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Properties

Label 16.0.85974764485...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{8}\cdot 89^{6}\cdot 97^{4}$
Root discriminant $203.42$
Ramified primes $5, 29, 89, 97$
Class number $51580848$ (GRH)
Class group $[2, 2, 2, 6447606]$ (GRH)
Galois group $(C_2^2\times D_4).C_2^3$ (as 16T600)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![37766617552225, 32381728258600, 12276401600505, 2123568115990, 1534675220624, -48736060670, 95094647211, -6177080710, 2765414123, -145621974, 39076255, -1360568, 264016, -5496, 835, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 835*x^14 - 5496*x^13 + 264016*x^12 - 1360568*x^11 + 39076255*x^10 - 145621974*x^9 + 2765414123*x^8 - 6177080710*x^7 + 95094647211*x^6 - 48736060670*x^5 + 1534675220624*x^4 + 2123568115990*x^3 + 12276401600505*x^2 + 32381728258600*x + 37766617552225)
 
gp: K = bnfinit(x^16 - 8*x^15 + 835*x^14 - 5496*x^13 + 264016*x^12 - 1360568*x^11 + 39076255*x^10 - 145621974*x^9 + 2765414123*x^8 - 6177080710*x^7 + 95094647211*x^6 - 48736060670*x^5 + 1534675220624*x^4 + 2123568115990*x^3 + 12276401600505*x^2 + 32381728258600*x + 37766617552225, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 835 x^{14} - 5496 x^{13} + 264016 x^{12} - 1360568 x^{11} + 39076255 x^{10} - 145621974 x^{9} + 2765414123 x^{8} - 6177080710 x^{7} + 95094647211 x^{6} - 48736060670 x^{5} + 1534675220624 x^{4} + 2123568115990 x^{3} + 12276401600505 x^{2} + 32381728258600 x + 37766617552225 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8597476448565520621040798398594140625=5^{8}\cdot 29^{8}\cdot 89^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $203.42$
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:  $5, 29, 89, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{3}{10} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{11} - \frac{1}{5} a^{9} + \frac{1}{10} a^{8} + \frac{1}{10} a^{7} + \frac{1}{10} a^{6} - \frac{1}{2} a^{5} + \frac{3}{10} a^{4} + \frac{1}{10} a^{3} - \frac{1}{5} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{12} + \frac{1}{5} a^{9} - \frac{1}{2} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{3}{10} a^{2}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{9} + \frac{1}{10} a^{8} - \frac{1}{2} a^{7} + \frac{3}{10} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{3}{10} a^{2} - \frac{1}{2}$, $\frac{1}{85690} a^{14} + \frac{213}{85690} a^{13} - \frac{3307}{85690} a^{12} + \frac{7}{4510} a^{11} + \frac{3783}{85690} a^{10} - \frac{175}{1558} a^{9} - \frac{989}{4510} a^{8} - \frac{251}{2090} a^{7} + \frac{28161}{85690} a^{6} - \frac{28801}{85690} a^{5} + \frac{25697}{85690} a^{4} - \frac{2653}{85690} a^{3} + \frac{6291}{17138} a^{2} + \frac{3057}{17138} a - \frac{327}{902}$, $\frac{1}{32479709955377686250629838323167555869514294951617949769804703142404465778880753730418125450} a^{15} + \frac{35226152084336447568820643397657657378352773160715032814864708244100414189709172109559}{32479709955377686250629838323167555869514294951617949769804703142404465778880753730418125450} a^{14} - \frac{241294407184929426641181977857892817914108308458823527191579396740357393142708495591664206}{16239854977688843125314919161583777934757147475808974884902351571202232889440376865209062725} a^{13} - \frac{1096505570906714767422783013639716265018111722059033049356944612218699860927993721635851}{59595798083261809634183189583793680494521642113060458293219638793402689502533493083336010} a^{12} - \frac{18732075808925325186802180049946189021150813577804569252845635139190522911877444899151649}{32479709955377686250629838323167555869514294951617949769804703142404465778880753730418125450} a^{11} + \frac{1290239413466040454109320899482440169143491023719439180374966339747447335843818723581297739}{32479709955377686250629838323167555869514294951617949769804703142404465778880753730418125450} a^{10} - \frac{4623884744963458257403057928793492010728034690012390393992218015006682986173754864201155227}{32479709955377686250629838323167555869514294951617949769804703142404465778880753730418125450} a^{9} - \frac{3924533346018018537928605029955135017456090604001367661094579107323041613918277498099885839}{16239854977688843125314919161583777934757147475808974884902351571202232889440376865209062725} a^{8} + \frac{9615302331744999064120196814875136478941625415985772605372605325672933131604011039437354777}{32479709955377686250629838323167555869514294951617949769804703142404465778880753730418125450} a^{7} + \frac{9408411902699949355344933463175784973622127761502805822186430097112745263720532273798545749}{32479709955377686250629838323167555869514294951617949769804703142404465778880753730418125450} a^{6} - \frac{10107315330293602191699121505044707519646557070583485829468717344919979472130903155521541}{72017095244739880821795650383963538513335465524651773325509319606218327669358655721547950} a^{5} + \frac{245668300401667987059254807357068464119576275749904579472327477259543575459756987784001443}{792188047692138689039752154223598923646690120771169506580602515668401604362945212937027450} a^{4} - \frac{621027580940421466568890963645391572351609662354437349348672853800914337384357834574711894}{3247970995537768625062983832316755586951429495161794976980470314240446577888075373041812545} a^{3} + \frac{766683841105458202121304790734014860967683622373803833289184376896726506463067567170443277}{6495941991075537250125967664633511173902858990323589953960940628480893155776150746083625090} a^{2} - \frac{154331045315058722668414353628408292329908849196812526264852969813841799938930138449831919}{1299188398215107450025193532926702234780571798064717990792188125696178631155230149216725018} a - \frac{16185723022715712600924558591457663185811669928346499630848479703124902084787468146679950}{34189168374081775000662987708597427231067678896439947126110213834109963977769214453071711}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6447606}$, which has order $51580848$ (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:  $7$
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:  \( 13998.0176198 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times D_4).C_2^3$ (as 16T600):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $(C_2^2\times D_4).C_2^3$
Character table for $(C_2^2\times D_4).C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.64525.1, 4.4.2225.1, 4.4.725.1, 8.8.4163475625.1

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

Sibling fields

Degree 16 siblings: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$97$97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$