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Properties

Label 16.8.56000759063...8913.1
Degree $16$
Signature $[8, 4]$
Discriminant $17^{15}\cdot 89^{14}$
Root discriminant $723.21$
Ramified primes $17, 89$
Class number $16$ (GRH)
Class group $[2, 8]$ (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4125171948377, 14170208488667, -13423616432713, 3751528448186, 493775735297, -393273090724, 37089111838, 7553415536, -1159905613, -41911767, 10413130, -715973, 78065, 2719, -662, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 662*x^14 + 2719*x^13 + 78065*x^12 - 715973*x^11 + 10413130*x^10 - 41911767*x^9 - 1159905613*x^8 + 7553415536*x^7 + 37089111838*x^6 - 393273090724*x^5 + 493775735297*x^4 + 3751528448186*x^3 - 13423616432713*x^2 + 14170208488667*x - 4125171948377)
 
gp: K = bnfinit(x^16 - x^15 - 662*x^14 + 2719*x^13 + 78065*x^12 - 715973*x^11 + 10413130*x^10 - 41911767*x^9 - 1159905613*x^8 + 7553415536*x^7 + 37089111838*x^6 - 393273090724*x^5 + 493775735297*x^4 + 3751528448186*x^3 - 13423616432713*x^2 + 14170208488667*x - 4125171948377, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 662 x^{14} + 2719 x^{13} + 78065 x^{12} - 715973 x^{11} + 10413130 x^{10} - 41911767 x^{9} - 1159905613 x^{8} + 7553415536 x^{7} + 37089111838 x^{6} - 393273090724 x^{5} + 493775735297 x^{4} + 3751528448186 x^{3} - 13423616432713 x^{2} + 14170208488667 x - 4125171948377 \)

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:  \(5600075906386661768959437042812533816731958913=17^{15}\cdot 89^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $723.21$
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:  $17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{19} a^{13} - \frac{6}{19} a^{12} + \frac{8}{19} a^{11} - \frac{9}{19} a^{9} + \frac{8}{19} a^{8} + \frac{4}{19} a^{7} - \frac{5}{19} a^{5} - \frac{3}{19} a^{4} + \frac{3}{19} a^{3} + \frac{6}{19} a^{2} - \frac{5}{19} a - \frac{5}{19}$, $\frac{1}{413459} a^{14} - \frac{495}{21761} a^{13} + \frac{155582}{413459} a^{12} - \frac{156075}{413459} a^{11} + \frac{46009}{413459} a^{10} - \frac{149937}{413459} a^{9} + \frac{79472}{413459} a^{8} - \frac{151843}{413459} a^{7} - \frac{102206}{413459} a^{6} - \frac{103868}{413459} a^{5} + \frac{28732}{413459} a^{4} - \frac{24771}{413459} a^{3} + \frac{33433}{413459} a^{2} + \frac{140508}{413459} a - \frac{28340}{413459}$, $\frac{1}{202208359475821148976664124878653508463207194519729400700394541422033664029322766212835727589} a^{15} + \frac{148769998899969594358261220477852390163056288210047092111974810550426237859749619231728}{202208359475821148976664124878653508463207194519729400700394541422033664029322766212835727589} a^{14} - \frac{3970453564274667620002311083669329889938638307568345341323134602955678433709179213989273192}{202208359475821148976664124878653508463207194519729400700394541422033664029322766212835727589} a^{13} + \frac{63387467570054667126873555849059115232607985073009685615263762798101512310255649031834239184}{202208359475821148976664124878653508463207194519729400700394541422033664029322766212835727589} a^{12} - \frac{9062502771229018197577395430454551385231439023277080125982691704971504322934697481425381603}{202208359475821148976664124878653508463207194519729400700394541422033664029322766212835727589} a^{11} - \frac{9517658873502579674735274053938527955887220797117704382335194942241531509913550802419523081}{202208359475821148976664124878653508463207194519729400700394541422033664029322766212835727589} a^{10} - \frac{31800218074403221369005352955090362097325467669107021322740092140338458522491776076174308579}{202208359475821148976664124878653508463207194519729400700394541422033664029322766212835727589} a^{9} - \frac{9209923845938419024074348139035459753834494424735271935748067781146663437224030165311637906}{202208359475821148976664124878653508463207194519729400700394541422033664029322766212835727589} a^{8} - \frac{1970331686127567676123756216913736303167952282337557607544748998490864001617274979732444001}{4302305520762152105886470742099010818366110521696370227667968966426248170836654600273100587} a^{7} + \frac{3158545991693872250355463657647659206156728128674353681350506125352379749772939129330853619}{10642545235569534156666532888350184655958273395775231615810239022212298106806461379622933031} a^{6} + \frac{32789453321411597515610121155733211595308419321857019254588614279796468075919844075460457390}{202208359475821148976664124878653508463207194519729400700394541422033664029322766212835727589} a^{5} - \frac{100717987793781468828375078448397921679950841192580915693089953911067903808854447495691800406}{202208359475821148976664124878653508463207194519729400700394541422033664029322766212835727589} a^{4} - \frac{80659075007504848658477739064586612904420027302816748149986344401323416616596234243723779672}{202208359475821148976664124878653508463207194519729400700394541422033664029322766212835727589} a^{3} + \frac{47394055154006166476396919732222378730791487415462423192644185498814815849872815764028380404}{202208359475821148976664124878653508463207194519729400700394541422033664029322766212835727589} a^{2} + \frac{17257860528173236436071481642035794445543234854102203670322526874671988104578758798151139618}{202208359475821148976664124878653508463207194519729400700394541422033664029322766212835727589} a - \frac{3763814060985823085921877153547655074732143801437185935636592990669661945135850840360875561}{202208359475821148976664124878653508463207194519729400700394541422033664029322766212835727589}$

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

Class group and class number

$C_{2}\times C_{8}$, which has order $16$ (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:  \( 7990596409790000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.38915873.1, 8.8.203930643438763634753.2

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

Sibling fields

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$89$89.8.7.2$x^{8} - 801$$8$$1$$7$$C_8$$[\ ]_{8}$
89.8.7.4$x^{8} - 64881$$8$$1$$7$$C_8$$[\ ]_{8}$