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Properties

Label 16.8.15712755557...5489.3
Degree $16$
Signature $[8, 4]$
Discriminant $23^{10}\cdot 41^{14}$
Root discriminant $182.92$
Ramified primes $23, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T817

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3005776, 16252912, -7903064, -19052742, 14422779, -3038946, -3940881, 4697578, -1737261, 372432, -70511, -3666, 1885, -616, 108, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 108*x^14 - 616*x^13 + 1885*x^12 - 3666*x^11 - 70511*x^10 + 372432*x^9 - 1737261*x^8 + 4697578*x^7 - 3940881*x^6 - 3038946*x^5 + 14422779*x^4 - 19052742*x^3 - 7903064*x^2 + 16252912*x - 3005776)
 
gp: K = bnfinit(x^16 - 8*x^15 + 108*x^14 - 616*x^13 + 1885*x^12 - 3666*x^11 - 70511*x^10 + 372432*x^9 - 1737261*x^8 + 4697578*x^7 - 3940881*x^6 - 3038946*x^5 + 14422779*x^4 - 19052742*x^3 - 7903064*x^2 + 16252912*x - 3005776, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 108 x^{14} - 616 x^{13} + 1885 x^{12} - 3666 x^{11} - 70511 x^{10} + 372432 x^{9} - 1737261 x^{8} + 4697578 x^{7} - 3940881 x^{6} - 3038946 x^{5} + 14422779 x^{4} - 19052742 x^{3} - 7903064 x^{2} + 16252912 x - 3005776 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1571275555715210001755383712793895489=23^{10}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $182.92$
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:  $23, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{92} a^{12} - \frac{3}{46} a^{11} - \frac{9}{92} a^{10} + \frac{2}{23} a^{9} + \frac{1}{92} a^{8} + \frac{5}{23} a^{7} - \frac{3}{46} a^{6} + \frac{43}{92} a^{5} - \frac{25}{92} a^{4} + \frac{37}{92} a^{3} - \frac{17}{46} a^{2} - \frac{15}{46} a + \frac{8}{23}$, $\frac{1}{92} a^{13} + \frac{1}{92} a^{11} + \frac{3}{92} a^{9} + \frac{3}{92} a^{8} + \frac{11}{46} a^{7} - \frac{4}{23} a^{6} - \frac{5}{23} a^{5} - \frac{11}{23} a^{4} + \frac{27}{92} a^{3} - \frac{27}{92} a^{2} - \frac{5}{46} a + \frac{2}{23}$, $\frac{1}{335948386539112177386008} a^{14} - \frac{7}{335948386539112177386008} a^{13} + \frac{1729438789901027853343}{335948386539112177386008} a^{12} - \frac{10376632739406167119967}{335948386539112177386008} a^{11} - \frac{7665362417363007089967}{167974193269556088693004} a^{10} + \frac{1899282174315257069765}{167974193269556088693004} a^{9} + \frac{11352933589101465369315}{335948386539112177386008} a^{8} - \frac{14371887312100695940787}{335948386539112177386008} a^{7} - \frac{36750092244475762755069}{167974193269556088693004} a^{6} - \frac{76707874166306041329191}{167974193269556088693004} a^{5} - \frac{100326220756191381345483}{335948386539112177386008} a^{4} + \frac{122948254867623008045785}{335948386539112177386008} a^{3} - \frac{38688272198210301133839}{83987096634778044346502} a^{2} + \frac{5787113640307609812009}{41993548317389022173251} a - \frac{18696087885383145505198}{41993548317389022173251}$, $\frac{1}{4638439372945521833168612456} a^{15} + \frac{862}{579804921618190229146076557} a^{14} - \frac{6645070753444007777864169}{2319219686472760916584306228} a^{13} - \frac{1255269030720052998748117}{2319219686472760916584306228} a^{12} + \frac{276415559994050539442580523}{4638439372945521833168612456} a^{11} - \frac{237338632662612197412896643}{2319219686472760916584306228} a^{10} - \frac{70275634626968760909181241}{4638439372945521833168612456} a^{9} + \frac{30290412824952775894049535}{579804921618190229146076557} a^{8} + \frac{550777489653262007637162681}{4638439372945521833168612456} a^{7} + \frac{459651191274750944310600045}{2319219686472760916584306228} a^{6} - \frac{37730881845202181617983589}{201671277084587905789939672} a^{5} + \frac{141611022353598738150286040}{579804921618190229146076557} a^{4} - \frac{1246654443033810431925476199}{4638439372945521833168612456} a^{3} + \frac{1081094363030700937098017689}{2319219686472760916584306228} a^{2} - \frac{295184881280285262255718823}{1159609843236380458292153114} a + \frac{248050272443147894219007534}{579804921618190229146076557}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 726736040744 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T817:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n817
Character table for t16n817 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.54500230757132921.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$