Properties

Label 16.0.462...081.1
Degree $16$
Signature $[0, 8]$
Discriminant $4.627\times 10^{44}$
Root discriminant $618.85$
Ramified primes $29, 41$
Class number $13506148$ (GRH)
Class group $[13506148]$ (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 96*x^14 + 91*x^13 + 2772*x^12 + 129667*x^11 - 310036*x^10 + 576733*x^9 - 5430577*x^8 - 152310452*x^7 + 3090172208*x^6 + 409621632*x^5 - 96715161760*x^4 + 166203665088*x^3 - 59137532864*x^2 + 3645586489344*x + 106045326871552)
 
gp: K = bnfinit(x^16 - 7*x^15 + 96*x^14 + 91*x^13 + 2772*x^12 + 129667*x^11 - 310036*x^10 + 576733*x^9 - 5430577*x^8 - 152310452*x^7 + 3090172208*x^6 + 409621632*x^5 - 96715161760*x^4 + 166203665088*x^3 - 59137532864*x^2 + 3645586489344*x + 106045326871552, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![106045326871552, 3645586489344, -59137532864, 166203665088, -96715161760, 409621632, 3090172208, -152310452, -5430577, 576733, -310036, 129667, 2772, 91, 96, -7, 1]);
 

\( x^{16} - 7 x^{15} + 96 x^{14} + 91 x^{13} + 2772 x^{12} + 129667 x^{11} - 310036 x^{10} + 576733 x^{9} - 5430577 x^{8} - 152310452 x^{7} + 3090172208 x^{6} + 409621632 x^{5} - 96715161760 x^{4} + 166203665088 x^{3} - 59137532864 x^{2} + 3645586489344 x + 106045326871552 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(462732306245995722656474121747020143758680081\)\(\medspace = 29^{14}\cdot 41^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $618.85$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $29, 41$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{9} + \frac{1}{32} a^{8} - \frac{1}{16} a^{7} + \frac{3}{32} a^{6} - \frac{5}{32} a^{5} - \frac{5}{32} a^{4} + \frac{1}{4} a$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{8} + \frac{1}{32} a^{7} - \frac{1}{16} a^{6} + \frac{3}{16} a^{5} - \frac{5}{32} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{64} a^{12} - \frac{1}{64} a^{9} + \frac{1}{64} a^{8} - \frac{1}{32} a^{7} + \frac{3}{32} a^{6} + \frac{11}{64} a^{5} - \frac{1}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{256} a^{13} - \frac{1}{128} a^{11} - \frac{3}{256} a^{10} + \frac{3}{256} a^{9} - \frac{1}{128} a^{8} - \frac{3}{32} a^{7} + \frac{9}{256} a^{6} - \frac{1}{128} a^{5} + \frac{11}{64} a^{4} + \frac{3}{32} a^{3} + \frac{7}{16} a^{2} - \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{42496} a^{14} + \frac{1}{2656} a^{13} + \frac{31}{21248} a^{12} + \frac{397}{42496} a^{11} + \frac{259}{42496} a^{10} - \frac{33}{21248} a^{9} - \frac{167}{5312} a^{8} - \frac{3815}{42496} a^{7} - \frac{1953}{21248} a^{6} - \frac{2129}{10624} a^{5} + \frac{631}{5312} a^{4} - \frac{1217}{2656} a^{3} - \frac{57}{1328} a^{2} + \frac{11}{332} a + \frac{15}{83}$, $\frac{1}{185439383371689668992192314149241249148950325854506670841163521593229713928830464} a^{15} + \frac{75185281066047901430433462306014876414603676093751988620658453595572061}{11996337389810432720416115548534173188572281398273170581004238685032327204608} a^{14} - \frac{16768324717392381453764869830627096467106158826132072940341523631771158448907}{23179922921461208624024039268655156143618790731813333855145440199153714241103808} a^{13} + \frac{978263215920264487131534053707096576465493925159169569692152035270828424988153}{185439383371689668992192314149241249148950325854506670841163521593229713928830464} a^{12} - \frac{2456712196914064324207458017166564485633343823732816616619790338061594847742175}{185439383371689668992192314149241249148950325854506670841163521593229713928830464} a^{11} + \frac{204763768564060105052383746200660813264127176309412588877175287929470156211679}{92719691685844834496096157074620624574475162927253335420581760796614856964415232} a^{10} - \frac{36327576303825580801569960525217085369792855098749500654218631679711729687141}{667048141624782981986303288306623198377519157750023995831523458968452208377088} a^{9} - \frac{1740550021395369020502777250580149667307672808395644942456036592615353604704283}{185439383371689668992192314149241249148950325854506670841163521593229713928830464} a^{8} + \frac{2360282513747756743736718860987036969779184790658037110220617675615862929440783}{23179922921461208624024039268655156143618790731813333855145440199153714241103808} a^{7} + \frac{7539731730694065960966883646051285896234094017679201639239351852897310096122961}{92719691685844834496096157074620624574475162927253335420581760796614856964415232} a^{6} - \frac{7176155105599101490326383180711536174385631740713410148501888707737983660700585}{46359845842922417248048078537310312287237581463626667710290880398307428482207616} a^{5} - \frac{3514558090304022124609452645459204308518129033362090174374352562574962196897227}{23179922921461208624024039268655156143618790731813333855145440199153714241103808} a^{4} - \frac{2229414973943189936734970279964073918650422010753248672623527680109484120659197}{11589961460730604312012019634327578071809395365906666927572720099576857120551904} a^{3} + \frac{355392874290803703132637389127591770465544520892592475178505614540735630830409}{5794980730365302156006009817163789035904697682953333463786360049788428560275952} a^{2} + \frac{236532812359296051931925351051131368625621165438737830980229839314601897786269}{2897490365182651078003004908581894517952348841476666731893180024894214280137976} a - \frac{22818456150029138843268218952803312565115226363638851830151999894729910633811}{724372591295662769500751227145473629488087210369166682973295006223553570034494}$

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

Class group and class number

$C_{13506148}$, which has order $13506148$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 276024556298 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 276024556298 \cdot 13506148}{2\sqrt{462732306245995722656474121747020143758680081}}\approx 210.486024391043$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.57962561.1, 8.8.115844383968839978801.2

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

Sibling fields

Galois closure: Deg 32

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ $16$ $16$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
29Data not computed
41Data not computed