801.2

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 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$53$53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$