↑

Properties

Label 16.8.64479545841...7129.4
Degree $16$
Signature $[8, 4]$
Discriminant $61^{14}\cdot 97^{14}$
Root discriminant $1997.97$
Ramified primes $61, 97$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22791644861288076339, -13771872983608380722, -1617614637177574531, 194886203868804103, 36797324745255904, 3073967109999834, 174686723625779, 2693259145663, -201693621442, -17203793113, -964537235, -21555312, -557765, -7595, 554, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 554*x^14 - 7595*x^13 - 557765*x^12 - 21555312*x^11 - 964537235*x^10 - 17203793113*x^9 - 201693621442*x^8 + 2693259145663*x^7 + 174686723625779*x^6 + 3073967109999834*x^5 + 36797324745255904*x^4 + 194886203868804103*x^3 - 1617614637177574531*x^2 - 13771872983608380722*x + 22791644861288076339)
 
gp: K = bnfinit(x^16 - 3*x^15 + 554*x^14 - 7595*x^13 - 557765*x^12 - 21555312*x^11 - 964537235*x^10 - 17203793113*x^9 - 201693621442*x^8 + 2693259145663*x^7 + 174686723625779*x^6 + 3073967109999834*x^5 + 36797324745255904*x^4 + 194886203868804103*x^3 - 1617614637177574531*x^2 - 13771872983608380722*x + 22791644861288076339, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 554 x^{14} - 7595 x^{13} - 557765 x^{12} - 21555312 x^{11} - 964537235 x^{10} - 17203793113 x^{9} - 201693621442 x^{8} + 2693259145663 x^{7} + 174686723625779 x^{6} + 3073967109999834 x^{5} + 36797324745255904 x^{4} + 194886203868804103 x^{3} - 1617614637177574531 x^{2} - 13771872983608380722 x + 22791644861288076339 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(64479545841814254728122854912988442937092177855107129=61^{14}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1997.97$
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:  $61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{1}{9} a^{5} + \frac{4}{9} a^{2} - \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{27} a^{10} + \frac{1}{27} a^{9} - \frac{2}{27} a^{8} - \frac{1}{9} a^{7} + \frac{4}{27} a^{6} + \frac{2}{27} a^{5} + \frac{2}{9} a^{4} - \frac{8}{27} a^{3} - \frac{4}{9} a^{2} - \frac{10}{27} a - \frac{2}{9}$, $\frac{1}{27} a^{11} - \frac{4}{27} a^{8} - \frac{2}{27} a^{7} - \frac{2}{27} a^{6} - \frac{2}{27} a^{5} - \frac{5}{27} a^{4} - \frac{4}{27} a^{3} + \frac{5}{27} a^{2} - \frac{11}{27} a - \frac{1}{9}$, $\frac{1}{81} a^{12} + \frac{1}{81} a^{11} - \frac{1}{81} a^{9} + \frac{5}{81} a^{7} - \frac{4}{81} a^{6} - \frac{4}{81} a^{5} - \frac{2}{9} a^{4} + \frac{28}{81} a^{3} + \frac{11}{27} a^{2} - \frac{2}{81} a - \frac{13}{27}$, $\frac{1}{243} a^{13} + \frac{2}{243} a^{11} - \frac{1}{243} a^{10} - \frac{8}{243} a^{9} + \frac{2}{243} a^{8} + \frac{4}{81} a^{7} - \frac{2}{81} a^{6} - \frac{2}{243} a^{5} + \frac{4}{243} a^{4} + \frac{74}{243} a^{3} - \frac{29}{243} a^{2} - \frac{25}{243} a - \frac{8}{81}$, $\frac{1}{588303} a^{14} + \frac{581}{588303} a^{13} + \frac{2441}{588303} a^{12} - \frac{220}{65367} a^{11} - \frac{3451}{588303} a^{10} + \frac{18133}{588303} a^{9} - \frac{90392}{588303} a^{8} + \frac{2911}{21789} a^{7} - \frac{34754}{588303} a^{6} - \frac{14930}{196101} a^{5} - \frac{173822}{588303} a^{4} + \frac{193004}{588303} a^{3} - \frac{18422}{588303} a^{2} + \frac{285763}{588303} a - \frac{65443}{196101}$, $\frac{1}{49911896197846562118800200647332108788083396965219699643748883473340685527086629168568029914782178970163757917988404938786909} a^{15} + \frac{33387924870272204797711882943729944695554971351322717150173504847797177841492639822654942104816349858740377903589615930}{49911896197846562118800200647332108788083396965219699643748883473340685527086629168568029914782178970163757917988404938786909} a^{14} + \frac{6202597419561474286101321455482197966411583241534185265792367292128800629667917606780778517799772339837176998757373261744}{5545766244205173568755577849703567643120377440579966627083209274815631725231847685396447768309130996684861990887600548754101} a^{13} - \frac{148947112607650869922705604667912292776565028963035451582074622824733779547230741483107064194716483237726038057585477997