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Properties

Label 16.0.24761575488...1809.2
Degree $16$
Signature $[0, 8]$
Discriminant $41^{14}\cdot 97^{14}$
Root discriminant $1411.27$
Ramified primes $41, 97$
Class number $91117192068$ (GRH)
Class group $[21, 4338913908]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![99902637257197312, 2899855979953376, 5228711472095080, 1030855819624916, 215319701431098, 28963391474261, 3045583313341, 150438196003, 11790894313, -31586510, 3847802, -3860358, -24628, -7127, 369, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 369*x^14 - 7127*x^13 - 24628*x^12 - 3860358*x^11 + 3847802*x^10 - 31586510*x^9 + 11790894313*x^8 + 150438196003*x^7 + 3045583313341*x^6 + 28963391474261*x^5 + 215319701431098*x^4 + 1030855819624916*x^3 + 5228711472095080*x^2 + 2899855979953376*x + 99902637257197312)
 
gp: K = bnfinit(x^16 - x^15 + 369*x^14 - 7127*x^13 - 24628*x^12 - 3860358*x^11 + 3847802*x^10 - 31586510*x^9 + 11790894313*x^8 + 150438196003*x^7 + 3045583313341*x^6 + 28963391474261*x^5 + 215319701431098*x^4 + 1030855819624916*x^3 + 5228711472095080*x^2 + 2899855979953376*x + 99902637257197312, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 369 x^{14} - 7127 x^{13} - 24628 x^{12} - 3860358 x^{11} + 3847802 x^{10} - 31586510 x^{9} + 11790894313 x^{8} + 150438196003 x^{7} + 3045583313341 x^{6} + 28963391474261 x^{5} + 215319701431098 x^{4} + 1030855819624916 x^{3} + 5228711472095080 x^{2} + 2899855979953376 x + 99902637257197312 \)

