↑

Properties

Label 16.12.1274602568...9401.2
Degree $16$
Signature $[12, 2]$
Discriminant $31^{10}\cdot 41^{15}$
Root discriminant $278.03$
Ramified primes $31, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1223

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![26094979, 18504933, -286455995, -244482346, 202450036, 118654269, -64725915, -6540911, 5199015, -472409, -35277, 30525, -5859, -100, 63, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 63*x^14 - 100*x^13 - 5859*x^12 + 30525*x^11 - 35277*x^10 - 472409*x^9 + 5199015*x^8 - 6540911*x^7 - 64725915*x^6 + 118654269*x^5 + 202450036*x^4 - 244482346*x^3 - 286455995*x^2 + 18504933*x + 26094979)
 
gp: K = bnfinit(x^16 - 6*x^15 + 63*x^14 - 100*x^13 - 5859*x^12 + 30525*x^11 - 35277*x^10 - 472409*x^9 + 5199015*x^8 - 6540911*x^7 - 64725915*x^6 + 118654269*x^5 + 202450036*x^4 - 244482346*x^3 - 286455995*x^2 + 18504933*x + 26094979, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 63 x^{14} - 100 x^{13} - 5859 x^{12} + 30525 x^{11} - 35277 x^{10} - 472409 x^{9} + 5199015 x^{8} - 6540911 x^{7} - 64725915 x^{6} + 118654269 x^{5} + 202450036 x^{4} - 244482346 x^{3} - 286455995 x^{2} + 18504933 x + 26094979 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1274602568003244304847660149105336749401=31^{10}\cdot 41^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $278.03$
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:  $31, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{16204235523355324533161786407091174203735126142204191522557504233} a^{15} + \frac{5505529873774142102196682358104440222190046599192921673187981187}{16204235523355324533161786407091174203735126142204191522557504233} a^{14} + \frac{1035284105991012233945324638994533525411521394384671946136376841}{16204235523355324533161786407091174203735126142204191522557504233} a^{13} + \frac{1896625849604030804580884517165398932397828659136029109023447782}{16204235523355324533161786407091174203735126142204191522557504233} a^{12} + \frac{3458379470229810640447762771612528614651974732716674418329437045}{16204235523355324533161786407091174203735126142204191522557504233} a^{11} + \frac{256002099724468391721400751424045434680069657553546093572975863}{704531979276318457963555930743094530597179397487138761850326271} a^{10} + \frac{7101747739282608949386232387349641250283042553310679993730760964}{16204235523355324533161786407091174203735126142204191522557504233} a^{9} - \frac{1504615170035509776431883850370904280786289446706393946879024608}{16204235523355324533161786407091174203735126142204191522557504233} a^{8} - \frac{5622831602198155873988244641996117219049119714689082192818636815}{16204235523355324533161786407091174203735126142204191522557504233} a^{7} + \frac{6487393274085957287942239875134270038962529679924727336244444056}{16204235523355324533161786407091174203735126142204191522557504233} a^{6} + \frac{1384722485833337890873251757385922664629177285887617607730341401}{16204235523355324533161786407091174203735126142204191522557504233} a^{5} + \frac{7270786561803521795745355577107533451545600792482494118072452482}{16204235523355324533161786407091174203735126142204191522557504233} a^{4} - \frac{3959824836827293903844921426919421278428432938550270676119817433}{16204235523355324533161786407091174203735126142204191522557504233} a^{3} + \frac{32280141836126817416007892950833028109827809900720123547651278}{16204235523355324533161786407091174203735126142204191522557504233} a^{2} - \frac{4752298069515842211475263763623546337805256714704461543957108376}{16204235523355324533161786407091174203735126142204191522557504233} a - \frac{1492974846320519964350555010250687691394606169383102557517041686}{16204235523355324533161786407091174203735126142204191522557504233}$

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:  $13$
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:  \( 105585113967000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1223:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1223
Character table for t16n1223 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.179859661768855001.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ $16$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{12}$

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
31Data not computed
41Data not computed