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Properties

Label 16.16.1622511630...0625.3
Degree $16$
Signature $[16, 0]$
Discriminant $5^{14}\cdot 149^{14}$
Root discriminant $325.94$
Ramified primes $5, 149$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![35867779776, -36379432800, -27965699712, 17851139640, 10995877008, -1541936250, -1351490139, 21744135, 75196675, 1841560, -2207799, -76475, 35573, 1060, -297, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 297*x^14 + 1060*x^13 + 35573*x^12 - 76475*x^11 - 2207799*x^10 + 1841560*x^9 + 75196675*x^8 + 21744135*x^7 - 1351490139*x^6 - 1541936250*x^5 + 10995877008*x^4 + 17851139640*x^3 - 27965699712*x^2 - 36379432800*x + 35867779776)
 
gp: K = bnfinit(x^16 - 5*x^15 - 297*x^14 + 1060*x^13 + 35573*x^12 - 76475*x^11 - 2207799*x^10 + 1841560*x^9 + 75196675*x^8 + 21744135*x^7 - 1351490139*x^6 - 1541936250*x^5 + 10995877008*x^4 + 17851139640*x^3 - 27965699712*x^2 - 36379432800*x + 35867779776, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 297 x^{14} + 1060 x^{13} + 35573 x^{12} - 76475 x^{11} - 2207799 x^{10} + 1841560 x^{9} + 75196675 x^{8} + 21744135 x^{7} - 1351490139 x^{6} - 1541936250 x^{5} + 10995877008 x^{4} + 17851139640 x^{3} - 27965699712 x^{2} - 36379432800 x + 35867779776 \)

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:  \(16225116300501260546649351778570556640625=5^{14}\cdot 149^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $325.94$
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:  $5, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{6} - \frac{1}{6} a^{5} - \frac{1}{4} a^{4} - \frac{5}{12} a^{2} + \frac{1}{3} a$, $\frac{1}{12} a^{7} - \frac{1}{12} a^{5} + \frac{1}{12} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{72} a^{8} - \frac{1}{72} a^{7} - \frac{1}{36} a^{6} - \frac{5}{24} a^{5} - \frac{1}{36} a^{4} - \frac{7}{72} a^{3} - \frac{23}{72} a^{2} + \frac{1}{12} a - \frac{1}{2}$, $\frac{1}{144} a^{9} - \frac{1}{144} a^{8} - \frac{1}{72} a^{7} - \frac{1}{48} a^{6} + \frac{5}{72} a^{5} + \frac{29}{144} a^{4} - \frac{23}{144} a^{3} - \frac{1}{8} a^{2} + \frac{1}{12} a$, $\frac{1}{432} a^{10} - \frac{1}{144} a^{8} - \frac{17}{432} a^{7} - \frac{17}{432} a^{6} + \frac{11}{48} a^{5} + \frac{1}{72} a^{4} + \frac{19}{432} a^{3} - \frac{29}{72} a^{2} - \frac{5}{12} a$, $\frac{1}{432} a^{11} - \frac{1}{216} a^{8} - \frac{5}{432} a^{7} - \frac{1}{24} a^{6} + \frac{5}{24} a^{5} + \frac{35}{216} a^{4} + \frac{11}{48} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{2592} a^{12} - \frac{1}{1296} a^{11} - \frac{1}{1296} a^{9} - \frac{1}{2592} a^{8} - \frac{11}{648} a^{7} + \frac{1}{48} a^{6} - \frac{289}{1296} a^{5} + \frac{391}{2592} a^{4} - \frac{17}{144} a^{3} + \frac{1}{12} a^{2} + \frac{1}{12} a - \frac{1}{2}$, $\frac{1}{23328} a^{13} + \frac{1}{5832} a^{12} + \frac{1}{1296} a^{11} + \frac{1}{1458} a^{10} + \frac{77}{23328} a^{9} - \frac{19}{11664} a^{8} - \frac{11}{1296} a^{7} + \frac{395}{11664} a^{6} - \frac{4391}{23328} a^{5} - \frac{1}{432} a^{4} + \frac{29}{432} a^{3} - \frac{5}{18} a^{2} - \frac{2}{9} a - \frac{1}{2}$, $\frac{1}{139968} a^{14} + \frac{1}{139968} a^{13} - \frac{7}{46656} a^{12} + \frac{35}{69984} a^{11} + \frac{29}{139968} a^{10} - \frac{53}{139968} a^{9} - \frac{109}{46656} a^{8} + \frac{17}{69984} a^{7} - \frac{3845}{139968} a^{6} - \frac{3169}{46656} a^{5} + \frac{635}{5184} a^{4} - \frac{41}{216} a^{3} + \frac{23}{432} a^{2} + \frac{4}{9} a - \frac{1}{4}$, $\frac{1}{32332996794059300542423079043391627444608} a^{15} - \frac{51681520099052822561089166571166499}{32332996794059300542423079043391627444608} a^{14} - \frac{153999830241633732385107321765148505}{10777665598019766847474359681130542481536} a^{13} + \frac{991661765542277941703874524846527673}{16166498397029650271211539521695813722304} a^{12} + \frac{18711701814888095193065826451530898829}{32332996794059300542423079043391627444608} a^{11} + \frac{19950289336599162615541962425582802031}{32332996794059300542423079043391627444608} a^{10} - \frac{2576336151887256170889496155416550599}{10777665598019766847474359681130542481536} a^{9} - \frac{49248652108235482315103620807557649855}{16166498397029650271211539521695813722304} a^{8} - \frac{1471302090632274629878761764982521807}{751930158001379082381932070776549475456} a^{7} - \frac{357567962365015607454801056224715850737}{10777665598019766847474359681130542481536} a^{6} - \frac{292604442863770059822559876439671094167}{1197518399779974094163817742347838053504} a^{5} + \frac{1937971938889525047218641927916364253}{8316099998472042320582067655193319816} a^{4} + \frac{15766672341136620791563467050445317693}{99793199981664507846984811862319837792} a^{3} - \frac{285313259435103239863202100851147732}{1039512499809005290072758456899164977} a^{2} + \frac{388027396928460707119911303791494013}{924011110941338035620229739465924424} a - \frac{270447633082960542741678644961101}{12833487651963028272503190825915617}$

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:  \( 5709703146950000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

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 $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{745}) \), \(\Q(\sqrt{149}) \), 4.4.413493625.2, 4.4.413493625.1, \(\Q(\sqrt{5}, \sqrt{149})\), 8.8.127377848547152265625.2 x2, 8.8.127377848547152265625.1 x2, 8.8.170976977915640625.1

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$149$149.8.7.1$x^{8} - 149$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
149.8.7.1$x^{8} - 149$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$