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Properties

Label 16.8.34571164844...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{40}\cdot 5^{15}\cdot 101^{3}$
Root discriminant $60.77$
Ramified primes $2, 5, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1604

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![281, -1244, -26360, 106260, 209820, 20428, 95728, -202360, 56705, -3060, 3748, -752, 110, -120, 20, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 20*x^14 - 120*x^13 + 110*x^12 - 752*x^11 + 3748*x^10 - 3060*x^9 + 56705*x^8 - 202360*x^7 + 95728*x^6 + 20428*x^5 + 209820*x^4 + 106260*x^3 - 26360*x^2 - 1244*x + 281)
 
gp: K = bnfinit(x^16 - 4*x^15 + 20*x^14 - 120*x^13 + 110*x^12 - 752*x^11 + 3748*x^10 - 3060*x^9 + 56705*x^8 - 202360*x^7 + 95728*x^6 + 20428*x^5 + 209820*x^4 + 106260*x^3 - 26360*x^2 - 1244*x + 281, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 20 x^{14} - 120 x^{13} + 110 x^{12} - 752 x^{11} + 3748 x^{10} - 3060 x^{9} + 56705 x^{8} - 202360 x^{7} + 95728 x^{6} + 20428 x^{5} + 209820 x^{4} + 106260 x^{3} - 26360 x^{2} - 1244 x + 281 \)

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:  \(34571164844032000000000000000=2^{40}\cdot 5^{15}\cdot 101^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.77$
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:  $2, 5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{31521830397036271086501002806244662365771139} a^{15} + \frac{2575427039467747521815079096925151358319622}{31521830397036271086501002806244662365771139} a^{14} - \frac{9879387671271697106832779018252682650752775}{31521830397036271086501002806244662365771139} a^{13} - \frac{7041190364495979704522691388908751925851975}{31521830397036271086501002806244662365771139} a^{12} + \frac{11581208696683063922055970973569821865776317}{31521830397036271086501002806244662365771139} a^{11} - \frac{4413409608123113118723940359605925416453425}{31521830397036271086501002806244662365771139} a^{10} + \frac{6007102905964795314920361958096267099025154}{31521830397036271086501002806244662365771139} a^{9} - \frac{14205041115539229869956956964592576648340363}{31521830397036271086501002806244662365771139} a^{8} + \frac{13398261816834446714369536670247358045842385}{31521830397036271086501002806244662365771139} a^{7} - \frac{6097164976729615744932139128022100506533319}{31521830397036271086501002806244662365771139} a^{6} + \frac{13680004646856798854687189514631250464056273}{31521830397036271086501002806244662365771139} a^{5} + \frac{11103842060760110634930088830315739190673681}{31521830397036271086501002806244662365771139} a^{4} + \frac{2888138734807432825668945774303354292391830}{31521830397036271086501002806244662365771139} a^{3} - \frac{7990374813236320639483735279709067761719474}{31521830397036271086501002806244662365771139} a^{2} - \frac{2231604535677117882374429454073756505934396}{31521830397036271086501002806244662365771139} a - \frac{7861079650965433134351454495909671228221017}{31521830397036271086501002806244662365771139}$

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:  $11$
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:  \( 192268026.256 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1604:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 73 conjugacy class representatives for t16n1604 are not computed
Character table for t16n1604 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.5120000000.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ $16$ $16$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
2Data not computed
5Data not computed
101Data not computed