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Properties

Label 16.4.16176501932...5625.2
Degree $16$
Signature $[4, 6]$
Discriminant $5^{12}\cdot 13^{2}\cdot 29^{8}\cdot 97^{8}$
Root discriminant $244.37$
Ramified primes $5, 13, 29, 97$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 16T1220

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![926982331, -1460860467, 2780443254, -2189230097, 878892788, -36928272, 2512621, 29169414, -9369992, 2570953, -627886, 75084, -13097, 486, -24, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 24*x^14 + 486*x^13 - 13097*x^12 + 75084*x^11 - 627886*x^10 + 2570953*x^9 - 9369992*x^8 + 29169414*x^7 + 2512621*x^6 - 36928272*x^5 + 878892788*x^4 - 2189230097*x^3 + 2780443254*x^2 - 1460860467*x + 926982331)
 
gp: K = bnfinit(x^16 - 6*x^15 - 24*x^14 + 486*x^13 - 13097*x^12 + 75084*x^11 - 627886*x^10 + 2570953*x^9 - 9369992*x^8 + 29169414*x^7 + 2512621*x^6 - 36928272*x^5 + 878892788*x^4 - 2189230097*x^3 + 2780443254*x^2 - 1460860467*x + 926982331, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 24 x^{14} + 486 x^{13} - 13097 x^{12} + 75084 x^{11} - 627886 x^{10} + 2570953 x^{9} - 9369992 x^{8} + 29169414 x^{7} + 2512621 x^{6} - 36928272 x^{5} + 878892788 x^{4} - 2189230097 x^{3} + 2780443254 x^{2} - 1460860467 x + 926982331 \)

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:  \(161765019327832493697466398205322265625=5^{12}\cdot 13^{2}\cdot 29^{8}\cdot 97^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $244.37$
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:  $5, 13, 29, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{5} - \frac{2}{5}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{6} - \frac{2}{5} a$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{9} - \frac{2}{5} a^{7} + \frac{1}{10} a^{6} - \frac{1}{5} a^{5} - \frac{3}{10} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{3}{10}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{10} - \frac{1}{10} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{3}{10} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{3}{10} a + \frac{2}{5}$, $\frac{1}{30} a^{14} - \frac{1}{30} a^{12} - \frac{1}{10} a^{11} + \frac{1}{15} a^{10} - \frac{1}{10} a^{9} - \frac{1}{10} a^{8} + \frac{2}{15} a^{7} - \frac{11}{30} a^{6} + \frac{11}{30} a^{5} + \frac{1}{5} a^{4} + \frac{13}{30} a^{3} - \frac{1}{30} a^{2} + \frac{1}{5} a - \frac{1}{6}$, $\frac{1}{20992855676095192573987018947949970264866992688434512959736471708070} a^{15} + \frac{140438824671772411751622235316894504419775772846650926196423951171}{20992855676095192573987018947949970264866992688434512959736471708070} a^{14} - \frac{181598377618250772934728560315326982553078894408324320562595147493}{4198571135219038514797403789589994052973398537686902591947294341614} a^{13} - \frac{56650785145756529361598488067605311915490699114308525912376062623}{10496427838047596286993509473974985132433496344217256479868235854035} a^{12} - \frac{824688609454274224964204209905424806531642556021168899469206049849}{20992855676095192573987018947949970264866992688434512959736471708070} a^{11} + \frac{37862795771030279507569562357136427131160034485117327649984567603}{4198571135219038514797403789589994052973398537686902591947294341614} a^{10} + \frac{266918046520673241035836337688938128914408245606740890025949925059}{3498809279349198762331169824658328377477832114739085493289411951345} a^{9} + \frac{146933168332399171247015205064979133439225972199213410791857143691}{1908441425099562961271547177086360933169726608039501178157861064370} a^{8} - \frac{38833838350773895873565933194212351807420568381462327546131234131}{538278350669107501897103049947435134996589556113705460506063377130} a^{7} - \frac{3257568814392129050913951801475534152294467427762617241132710191201}{10496427838047596286993509473974985132433496344217256479868235854035} a^{6} + \frac{7680133383179465034566035837574171303325023671285648260909695707447}{20992855676095192573987018947949970264866992688434512959736471708070} a^{5} - \frac{225759939714355603384418854631116144019591495599352260899989277413}{20992855676095192573987018947949970264866992688434512959736471708070} a^{4} - \frac{3070984978498323286617649290786007440635488932975153138530761713382}{10496427838047596286993509473974985132433496344217256479868235854035} a^{3} - \frac{17222238319293951235962851670351381899771137844543259950902994003}{677188892777264276580226417675805492415064280272081063217305538970} a^{2} + \frac{720948485717083119878788079567721628863059287239278536227722440793}{4198571135219038514797403789589994052973398537686902591947294341614} a + \frac{143039885743479656243018568618108114006317239773845415670961499959}{381688285019912592254309435417272186633945321607900235631572212874}$

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:  $9$
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:  \( 1788699334250 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1220:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1220 are not computed
Character table for t16n1220 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2813}) \), \(\Q(\sqrt{14065}) \), 4.4.725.1, 4.4.6821525.1, \(\Q(\sqrt{5}, \sqrt{2813})\), 8.8.39134423996850625.3

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$97$97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$