825}{49911896197846562118800200647332108788083396965219699643748883473340685527086629168568029914782178970163757917988404938786909} a^{12} + \frac{96623366320109786644005572679255072965626857524681187367261056071066428092647294860603033959705720587030348351777519891153}{49911896197846562118800200647332108788083396965219699643748883473340685527086629168568029914782178970163757917988404938786909} a^{11} - \frac{802026010290222404955878626583581908377304365411744809578623374504855704197940943609605918785021761846278620664590079482269}{49911896197846562118800200647332108788083396965219699643748883473340685527086629168568029914782178970163757917988404938786909} a^{10} + \frac{552384814493772060621648336164904246289826292367773978201494067255036609294240875567747747458402593902859410450272908839752}{16637298732615520706266733549110702929361132321739899881249627824446895175695543056189343304927392990054585972662801646262303} a^{9} + \frac{4950694160211278850327580257876744726032987170051851462765120032847770609687860365625383133494969353200316052213550476294362}{49911896197846562118800200647332108788083396965219699643748883473340685527086629168568029914782178970163757917988404938786909} a^{8} + \frac{7641738624330029660333073821116037624843005600939226631593706056734569544320327998070677221937166738336394056342778647530581}{49911896197846562118800200647332108788083396965219699643748883473340685527086629168568029914782178970163757917988404938786909} a^{7} + \frac{7056447792332701493002239670790768054516655237564837946666003865914656531920168602843251611005017105729944956952280502087955}{49911896197846562118800200647332108788083396965219699643748883473340685527086629168568029914782178970163757917988404938786909} a^{6} + \frac{7731154508566821924801918941282427099259525913674500820193603974686216241564717441743822653589835264909989785933633841800800}{49911896197846562118800200647332108788083396965219699643748883473340685527086629168568029914782178970163757917988404938786909} a^{5} + \frac{91014585539911333050759430655809750039473071930661835103003931837008932024224924584615827836716796039490182062058923215267}{49911896197846562118800200647332108788083396965219699643748883473340685527086629168568029914782178970163757917988404938786909} a^{4} + \frac{1927573608656166325934332951359092924892056835709002880617021058553108161318191213252698012836316593002622227685845485615084}{49911896197846562118800200647332108788083396965219699643748883473340685527086629168568029914782178970163757917988404938786909} a^{3} + \frac{376610602999890874125758952346490824956472237380463356598792090320905178445825603418707235908992184974140969564711359352384}{5545766244205173568755577849703567643120377440579966627083209274815631725231847685396447768309130996684861990887600548754101} a^{2} - \frac{8598853141894164186114771243178999628533549085408190376096218074436181331498355107727135899502553613127650327986675949636962}{49911896197846562118800200647332108788083396965219699643748883473340685527086629168568029914782178970163757917988404938786909} a - \frac{4385723710684867723259830239700301031778466104766923133821258230461357090613247462879183605586982628007395415059746825471180}{16637298732615520706266733549110702929361132321739899881249627824446895175695543056189343304927392990054585972662801646262303}$

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:  $11$
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:  \( 1709879645410000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{97}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{5917}) \), \(\Q(\sqrt{61}, \sqrt{97})\), 8.8.42915029526174817225369.1

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

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$61$61.8.7.1$x^{8} - 61$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
61.8.7.1$x^{8} - 61$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$97$97.8.7.2$x^{8} - 2425$$8$$1$$7$$C_8$$[\ ]_{8}$
97.8.7.4$x^{8} - 1515625$$8$$1$$7$$C_8$$[\ ]_{8}$