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Properties

Label 16.8.16808696443...8353.3
Degree $16$
Signature $[8, 4]$
Discriminant $61^{12}\cdot 97^{15}$
Root discriminant $1590.73$
Ramified primes $61, 97$
Class number $80$ (GRH)
Class group $[2, 2, 20]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![165600259875929075027, -5596555580957524159, -1849537571150881115, 111422777813856136, 4581066558613275, -626802607944064, 3236328597138, -78298872864, 15042335559, 8144026557, -373958264, -21614001, 1485243, 20785, -2248, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 2248*x^14 + 20785*x^13 + 1485243*x^12 - 21614001*x^11 - 373958264*x^10 + 8144026557*x^9 + 15042335559*x^8 - 78298872864*x^7 + 3236328597138*x^6 - 626802607944064*x^5 + 4581066558613275*x^4 + 111422777813856136*x^3 - 1849537571150881115*x^2 - 5596555580957524159*x + 165600259875929075027)
 
gp: K = bnfinit(x^16 - 3*x^15 - 2248*x^14 + 20785*x^13 + 1485243*x^12 - 21614001*x^11 - 373958264*x^10 + 8144026557*x^9 + 15042335559*x^8 - 78298872864*x^7 + 3236328597138*x^6 - 626802607944064*x^5 + 4581066558613275*x^4 + 111422777813856136*x^3 - 1849537571150881115*x^2 - 5596555580957524159*x + 165600259875929075027, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 2248 x^{14} + 20785 x^{13} + 1485243 x^{12} - 21614001 x^{11} - 373958264 x^{10} + 8144026557 x^{9} + 15042335559 x^{8} - 78298872864 x^{7} + 3236328597138 x^{6} - 626802607944064 x^{5} + 4581066558613275 x^{4} + 111422777813856136 x^{3} - 1849537571150881115 x^{2} - 5596555580957524159 x + 165600259875929075027 \)

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:  \(1680869644357963641125481571233506843563004905118353=61^{12}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1590.73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{43} a^{13} - \frac{6}{43} a^{12} - \frac{17}{43} a^{11} + \frac{12}{43} a^{10} - \frac{12}{43} a^{9} - \frac{20}{43} a^{8} + \frac{6}{43} a^{7} - \frac{18}{43} a^{6} - \frac{17}{43} a^{4} + \frac{16}{43} a^{3} - \frac{21}{43} a^{2} - \frac{10}{43} a + \frac{11}{43}$, $\frac{1}{43} a^{14} - \frac{10}{43} a^{12} - \frac{4}{43} a^{11} + \frac{17}{43} a^{10} - \frac{6}{43} a^{9} + \frac{15}{43} a^{8} + \frac{18}{43} a^{7} + \frac{21}{43} a^{6} - \frac{17}{43} a^{5} - \frac{11}{43} a^{3} - \frac{7}{43} a^{2} - \frac{6}{43} a - \frac{20}{43}$, $\frac{1}{62735003465199055217183463694156530312970756481967974446412887359346359305128815988179521107031149989531406062597719104934862648587507} a^{15} + \frac{620822554366807029245242611923501290441969980801930853075198784414012650994310201320891967736477144783621908730858707859872022116763}{62735003465199055217183463694156530312970756481967974446412887359346359305128815988179521107031149989531406062597719104934862648587507} a^{14} + \frac{704544444287605071072997058684629684990923123134415602688092285415957166977779229392128368925199626137059924421423452388134298824937}{62735003465199055217183463694156530312970756481967974446412887359346359305128815988179521107031149989531406062597719104934862648587507} a^{13} - \frac{17406323051949070533748984009390994067866860200133451984244843677031595874202140901265868552360576823718400757565728315920060718966363}{62735003465199055217183463694156530312970756481967974446412887359346359305128815988179521107031149989531406062597719104934862648587507} a^{12} + \frac{7495076626029324881518761311559386468466740198334617209573215282303184964074822300602988443740295610289918759077972485727078111449254}{62735003465199055217183463694156530312970756481967974446412887359346359305128815988179521107031149989531406062597719104934862648587507} a^{11} - \frac{12525572037759696127397806535263192730832884844337567901764754354846278480096376887913911465684046855830524348588863793335445270318154}{62735003465199055217183463694156530312970756481967974446412887359346359305128815988179521107031149989531406062597719104934862648587507} a^{10} - \frac{3865729100181943471771136786113335911062512011505372148955035499840854152090930926744595794060519416525342877504847381516943071966602}{62735003465199055217183463694156530312970756481967974446412887359346359305128815988179521107031149989531406062597719104934862648587507} a^{9} + \frac{10603249952030875896197442561365933517830578181183732608785105103447558812992629474203214256929146899224875288148837945180272520585523}{62735003465199055217183463694156530312970756481967974446412887359346359305128815988179521107031149989531406062597719104934862648587507} a^{8} - \frac{18088709077868425446867952397642644813680457921370289222762884914447855532327323652678945404773937145164122993970664334134021296858794}{62735003465199055217183463694156530312970756481967974446412887359346359305128815988179521107031149989531406062597719104934862648587507} a^{7} + \frac{19000659016886520684098171007274107886512115156277445185386617502544683363584011228785376980678298007865976968371233246999743351556997}{62735003465199055217183463694156530312970756481967974446412887359346359305128815988179521107031149989531406062597719104934862648587507} a^{6} + \frac{14242355845279023438830732385908392718436634646791101235569346530556692478772175183634584258909317767481985824236387721918828969080699}{62735003465199055217183463694156530312970756481967974446412887359346359305128815988179521107031149989531406062597719104934862648587507} a^{5} - \frac{16414943777581315930274503052610568973964981287770337719264045939429665449853431650878234445269924755644683936800152231048124472190202}{62735003465199055217183463694156530312970756481967974446412887359346359305128815988179521107031149989531406062597719104934862648587507} a^{4} + \frac{8119890341485090132161339790734429414180920473264231812673657223234191672637919994990609332582333604349247412983416846765373978042709}{62735003465199055217183463694156530312970756481967974446412887359346359305128815988179521107031149989531406062597719104934862648587507} a^{3} + \frac{17698816991637046666884287517570839642286503138995052755417876395270853529132377111247752251727312720129454288171098370166614568831360}{62735003465199055217183463694156530312970756481967974446412887359346359305128815988179521107031149989531406062597719104934862648587507} a^{2} - \frac{20504678235917912564912607774309112563682142767661728793176159646928968043535738161858650767277094471321363001776795317521148049784458}{62735003465199055217183463694156530312970756481967974446412887359346359305128815988179521107031149989531406062597719104934862648587507} a + \frac{29374746792583411205737669975180811980956527170278123891028143263614944880436962502453334470278081843549346660481595064844423176770867}{62735003465199055217183463694156530312970756481967974446412887359346359305128815988179521107031149989531406062597719104934862648587507}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{20}$, which has order $80$ (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:  \( 553669793628000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.1118720199956720578033.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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