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Properties

Label 16.4.45453408428...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{8}\cdot 5^{8}\cdot 11^{6}\cdot 37^{6}$
Root discriminant $30.10$
Ramified primes $2, 5, 11, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1191

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-93649, 22975, -33080, 11562, 1031, -1104, 5639, -3492, 2773, -1814, 1110, -647, 286, -113, 33, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 33*x^14 - 113*x^13 + 286*x^12 - 647*x^11 + 1110*x^10 - 1814*x^9 + 2773*x^8 - 3492*x^7 + 5639*x^6 - 1104*x^5 + 1031*x^4 + 11562*x^3 - 33080*x^2 + 22975*x - 93649)
 
gp: K = bnfinit(x^16 - 7*x^15 + 33*x^14 - 113*x^13 + 286*x^12 - 647*x^11 + 1110*x^10 - 1814*x^9 + 2773*x^8 - 3492*x^7 + 5639*x^6 - 1104*x^5 + 1031*x^4 + 11562*x^3 - 33080*x^2 + 22975*x - 93649, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 33 x^{14} - 113 x^{13} + 286 x^{12} - 647 x^{11} + 1110 x^{10} - 1814 x^{9} + 2773 x^{8} - 3492 x^{7} + 5639 x^{6} - 1104 x^{5} + 1031 x^{4} + 11562 x^{3} - 33080 x^{2} + 22975 x - 93649 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(454534084285444900000000=2^{8}\cdot 5^{8}\cdot 11^{6}\cdot 37^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.10$
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:  $2, 5, 11, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{2} a^{12} + \frac{1}{4} a^{9} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{16404359272507730826410120120475496} a^{15} + \frac{151958126046785024428336900990675}{2050544909063466353301265015059437} a^{14} + \frac{2155476348225032454759186077739233}{16404359272507730826410120120475496} a^{13} + \frac{2822192369992044355090468272101451}{8202179636253865413205060060237748} a^{12} - \frac{878056993324414772609564293831801}{2050544909063466353301265015059437} a^{11} - \frac{4217061100384609896435632363016079}{16404359272507730826410120120475496} a^{10} - \frac{708184843135062332894593138313211}{16404359272507730826410120120475496} a^{9} - \frac{6863565235488936308663618760641371}{16404359272507730826410120120475496} a^{8} + \frac{511477855226750381920773179702121}{2050544909063466353301265015059437} a^{7} + \frac{791676873597968249774239957017513}{4101089818126932706602530030118874} a^{6} - \frac{1707315140490695598815827305208717}{16404359272507730826410120120475496} a^{5} - \frac{537157717762720449803725341683843}{16404359272507730826410120120475496} a^{4} - \frac{1581703490871139807423376736931315}{8202179636253865413205060060237748} a^{3} + \frac{21733698091177921462671648503847}{2050544909063466353301265015059437} a^{2} - \frac{620783971864553825737735693974861}{2050544909063466353301265015059437} a + \frac{21925762617972206238880504207903}{16404359272507730826410120120475496}$

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:  $9$
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:  \( 252894.9424 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1191:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1191 are not computed
Character table for t16n1191 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 8.4.103530625.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
5Data not computed
$11$11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$37$37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.8.6.2$x^{8} + 333 x^{4} + 34225$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$