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Properties

Label 16.4.57681033264...3953.4
Degree $16$
Signature $[4, 6]$
Discriminant $17^{15}\cdot 67^{4}$
Root discriminant $40.74$
Ramified primes $17, 67$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T841

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31247, -16142, 32891, -2310, 11899, -127, 684, -3606, 1, -60, -208, 138, -69, 26, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 4*x^14 + 26*x^13 - 69*x^12 + 138*x^11 - 208*x^10 - 60*x^9 + x^8 - 3606*x^7 + 684*x^6 - 127*x^5 + 11899*x^4 - 2310*x^3 + 32891*x^2 - 16142*x + 31247)
 
gp: K = bnfinit(x^16 - 2*x^15 + 4*x^14 + 26*x^13 - 69*x^12 + 138*x^11 - 208*x^10 - 60*x^9 + x^8 - 3606*x^7 + 684*x^6 - 127*x^5 + 11899*x^4 - 2310*x^3 + 32891*x^2 - 16142*x + 31247, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 4 x^{14} + 26 x^{13} - 69 x^{12} + 138 x^{11} - 208 x^{10} - 60 x^{9} + x^{8} - 3606 x^{7} + 684 x^{6} - 127 x^{5} + 11899 x^{4} - 2310 x^{3} + 32891 x^{2} - 16142 x + 31247 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2252870497972659603429586297933699507} a^{15} - \frac{318288795570154304249140997665428498}{2252870497972659603429586297933699507} a^{14} - \frac{839616531774346261309872079284021437}{2252870497972659603429586297933699507} a^{13} - \frac{494490230923600880358118662004220701}{2252870497972659603429586297933699507} a^{12} - \frac{1114614965187481315219202266559387753}{2252870497972659603429586297933699507} a^{11} + \frac{172253887250751649030910979009293680}{2252870497972659603429586297933699507} a^{10} - \frac{1077275670478459626996283615109504366}{2252870497972659603429586297933699507} a^{9} - \frac{94136616664207168223211231294992907}{2252870497972659603429586297933699507} a^{8} + \frac{762436818531767137198598174333626021}{2252870497972659603429586297933699507} a^{7} - \frac{482037586340300772738820155653723710}{2252870497972659603429586297933699507} a^{6} + \frac{345218788704438229456498325165814682}{2252870497972659603429586297933699507} a^{5} - \frac{136715858449323218148800463834623299}{2252870497972659603429586297933699507} a^{4} - \frac{1007882522297578134527588300671788784}{2252870497972659603429586297933699507} a^{3} + \frac{557417484203212907292454720245322061}{2252870497972659603429586297933699507} a^{2} + \frac{1005556970315189608202267890460828730}{2252870497972659603429586297933699507} a + \frac{304053046265997287403577033274044369}{2252870497972659603429586297933699507}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $9$
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:  \( 667439.049422 \) (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