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Properties

Label 16.0.37228626555...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{10}\cdot 5^{10}\cdot 41^{6}\cdot 97^{8}$
Root discriminant $167.18$
Ramified primes $2, 5, 41, 97$
Class number $58$ (GRH)
Class group $[58]$ (GRH)
Galois group 16T1220

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4908433609, -1353001077, 5298257835, -726339735, 1472323023, -18177834, 141296160, 6804645, 6060244, 495646, 158184, 8141, 4415, -157, 105, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 105*x^14 - 157*x^13 + 4415*x^12 + 8141*x^11 + 158184*x^10 + 495646*x^9 + 6060244*x^8 + 6804645*x^7 + 141296160*x^6 - 18177834*x^5 + 1472323023*x^4 - 726339735*x^3 + 5298257835*x^2 - 1353001077*x + 4908433609)
 
gp: K = bnfinit(x^16 - 4*x^15 + 105*x^14 - 157*x^13 + 4415*x^12 + 8141*x^11 + 158184*x^10 + 495646*x^9 + 6060244*x^8 + 6804645*x^7 + 141296160*x^6 - 18177834*x^5 + 1472323023*x^4 - 726339735*x^3 + 5298257835*x^2 - 1353001077*x + 4908433609, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 105 x^{14} - 157 x^{13} + 4415 x^{12} + 8141 x^{11} + 158184 x^{10} + 495646 x^{9} + 6060244 x^{8} + 6804645 x^{7} + 141296160 x^{6} - 18177834 x^{5} + 1472323023 x^{4} - 726339735 x^{3} + 5298257835 x^{2} - 1353001077 x + 4908433609 \)

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:  \(372286265552058761987916010000000000=2^{10}\cdot 5^{10}\cdot 41^{6}\cdot 97^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $167.18$
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:  $2, 5, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{36} a^{10} - \frac{1}{36} a^{9} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{5}{18} a^{6} - \frac{2}{9} a^{5} + \frac{1}{3} a^{4} + \frac{1}{36} a^{3} - \frac{13}{36} a^{2} + \frac{1}{6} a + \frac{1}{36}$, $\frac{1}{36} a^{11} + \frac{1}{36} a^{9} + \frac{2}{9} a^{7} - \frac{4}{9} a^{6} - \frac{2}{9} a^{5} + \frac{7}{36} a^{4} - \frac{1}{2} a^{3} + \frac{11}{36} a^{2} + \frac{1}{36} a + \frac{13}{36}$, $\frac{1}{14760} a^{12} - \frac{7}{820} a^{11} + \frac{79}{7380} a^{10} + \frac{1163}{14760} a^{9} - \frac{643}{7380} a^{8} + \frac{53}{2460} a^{7} + \frac{383}{1476} a^{6} - \frac{7111}{14760} a^{5} + \frac{463}{2460} a^{4} + \frac{20}{41} a^{3} - \frac{919}{2460} a^{2} + \frac{4621}{14760} a + \frac{1969}{14760}$, $\frac{1}{162360} a^{13} + \frac{1}{40590} a^{12} + \frac{49}{13530} a^{11} - \frac{419}{54120} a^{10} + \frac{5047}{81180} a^{9} - \frac{391}{7380} a^{8} + \frac{183}{1804} a^{7} + \frac{80789}{162360} a^{6} + \frac{8716}{20295} a^{5} + \frac{2107}{16236} a^{4} + \frac{12193}{81180} a^{3} + \frac{66391}{162360} a^{2} + \frac{2141}{18040} a + \frac{1756}{4059}$, $\frac{1}{649440} a^{14} + \frac{1}{649440} a^{13} + \frac{1}{43296} a^{12} + \frac{1}{43296} a^{11} + \frac{1897}{649440} a^{10} - \frac{7949}{216480} a^{9} - \frac{55609}{324720} a^{8} + \frac{267901}{649440} a^{7} + \frac{258451}{649440} a^{6} - \frac{9607}{72160} a^{5} + \frac{18511}{81180} a^{4} + \frac{28007}{72160} a^{3} - \frac{20747}{64944} a^{2} - \frac{79039}{324720} a + \frac{73751}{649440}$, $\frac{1}{1205107323269918454180298866730009274648210421246240} a^{15} + \frac{3910919182722091835865218350236408814610833}{33475203424164401505008301853611368740228067256840} a^{14} - \frac{4923466516970200104201882288344319660841977}{4463360456555253534001106913814849165363742300912} a^{13} + \frac{2320318235632574602847001671682215296566931357}{150638415408739806772537358341251159331026302655780} a^{12} + \frac{7398969927594899163642486272259153967218524580491}{602553661634959227090149433365004637324105210623120} a^{11} + \frac{145217861625929050914540072681944870967947885453}{33475203424164401505008301853611368740228067256840} a^{10} + \frac{2348359828401005433169619663782593719094889192669}{36518403735452074369099965658485129534794255189280} a^{9} - \frac{22453538592362224608631695073154600194340712015517}{241021464653983690836059773346001854929642084249248} a^{8} + \frac{12835727857253185375397191093216516205840935468421}{120510732326991845418029886673000927464821042124624} a^{7} + \frac{248615767311482854821585577170901361320104740029513}{602553661634959227090149433365004637324105210623120} a^{6} - \frac{28681014489012888213654187898557609837325563748947}{63426701224732550220015729827895224981484759012960} a^{5} - \frac{53469819509823502261509070175607678412137768228169}{1205107323269918454180298866730009274648210421246240} a^{4} - \frac{136163212953492506740075236993836539405591353523081}{1205107323269918454180298866730009274648210421246240} a^{3} + \frac{24620969686355633399496622417804350446329852108079}{75319207704369903386268679170625579665513151327890} a^{2} + \frac{46422400640652106048887471365531651815952436219571}{109555211206356223107299896975455388604382765567840} a - \frac{8073626007500198390227782743026378413718363403413}{63426701224732550220015729827895224981484759012960}$

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

Class group and class number

$C_{58}$, which has order $58$ (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:  \( 7669645550.83 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{97}) \), \(\Q(\sqrt{485}) \), 4.0.9644225.1, 4.0.1025.1, \(\Q(\sqrt{5}, \sqrt{97})\), 8.0.93011075850625.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.6.1$x^{8} - 9881 x^{4} + 34857216$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$97$97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$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}$