↑

Properties

Label 16.0.22368513682...8125.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 13^{6}\cdot 29^{7}\cdot 1049^{2}$
Root discriminant $91.07$
Ramified primes $5, 13, 29, 1049$
Class number $103168$ (GRH)
Class group $[2, 2, 2, 2, 6448]$ (GRH)
Galois group 16T1558

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![431724841, 13494377, 207616495, 18571065, 40963734, 5112247, 5523873, 192135, 763006, -82748, 80393, -10309, 4479, -436, 114, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 114*x^14 - 436*x^13 + 4479*x^12 - 10309*x^11 + 80393*x^10 - 82748*x^9 + 763006*x^8 + 192135*x^7 + 5523873*x^6 + 5112247*x^5 + 40963734*x^4 + 18571065*x^3 + 207616495*x^2 + 13494377*x + 431724841)
 
gp: K = bnfinit(x^16 - 6*x^15 + 114*x^14 - 436*x^13 + 4479*x^12 - 10309*x^11 + 80393*x^10 - 82748*x^9 + 763006*x^8 + 192135*x^7 + 5523873*x^6 + 5112247*x^5 + 40963734*x^4 + 18571065*x^3 + 207616495*x^2 + 13494377*x + 431724841, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 114 x^{14} - 436 x^{13} + 4479 x^{12} - 10309 x^{11} + 80393 x^{10} - 82748 x^{9} + 763006 x^{8} + 192135 x^{7} + 5523873 x^{6} + 5112247 x^{5} + 40963734 x^{4} + 18571065 x^{3} + 207616495 x^{2} + 13494377 x + 431724841 \)

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:  \(22368513682624478383882080078125=5^{12}\cdot 13^{6}\cdot 29^{7}\cdot 1049^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.07$
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:  $5, 13, 29, 1049$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{19} a^{13} - \frac{5}{19} a^{12} + \frac{4}{19} a^{11} + \frac{7}{19} a^{10} - \frac{7}{19} a^{9} + \frac{5}{19} a^{8} + \frac{9}{19} a^{7} + \frac{7}{19} a^{6} + \frac{4}{19} a^{5} + \frac{3}{19} a^{4} - \frac{9}{19} a^{3} + \frac{9}{19} a^{2} - \frac{7}{19} a + \frac{2}{19}$, $\frac{1}{779} a^{14} - \frac{11}{779} a^{13} + \frac{91}{779} a^{12} - \frac{74}{779} a^{11} + \frac{293}{779} a^{10} + \frac{66}{779} a^{9} + \frac{55}{779} a^{8} + \frac{143}{779} a^{7} - \frac{9}{41} a^{6} - \frac{40}{779} a^{5} - \frac{8}{779} a^{4} + \frac{329}{779} a^{3} - \frac{346}{779} a^{2} + \frac{291}{779} a - \frac{12}{779}$, $\frac{1}{226247998505190003599749141854901461379035831355212251} a^{15} + \frac{77018414513879691394854949408798657846706033217446}{226247998505190003599749141854901461379035831355212251} a^{14} + \frac{2973929394398514196926370283195815138563786506834551}{226247998505190003599749141854901461379035831355212251} a^{13} - \frac{65395049074378441793704151842983329240758055946061888}{226247998505190003599749141854901461379035831355212251} a^{12} + \frac{30836400228961370392282812095856435564165043338318885}{226247998505190003599749141854901461379035831355212251} a^{11} + \frac{106253643517060858541899243967387028168036034525835292}{226247998505190003599749141854901461379035831355212251} a^{10} + \frac{10361215211169006114084287565607171373129795175143178}{226247998505190003599749141854901461379035831355212251} a^{9} + \frac{13784523674227145084983114854513008707979330041462839}{226247998505190003599749141854901461379035831355212251} a^{8} + \frac{23726208219942550886267729711190891733967073814237465}{226247998505190003599749141854901461379035831355212251} a^{7} - \frac{3290830942354312123414166891076112959784519679159768}{226247998505190003599749141854901461379035831355212251} a^{6} - \frac{1106208105776654820013091070655548271104937163368016}{11907789395010000189460481150257971651528201650274329} a^{5} - \frac{77465224311523200137264174064140008415994873824342794}{226247998505190003599749141854901461379035831355212251} a^{4} + \frac{85480690878158943915228169925179585404339739750465438}{226247998505190003599749141854901461379035831355212251} a^{3} - \frac{12228706917523831778137000123603354840046191307178}{134591313804396194883848388967817645079735771180971} a^{2} + \frac{10063358587979342842868447625433757514219289227209620}{226247998505190003599749141854901461379035831355212251} a + \frac{12740274976849544590164854100087030445818541803526938}{226247998505190003599749141854901461379035831355212251}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{6448}$, which has order $103168$ (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:  \( 10968.6213178 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1558:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 94 conjugacy class representatives for t16n1558 are not computed
Character table for t16n1558 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.8.2576088125.1

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 $16$ $16$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ R $16$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$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$13.8.0.1$x^{8} + 4 x^{2} - x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
13.8.6.4$x^{8} - 13 x^{4} + 338$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
$29$29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
29.8.7.3$x^{8} + 58$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
1049Data not computed