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Properties

Label 16.12.5768103326...3953.3
Degree $16$
Signature $[12, 2]$
Discriminant $17^{15}\cdot 67^{4}$
Root discriminant $40.74$
Ramified primes $17, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T841

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![919, 4551, 1997, -15255, -11054, 17904, 8417, -8748, 339, 1093, -1062, 414, -13, -20, 15, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 15*x^14 - 20*x^13 - 13*x^12 + 414*x^11 - 1062*x^10 + 1093*x^9 + 339*x^8 - 8748*x^7 + 8417*x^6 + 17904*x^5 - 11054*x^4 - 15255*x^3 + 1997*x^2 + 4551*x + 919)
 
gp: K = bnfinit(x^16 - 7*x^15 + 15*x^14 - 20*x^13 - 13*x^12 + 414*x^11 - 1062*x^10 + 1093*x^9 + 339*x^8 - 8748*x^7 + 8417*x^6 + 17904*x^5 - 11054*x^4 - 15255*x^3 + 1997*x^2 + 4551*x + 919, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 15 x^{14} - 20 x^{13} - 13 x^{12} + 414 x^{11} - 1062 x^{10} + 1093 x^{9} + 339 x^{8} - 8748 x^{7} + 8417 x^{6} + 17904 x^{5} - 11054 x^{4} - 15255 x^{3} + 1997 x^{2} + 4551 x + 919 \)

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:  \(57681033264163530732453953=17^{15}\cdot 67^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.74$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{14608390064245717801811298107} a^{15} + \frac{5601526280458794897379425746}{14608390064245717801811298107} a^{14} - \frac{6910788647119937987323214189}{14608390064245717801811298107} a^{13} - \frac{2170825422349634433209753274}{14608390064245717801811298107} a^{12} + \frac{964522234518397425339984703}{14608390064245717801811298107} a^{11} - \frac{193393549967476822941436540}{14608390064245717801811298107} a^{10} + \frac{1663385824139108149053622075}{14608390064245717801811298107} a^{9} - \frac{7074163327891958476570869324}{14608390064245717801811298107} a^{8} + \frac{5154258898851696978582648795}{14608390064245717801811298107} a^{7} - \frac{6739908207008497541525953847}{14608390064245717801811298107} a^{6} - \frac{551792358163178325694651149}{14608390064245717801811298107} a^{5} + \frac{7281134360442503288688538778}{14608390064245717801811298107} a^{4} + \frac{5825060019043657671013596192}{14608390064245717801811298107} a^{3} + \frac{6369659658897645395941617851}{14608390064245717801811298107} a^{2} - \frac{3025056615446441594118764891}{14608390064245717801811298107} a - \frac{7161571655379962894122315680}{14608390064245717801811298107}$

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:  \( 13552105.3628 \) (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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
67Data not computed