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Properties

Label 16.16.1243616029...7041.1
Degree $16$
Signature $[16, 0]$
Discriminant $37^{14}\cdot 53^{14}$
Root discriminant $760.18$
Ramified primes $37, 53$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $OD_{16}.C_2$ (as 16T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8585514871439, 431756362246264, 377047001834358, 92029351276520, -7225988019316, -5748766657820, -423891709799, 97377694420, 13086980592, -529773288, -126688589, 450094, 567046, 3400, -1220, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 1220*x^14 + 3400*x^13 + 567046*x^12 + 450094*x^11 - 126688589*x^10 - 529773288*x^9 + 13086980592*x^8 + 97377694420*x^7 - 423891709799*x^6 - 5748766657820*x^5 - 7225988019316*x^4 + 92029351276520*x^3 + 377047001834358*x^2 + 431756362246264*x + 8585514871439)
 
gp: K = bnfinit(x^16 - 6*x^15 - 1220*x^14 + 3400*x^13 + 567046*x^12 + 450094*x^11 - 126688589*x^10 - 529773288*x^9 + 13086980592*x^8 + 97377694420*x^7 - 423891709799*x^6 - 5748766657820*x^5 - 7225988019316*x^4 + 92029351276520*x^3 + 377047001834358*x^2 + 431756362246264*x + 8585514871439, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 1220 x^{14} + 3400 x^{13} + 567046 x^{12} + 450094 x^{11} - 126688589 x^{10} - 529773288 x^{9} + 13086980592 x^{8} + 97377694420 x^{7} - 423891709799 x^{6} - 5748766657820 x^{5} - 7225988019316 x^{4} + 92029351276520 x^{3} + 377047001834358 x^{2} + 431756362246264 x + 8585514871439 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12436160298838170417198712919084997051554477041=37^{14}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $760.18$
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:  $37, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{8} a^{10} + \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{3}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{3}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} + \frac{3}{8} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{88} a^{13} + \frac{1}{22} a^{12} + \frac{3}{88} a^{11} - \frac{1}{11} a^{9} - \frac{7}{88} a^{8} - \frac{13}{88} a^{7} - \frac{7}{88} a^{6} - \frac{1}{88} a^{5} + \frac{5}{22} a^{4} + \frac{5}{44} a^{3} + \frac{1}{88} a^{2} + \frac{1}{22} a + \frac{3}{8}$, $\frac{1}{1311112} a^{14} - \frac{23}{163889} a^{13} + \frac{20193}{327778} a^{12} + \frac{5831}{163889} a^{11} - \frac{15897}{327778} a^{10} - \frac{17863}{1311112} a^{9} + \frac{253445}{1311112} a^{8} + \frac{47087}{327778} a^{7} - \frac{156593}{655556} a^{6} + \frac{67655}{655556} a^{5} - \frac{72099}{655556} a^{4} + \frac{222059}{1311112} a^{3} + \frac{78659}{327778} a^{2} - \frac{159447}{655556} a - \frac{5175}{59596}$, $\frac{1}{2715035636305123165816990342445509787993812502750939608975074252694122450484105464091696} a^{15} - \frac{400968107626757617048798972568089299041121055901716400971752694685548108235652603}{2715035636305123165816990342445509787993812502750939608975074252694122450484105464091696} a^{14} - \frac{4655654182717502742301724878019086941931164530616545108256933289957873906381960828555}{2715035636305123165816990342445509787993812502750939608975074252694122450484105464091696} a^{13} + \frac{69332259838180575259830528819813566942479110292562137757567081300112981611076538443299}{2715035636305123165816990342445509787993812502750939608975074252694122450484105464091696} a^{12} - \frac{1524346525877718363234090071265704646309776117097773267090859328333983961069024124175}{2715035636305123165816990342445509787993812502750939608975074252694122450484105464091696} a^{11} - \frac{108950938796806100184619385328221308245283663654412135297116962685152411697126791941013}{2715035636305123165816990342445509787993812502750939608975074252694122450484105464091696} a^{10} - \frac{33114557812234618696127505179238688620808818444708019708540902854235688722512606281949}{169689727269070197863561896402844361749613281421933725560942140793382653155256591505731} a^{9} + \frac{333045586926645131674146862886396668435589367055572840057012420135830167701094927824433}{1357517818152561582908495171222754893996906251375469804487537126347061225242052732045848} a^{8} + \frac{16669580473176791148947306860330116688392383729447284010527455849457795744110134072943}{339379454538140395727123792805688723499226562843867451121884281586765306310513183011462} a^{7} - \frac{30266075616918096676360564787124587982209001716953161702889584239332745995855048736063}{1357517818152561582908495171222754893996906251375469804487537126347061225242052732045848} a^{6} - \frac{994356171711653901045866633193664171032740847464318682412751795706094049785190567402737}{2715035636305123165816990342445509787993812502750939608975074252694122450484105464091696} a^{5} - \frac{44565722646023814740548047544458959023191117129224158066674075575174592608773130183349}{2715035636305123165816990342445509787993812502750939608975074252694122450484105464091696} a^{4} - \frac{1089880966729164069525651083180856664683687645001865718521483626764640592222430780656903}{2715035636305123165816990342445509787993812502750939608975074252694122450484105464091696} a^{3} + \frac{1091820180074700471006973147675289370243928512096855264158207438978465117603289723749045}{2715035636305123165816990342445509787993812502750939608975074252694122450484105464091696} a^{2} - \frac{1145183007362826669231746906209880620974994219036610892312127262438526624324566857433763}{2715035636305123165816990342445509787993812502750939608975074252694122450484105464091696} a - \frac{77370307728095068871833761642502944260022809878549073395338694577481521352967142340953}{246821421482283924165180940222319071635801136613721782634097659335829313680373224008336}$

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:  $15$
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:  \( 320799854911000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}.C_2$ (as 16T40):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $OD_{16}.C_2$
Character table for $OD_{16}.C_2$

Intermediate fields

\(\Q(\sqrt{1961}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{37}) \), 4.4.7541066681.2, 4.4.7541066681.1, \(\Q(\sqrt{37}, \sqrt{53})\), 8.8.56867686687288355761.3

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$37$37.8.7.1$x^{8} - 37$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.1$x^{8} - 37$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$53$53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$