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Properties

Label 16.4.40030106677...4057.4
Degree $16$
Signature $[4, 6]$
Discriminant $43^{6}\cdot 97^{15}$
Root discriminant $298.64$
Ramified primes $43, 97$
Class number $72$ (GRH)
Class group $[2, 6, 6]$ (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![-1711681339, -1289438969, -7887221570, -552382365, 2803253366, -620145820, -71808218, 114656041, -45091632, 8421507, -1139968, 66321, 8969, -421, 250, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 250*x^14 - 421*x^13 + 8969*x^12 + 66321*x^11 - 1139968*x^10 + 8421507*x^9 - 45091632*x^8 + 114656041*x^7 - 71808218*x^6 - 620145820*x^5 + 2803253366*x^4 - 552382365*x^3 - 7887221570*x^2 - 1289438969*x - 1711681339)
 
gp: K = bnfinit(x^16 - 4*x^15 + 250*x^14 - 421*x^13 + 8969*x^12 + 66321*x^11 - 1139968*x^10 + 8421507*x^9 - 45091632*x^8 + 114656041*x^7 - 71808218*x^6 - 620145820*x^5 + 2803253366*x^4 - 552382365*x^3 - 7887221570*x^2 - 1289438969*x - 1711681339, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 250 x^{14} - 421 x^{13} + 8969 x^{12} + 66321 x^{11} - 1139968 x^{10} + 8421507 x^{9} - 45091632 x^{8} + 114656041 x^{7} - 71808218 x^{6} - 620145820 x^{5} + 2803253366 x^{4} - 552382365 x^{3} - 7887221570 x^{2} - 1289438969 x - 1711681339 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4003010667744610631429360934274392034057=43^{6}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $298.64$
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:  $43, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{67513105599963908710761975226144362511300826911495934866912798691028791849554741} a^{15} + \frac{26983876235921660426161606139870716573128479535363886134443250619234578398831096}{67513105599963908710761975226144362511300826911495934866912798691028791849554741} a^{14} - \frac{32599745801988504508198935417226252713496495534241400772880723538250902269649187}{67513105599963908710761975226144362511300826911495934866912798691028791849554741} a^{13} - \frac{10752611189017252927254105287493542543973954751626028619315434053511833005890230}{67513105599963908710761975226144362511300826911495934866912798691028791849554741} a^{12} + \frac{21450626439229060407252927976888658225002766180196948063408877561164705790360358}{67513105599963908710761975226144362511300826911495934866912798691028791849554741} a^{11} - \frac{6252037707578431377924106034695355428168422255518013995303566876986817902771900}{67513105599963908710761975226144362511300826911495934866912798691028791849554741} a^{10} - \frac{32365855797499663472621291080946571630099146702409254548371968228343404566891092}{67513105599963908710761975226144362511300826911495934866912798691028791849554741} a^{9} + \frac{26777398907239004491845682103619088231030200230765237822546071598595400988402544}{67513105599963908710761975226144362511300826911495934866912798691028791849554741} a^{8} + \frac{831509475455106170345499278688169120727371367080040195213462082283492484981609}{67513105599963908710761975226144362511300826911495934866912798691028791849554741} a^{7} + \frac{30460772537153596683922872235098449966140955714020963315308081735079992447238564}{67513105599963908710761975226144362511300826911495934866912798691028791849554741} a^{6} + \frac{17975047777089019224170339893887799227480938206090916384933712185586613282910861}{67513105599963908710761975226144362511300826911495934866912798691028791849554741} a^{5} + \frac{28085884427495202492347005270019919092675571078001548681394097355665368488312068}{67513105599963908710761975226144362511300826911495934866912798691028791849554741} a^{4} - \frac{6689752156697736150948034857013691928671877275477615413184246403567738815703723}{67513105599963908710761975226144362511300826911495934866912798691028791849554741} a^{3} - \frac{30710016837805458680228922328846401643867626249416769263742852636302972076978003}{67513105599963908710761975226144362511300826911495934866912798691028791849554741} a^{2} - \frac{9736104001080101402769853426909170877198951163234383789305124760810738173838890}{67513105599963908710761975226144362511300826911495934866912798691028791849554741} a - \frac{26757149414572419498440684796111407188416918074559353017509913214888702690696628}{67513105599963908710761975226144362511300826911495934866912798691028791849554741}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{6}$, which has order $72$ (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:  $9$
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:  \( 264926220872 \) (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, 8.8.80798284478113.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
$43$43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97Data not computed