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Properties

Label 16.0.31236842166...5728.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 17^{15}\cdot 6529^{2}$
Root discriminant $60.38$
Ramified primes $2, 17, 6529$
Class number $7872$ (GRH)
Class group $[2, 4, 984]$ (GRH)
Galois group 16T841

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![24212557, -57494058, 76529340, -65201089, 43654491, -23104150, 10438655, -3803949, 1301332, -335097, 93145, -16657, 3940, -448, 93, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 93*x^14 - 448*x^13 + 3940*x^12 - 16657*x^11 + 93145*x^10 - 335097*x^9 + 1301332*x^8 - 3803949*x^7 + 10438655*x^6 - 23104150*x^5 + 43654491*x^4 - 65201089*x^3 + 76529340*x^2 - 57494058*x + 24212557)
 
gp: K = bnfinit(x^16 - 5*x^15 + 93*x^14 - 448*x^13 + 3940*x^12 - 16657*x^11 + 93145*x^10 - 335097*x^9 + 1301332*x^8 - 3803949*x^7 + 10438655*x^6 - 23104150*x^5 + 43654491*x^4 - 65201089*x^3 + 76529340*x^2 - 57494058*x + 24212557, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 93 x^{14} - 448 x^{13} + 3940 x^{12} - 16657 x^{11} + 93145 x^{10} - 335097 x^{9} + 1301332 x^{8} - 3803949 x^{7} + 10438655 x^{6} - 23104150 x^{5} + 43654491 x^{4} - 65201089 x^{3} + 76529340 x^{2} - 57494058 x + 24212557 \)

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:  \(31236842166910780816202985728=2^{8}\cdot 17^{15}\cdot 6529^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.38$
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:  $2, 17, 6529$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11747654052342334537005833015594945318733507878490317} a^{15} + \frac{253565812996571888252585517722168514209712447022609}{903665696334025733615833308891918870671808298345409} a^{14} - \frac{5104999973230851057175079231957399629347438455633256}{11747654052342334537005833015594945318733507878490317} a^{13} - \frac{2249905750537766895253445696410921109236184073044610}{11747654052342334537005833015594945318733507878490317} a^{12} + \frac{5189290037854464183415644268482471755873220338381676}{11747654052342334537005833015594945318733507878490317} a^{11} + \frac{5030226816310103531839078245997276601436894843025752}{11747654052342334537005833015594945318733507878490317} a^{10} + \frac{508613118449978353928655753206783230077546931932557}{11747654052342334537005833015594945318733507878490317} a^{9} - \frac{3617344202415600375704204381290075686941677739528478}{11747654052342334537005833015594945318733507878490317} a^{8} + \frac{4897547935555920189780138942020286632819989787794871}{11747654052342334537005833015594945318733507878490317} a^{7} + \frac{4042845617900678436137085010078800523848278350245514}{11747654052342334537005833015594945318733507878490317} a^{6} - \frac{298004607540661260352484589672945611270900854509928}{903665696334025733615833308891918870671808298345409} a^{5} + \frac{2798987149026231617168436820652809497866799546527217}{11747654052342334537005833015594945318733507878490317} a^{4} - \frac{3222915073780229830497409731487170024066487172954776}{11747654052342334537005833015594945318733507878490317} a^{3} + \frac{712808952228392910748636601495545136477106556636633}{11747654052342334537005833015594945318733507878490317} a^{2} - \frac{5452362367275129296214048741842284810653717377635628}{11747654052342334537005833015594945318733507878490317} a - \frac{1516845847788721774378555652208749970408126748418863}{11747654052342334537005833015594945318733507878490317}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{984}$, which has order $7872$ (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:  \( 3640.01221338 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T841:

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 t16n841
Character table for t16n841 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
17Data not computed
6529Data not computed