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:  \(247615754888863175950064888078950980890077741301809=41^{14}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1411.27$
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:  $41, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3977=41\cdot 97\)
Dirichlet character group:    $\lbrace$$\chi_{3977}(1408,·)$, $\chi_{3977}(1,·)$, $\chi_{3977}(3976,·)$, $\chi_{3977}(2569,·)$, $\chi_{3977}(2059,·)$, $\chi_{3977}(3348,·)$, $\chi_{3977}(2133,·)$, $\chi_{3977}(1239,·)$, $\chi_{3977}(2586,·)$, $\chi_{3977}(161,·)$, $\chi_{3977}(3816,·)$, $\chi_{3977}(1391,·)$, $\chi_{3977}(2738,·)$, $\chi_{3977}(1844,·)$, $\chi_{3977}(629,·)$, $\chi_{3977}(1918,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{32} a^{7} - \frac{1}{16} a^{4} + \frac{7}{32} a^{3} + \frac{1}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{128} a^{8} + \frac{1}{64} a^{6} - \frac{1}{16} a^{5} - \frac{7}{128} a^{4} + \frac{3}{16} a^{3} + \frac{1}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{256} a^{9} - \frac{1}{256} a^{8} + \frac{1}{128} a^{7} + \frac{3}{128} a^{6} + \frac{1}{256} a^{5} - \frac{1}{256} a^{4} - \frac{13}{64} a^{3} - \frac{1}{64} a^{2} - \frac{5}{16} a - \frac{1}{2}$, $\frac{1}{3584} a^{10} - \frac{1}{896} a^{9} - \frac{3}{3584} a^{8} - \frac{5}{448} a^{7} - \frac{17}{3584} a^{6} + \frac{5}{128} a^{5} - \frac{31}{512} a^{4} + \frac{31}{224} a^{3} - \frac{37}{896} a^{2} - \frac{69}{224} a + \frac{11}{28}$, $\frac{1}{7168} a^{11} + \frac{9}{7168} a^{9} - \frac{3}{896} a^{8} + \frac{103}{7168} a^{7} + \frac{1}{56} a^{6} - \frac{43}{1024} a^{5} + \frac{41}{896} a^{4} + \frac{431}{1792} a^{3} + \frac{69}{448} a^{2} + \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{86016} a^{12} - \frac{5}{86016} a^{11} - \frac{5}{86016} a^{10} - \frac{125}{86016} a^{9} - \frac{71}{86016} a^{8} - \frac{1171}{86016} a^{7} - \frac{757}{28672} a^{6} - \frac{677}{28672} a^{5} - \frac{25}{6144} a^{4} - \frac{3671}{21504} a^{3} - \frac{527}{10752} a^{2} - \frac{397}{2688} a + \frac{163}{336}$, $\frac{1}{688128} a^{13} + \frac{1}{344064} a^{12} + \frac{1}{21504} a^{11} + \frac{1}{21504} a^{10} + \frac{139}{344064} a^{9} + \frac{117}{57344} a^{8} - \frac{1339}{172032} a^{7} - \frac{275}{28672} a^{6} + \frac{2695}{98304} a^{5} + \frac{943}{49152} a^{4} + \frac{13619}{57344} a^{3} + \frac{5073}{28672} a^{2} - \frac{419}{21504} a - \frac{251}{2688}$, $\frac{1}{38535168} a^{14} - \frac{25}{38535168} a^{13} - \frac{83}{19267584} a^{12} - \frac{103}{2408448} a^{11} + \frac{1315}{19267584} a^{10} + \frac{11455}{6422528} a^{9} - \frac{7957}{2408448} a^{8} - \frac{48815}{3211264} a^{7} + \frac{338809}{38535168} a^{6} + \frac{1032791}{38535168} a^{5} + \frac{304645}{6422528} a^{4} - \frac{51567}{3211264} a^{3} + \frac{69091}{4816896} a^{2} + \frac{401609}{1204224} a + \frac{24483}{50176}$, $\frac{1}{88700100850430410663128215888579155457310709133324659755267612076025083035099391328256} a^{15} + \frac{277923726492599080060134803102825483739026346016704579620574282207679746990485}{44350050425215205331564107944289577728655354566662329877633806038012541517549695664128} a^{14} - \frac{35190362869029431045216127019213975049251161074107153793836629039118156344621}{60711910233011916949437519430923446582690423773665064856446004158812514055509508096} a^{13} - \frac{8005764075197170288072635439110922910307532790417332774836552636502738927886477}{2111907163105485968169719425918551320412159741269634756077800287524406738930937888768} a^{12} + \frac{431462540967253196542977525201233362417922377623451381724953083030133642073636137}{14783350141738401777188035981429859242885118188887443292544602012670847172516565221376} a^{11} - \frac{867521626318406678194199456263810250436820832719650015355513546630864277135270093}{22175025212607602665782053972144788864327677283331164938816903019006270758774847832064} a^{10} - \frac{61314871964830585021665414074762502875736851246302174723316342779855550853430876049}{44350050425215205331564107944289577728655354566662329877633806038012541517549695664128} a^{9} - \frac{5399130281332892492224814830157331117546335299253759320351272914559851456070285455}{3167860744658228952254579138877826980618239611904452134116700431286610108396406833152} a^{8} + \frac{132655978450849001859240004386549698825370612281100947839734600431875839371029803643}{12671442978632915809018316555511307922472958447617808536466801725146440433585627332608} a^{7} - \frac{162648129317294762332826373845462670150822239384010966523879177213497511064508671689}{6335721489316457904509158277755653961236479223808904268233400862573220216792813666304} a^{6} - \frac{2227382627273086631225197512683447766804266946970742096458204272956987592056758788029}{88700100850430410663128215888579155457310709133324659755267612076025083035099391328256} a^{5} - \frac{471247876571425495172737521009634671303436059517350041621253515044657258401684700271}{14783350141738401777188035981429859242885118188887443292544602012670847172516565221376} a^{4} - \frac{5434985113931063037925087829134021299806101761579852678257540819350632619788976366081}{22175025212607602665782053972144788864327677283331164938816903019006270758774847832064} a^{3} + \frac{2726574965466785428907245537420670566127967206662677138279107353520480727347299925677}{11087512606303801332891026986072394432163838641665582469408451509503135379387423916032} a^{2} + \frac{292909479684890666409015054996702987467376419704952384848522546060816615796427269827}{2771878151575950333222756746518098608040959660416395617352112877375783844846855979008} a - \frac{37017212179660842120464181927854456269318833527721289364819277754501604947484284855}{115494922982331263884281531104920775335039985850683150723004703223990993535285665792}$

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

Class group and class number

$C_{21}\times C_{4338913908}$, which has order $91117192068$ (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:  \( 1085553540910 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{3977}) \), \(\Q(\sqrt{97}) \), \(\Q(\sqrt{41}, \sqrt{97})\), 4.4.62902335833.2, 4.4.62902335833.1, 8.8.3956703853247515803889.1, 8.0.15735811224365370352066553.4, 8.0.15735811224365370352066553.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
41Data not computed
97Data not computed