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Properties

Label 16.8.87678952778...7713.6
Degree $16$
Signature $[8, 4]$
Discriminant $61^{4}\cdot 97^{15}$
Root discriminant $203.67$
Ramified primes $61, 97$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 16T1223

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![186691984, 499997728, -612399080, -54294058, 104056579, -18436972, 22961073, -1411152, 111047, 325596, -114969, -5750, -2574, -518, 56, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 56*x^14 - 518*x^13 - 2574*x^12 - 5750*x^11 - 114969*x^10 + 325596*x^9 + 111047*x^8 - 1411152*x^7 + 22961073*x^6 - 18436972*x^5 + 104056579*x^4 - 54294058*x^3 - 612399080*x^2 + 499997728*x + 186691984)
 
gp: K = bnfinit(x^16 - 4*x^15 + 56*x^14 - 518*x^13 - 2574*x^12 - 5750*x^11 - 114969*x^10 + 325596*x^9 + 111047*x^8 - 1411152*x^7 + 22961073*x^6 - 18436972*x^5 + 104056579*x^4 - 54294058*x^3 - 612399080*x^2 + 499997728*x + 186691984, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 56 x^{14} - 518 x^{13} - 2574 x^{12} - 5750 x^{11} - 114969 x^{10} + 325596 x^{9} + 111047 x^{8} - 1411152 x^{7} + 22961073 x^{6} - 18436972 x^{5} + 104056579 x^{4} - 54294058 x^{3} - 612399080 x^{2} + 499997728 x + 186691984 \)

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:  \(8767895277848913089642817986417897713=61^{4}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $203.67$
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:  $61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{13585134578603724751776545345956150440596340804681034600979340869704} a^{15} + \frac{180327964611091963143310913892545910098082410693684565491252811829}{1698141822325465593972068168244518805074542600585129325122417608713} a^{14} - \frac{250794207711833965000434357277754236859046905367155963731449712471}{1698141822325465593972068168244518805074542600585129325122417608713} a^{13} + \frac{618162542609618038436002129356524173357016316458303898242591844891}{6792567289301862375888272672978075220298170402340517300489670434852} a^{12} - \frac{38887009415697061321286792290546186653497481818290028815751185499}{6792567289301862375888272672978075220298170402340517300489670434852} a^{11} + \frac{1308087036071160417589549177724906875723321134196953172178507163299}{6792567289301862375888272672978075220298170402340517300489670434852} a^{10} - \frac{2352715241794078764713237446996719288525336495676705068217596880073}{13585134578603724751776545345956150440596340804681034600979340869704} a^{9} - \frac{350398777804143361543461513830917002541785851737096312675523204892}{1698141822325465593972068168244518805074542600585129325122417608713} a^{8} - \frac{2179826251256888160979339105526848263852983881249474871086700651897}{13585134578603724751776545345956150440596340804681034600979340869704} a^{7} + \frac{267464529911365584800164432272047744174813781251137221014500071703}{1698141822325465593972068168244518805074542600585129325122417608713} a^{6} - \frac{5727574294016753739669658431693785654887619004627294682125914825283}{13585134578603724751776545345956150440596340804681034600979340869704} a^{5} + \frac{1050195910705073200162930171050414152418932099269318479782804143101}{3396283644650931187944136336489037610149085201170258650244835217426} a^{4} + \frac{2334983073991786823098729824319076298981496477024148785179649163963}{13585134578603724751776545345956150440596340804681034600979340869704} a^{3} - \frac{1474478681113539235260428157326572732702944091654177161687356566241}{6792567289301862375888272672978075220298170402340517300489670434852} a^{2} + \frac{309223311088910426427689949364271903559704796090704441565283964539}{1698141822325465593972068168244518805074542600585129325122417608713} a - \frac{775328103792952116039597280230671050449488775600070256283783354908}{1698141822325465593972068168244518805074542600585129325122417608713}$

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

Class group and class number

$C_{4}$, 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:  $11$
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:  \( 792607167618 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1223:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1223
Character table for t16n1223 is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ $16$

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
61Data not computed
97Data not computed