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Properties

Label 16.16.4106143709...6689.1
Degree $16$
Signature $[16, 0]$
Discriminant $29^{14}\cdot 53^{14}$
Root discriminant $614.24$
Ramified primes $29, 53$
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![105590872531, -422001336482, 246801227608, 381451049084, -302059720208, -61744943186, 61671443021, -5603494212, -2229690096, 439415370, -4080691, -4207992, 220836, 11888, -954, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 954*x^14 + 11888*x^13 + 220836*x^12 - 4207992*x^11 - 4080691*x^10 + 439415370*x^9 - 2229690096*x^8 - 5603494212*x^7 + 61671443021*x^6 - 61744943186*x^5 - 302059720208*x^4 + 381451049084*x^3 + 246801227608*x^2 - 422001336482*x + 105590872531)
 
gp: K = bnfinit(x^16 - 6*x^15 - 954*x^14 + 11888*x^13 + 220836*x^12 - 4207992*x^11 - 4080691*x^10 + 439415370*x^9 - 2229690096*x^8 - 5603494212*x^7 + 61671443021*x^6 - 61744943186*x^5 - 302059720208*x^4 + 381451049084*x^3 + 246801227608*x^2 - 422001336482*x + 105590872531, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 954 x^{14} + 11888 x^{13} + 220836 x^{12} - 4207992 x^{11} - 4080691 x^{10} + 439415370 x^{9} - 2229690096 x^{8} - 5603494212 x^{7} + 61671443021 x^{6} - 61744943186 x^{5} - 302059720208 x^{4} + 381451049084 x^{3} + 246801227608 x^{2} - 422001336482 x + 105590872531 \)

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:  \(410614370925299326221754150224933859206386689=29^{14}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $614.24$
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:  $29, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{30164} a^{14} - \frac{275}{15082} a^{13} + \frac{2523}{30164} a^{12} + \frac{59}{15082} a^{11} + \frac{3235}{30164} a^{10} - \frac{1502}{7541} a^{9} + \frac{2263}{15082} a^{8} - \frac{885}{7541} a^{7} - \frac{3703}{15082} a^{6} + \frac{673}{15082} a^{5} - \frac{2509}{30164} a^{4} - \frac{2737}{7541} a^{3} + \frac{5109}{30164} a^{2} + \frac{2394}{7541} a + \frac{12623}{30164}$, $\frac{1}{52166843697053957687540440609439631195195271301660504529715191769396829733635581064} a^{15} + \frac{76492777086215913193421865239540116913278437620359199850437556937910058057153}{52166843697053957687540440609439631195195271301660504529715191769396829733635581064} a^{14} + \frac{240590004573835405970564970480035916192261842069223201981150091985393892850291289}{52166843697053957687540440609439631195195271301660504529715191769396829733635581064} a^{13} + \frac{1305407940642085129561717519395244882256618308033807062638182957752307611875408415}{52166843697053957687540440609439631195195271301660504529715191769396829733635581064} a^{12} - \frac{7159469165984655779486960373304631231694245399984192150582393136763831276320567971}{52166843697053957687540440609439631195195271301660504529715191769396829733635581064} a^{11} - \frac{3891248327741074185890109055039538349050496754924240248806390362804406043674496321}{52166843697053957687540440609439631195195271301660504529715191769396829733635581064} a^{10} + \frac{2517250259080483294263732784768250715790843770199169481274184487762174127679245789}{26083421848526978843770220304719815597597635650830252264857595884698414866817790532} a^{9} - \frac{479567466950271279836317436199930402607054188883035078915071320682977216109723}{3086065055433859304752747314803574964221206300382187915861050151999339193897041} a^{8} - \frac{1317990389291228141026944811895973331731808312945514613403189875282671317894345716}{6520855462131744710942555076179953899399408912707563066214398971174603716704447633} a^{7} + \frac{1679317758834223370882482983995190509572498233964164922139406394638168960731690180}{6520855462131744710942555076179953899399408912707563066214398971174603716704447633} a^{6} + \frac{19032314137520879380336516081496848488812585498028640941666206371167430140691176389}{52166843697053957687540440609439631195195271301660504529715191769396829733635581064} a^{5} + \frac{15423023417565577849301006869616047325730188425377508243952994743084285450689942309}{52166843697053957687540440609439631195195271301660504529715191769396829733635581064} a^{4} - \frac{17615207688251364095155722631802274016409561015129084181539712942909607662998875017}{52166843697053957687540440609439631195195271301660504529715191769396829733635581064} a^{3} - \frac{9254937940677367400126887680723545580041306936624131048269861145019718333728311099}{52166843697053957687540440609439631195195271301660504529715191769396829733635581064} a^{2} - \frac{5444487573479175953168427438654101292500751105052350716284511915824162692778659441}{52166843697053957687540440609439631195195271301660504529715191769396829733635581064} a - \frac{7661148038581981231244639155558290413918074157394476521717258971055874468433538333}{52166843697053957687540440609439631195195271301660504529715191769396829733635581064}$

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:  \( 26023114028100000 \) (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

\(\Q(\sqrt{1537}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{29}) \), 4.4.3630961153.1, 4.4.3630961153.2, \(\Q(\sqrt{29}, \sqrt{53})\), 8.8.13183878894595089409.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$29$29.8.7.1$x^{8} - 29$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.7.1$x^{8} - 29$$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}$