↑

Properties

Label 21.21.9835336749...2129.2
Degree $21$
Signature $[21, 0]$
Discriminant $7^{38}\cdot 31^{14}$
Root discriminant $333.78$
Ramified primes $7, 31$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_{21}$ (as 21T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-64432726399, 606314747948, 4902119383049, 10342548829935, 7568631421988, -388511337004, -2634639635474, -543595119755, 338432862404, 99101537213, -21936726567, -7503654305, 777025536, 295491070, -15018669, -6403488, 143990, 75383, -504, -441, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 441*x^19 - 504*x^18 + 75383*x^17 + 143990*x^16 - 6403488*x^15 - 15018669*x^14 + 295491070*x^13 + 777025536*x^12 - 7503654305*x^11 - 21936726567*x^10 + 99101537213*x^9 + 338432862404*x^8 - 543595119755*x^7 - 2634639635474*x^6 - 388511337004*x^5 + 7568631421988*x^4 + 10342548829935*x^3 + 4902119383049*x^2 + 606314747948*x - 64432726399)
 
gp: K = bnfinit(x^21 - 441*x^19 - 504*x^18 + 75383*x^17 + 143990*x^16 - 6403488*x^15 - 15018669*x^14 + 295491070*x^13 + 777025536*x^12 - 7503654305*x^11 - 21936726567*x^10 + 99101537213*x^9 + 338432862404*x^8 - 543595119755*x^7 - 2634639635474*x^6 - 388511337004*x^5 + 7568631421988*x^4 + 10342548829935*x^3 + 4902119383049*x^2 + 606314747948*x - 64432726399, 1)
 

Normalized defining polynomial

\( x^{21} - 441 x^{19} - 504 x^{18} + 75383 x^{17} + 143990 x^{16} - 6403488 x^{15} - 15018669 x^{14} + 295491070 x^{13} + 777025536 x^{12} - 7503654305 x^{11} - 21936726567 x^{10} + 99101537213 x^{9} + 338432862404 x^{8} - 543595119755 x^{7} - 2634639635474 x^{6} - 388511337004 x^{5} + 7568631421988 x^{4} + 10342548829935 x^{3} + 4902119383049 x^{2} + 606314747948 x - 64432726399 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[21, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(98353367498957471525704156941061377491148627676882129=7^{38}\cdot 31^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $333.78$
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:  $7, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1519=7^{2}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{1519}(1152,·)$, $\chi_{1519}(1,·)$, $\chi_{1519}(67,·)$, $\chi_{1519}(583,·)$, $\chi_{1519}(652,·)$, $\chi_{1519}(718,·)$, $\chi_{1519}(1234,·)$, $\chi_{1519}(149,·)$, $\chi_{1519}(1303,·)$, $\chi_{1519}(1369,·)$, $\chi_{1519}(218,·)$, $\chi_{1519}(284,·)$, $\chi_{1519}(800,·)$, $\chi_{1519}(869,·)$, $\chi_{1519}(935,·)$, $\chi_{1519}(1451,·)$, $\chi_{1519}(366,·)$, $\chi_{1519}(435,·)$, $\chi_{1519}(501,·)$, $\chi_{1519}(1017,·)$, $\chi_{1519}(1086,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{19} a^{12} - \frac{7}{19} a^{11} + \frac{4}{19} a^{10} - \frac{6}{19} a^{9} - \frac{3}{19} a^{8} + \frac{2}{19} a^{7} + \frac{2}{19} a^{6} - \frac{5}{19} a^{5} - \frac{3}{19} a^{4} - \frac{2}{19} a^{3} - \frac{9}{19} a^{2} - \frac{1}{19} a + \frac{8}{19}$, $\frac{1}{19} a^{13} - \frac{7}{19} a^{11} + \frac{3}{19} a^{10} - \frac{7}{19} a^{9} - \frac{3}{19} a^{7} + \frac{9}{19} a^{6} - \frac{4}{19} a^{4} - \frac{4}{19} a^{3} - \frac{7}{19} a^{2} + \frac{1}{19} a - \frac{1}{19}$, $\frac{1}{19} a^{14} - \frac{8}{19} a^{11} + \frac{2}{19} a^{10} - \frac{4}{19} a^{9} - \frac{5}{19} a^{8} + \frac{4}{19} a^{7} - \frac{5}{19} a^{6} - \frac{1}{19} a^{5} - \frac{6}{19} a^{4} - \frac{2}{19} a^{3} - \frac{5}{19} a^{2} - \frac{8}{19} a - \frac{1}{19}$, $\frac{1}{19} a^{15} + \frac{3}{19} a^{11} + \frac{9}{19} a^{10} + \frac{4}{19} a^{9} - \frac{1}{19} a^{8} - \frac{8}{19} a^{7} - \frac{4}{19} a^{6} - \frac{8}{19} a^{5} - \frac{7}{19} a^{4} - \frac{2}{19} a^{3} - \frac{4}{19} a^{2} - \frac{9}{19} a + \frac{7}{19}$, $\frac{1}{19} a^{16} - \frac{8}{19} a^{11} - \frac{8}{19} a^{10} - \frac{2}{19} a^{9} + \frac{1}{19} a^{8} + \frac{9}{19} a^{7} + \frac{5}{19} a^{6} + \frac{8}{19} a^{5} + \frac{7}{19} a^{4} + \frac{2}{19} a^{3} - \frac{1}{19} a^{2} - \frac{9}{19} a - \frac{5}{19}$, $\frac{1}{1273} a^{17} + \frac{3}{1273} a^{16} + \frac{17}{1273} a^{15} + \frac{33}{1273} a^{14} - \frac{9}{1273} a^{13} + \frac{5}{1273} a^{12} - \frac{26}{1273} a^{11} - \frac{314}{1273} a^{10} + \frac{125}{1273} a^{9} + \frac{7}{67} a^{8} - \frac{394}{1273} a^{7} - \frac{284}{1273} a^{6} + \frac{253}{1273} a^{5} - \frac{221}{1273} a^{4} - \frac{66}{1273} a^{3} - \frac{33}{1273} a^{2} - \frac{395}{1273} a - \frac{82}{1273}$, $\frac{1}{59233963} a^{18} + \frac{21988}{59233963} a^{17} - \frac{183938}{59233963} a^{16} + \frac{44808}{59233963} a^{15} - \frac{148385}{59233963} a^{14} - \frac{1246946}{59233963} a^{13} - \frac{295518}{59233963} a^{12} - \frac{17752198}{59233963} a^{11} + \frac{10223241}{59233963} a^{10} + \frac{25853744}{59233963} a^{9} + \frac{4015711}{59233963} a^{8} + \frac{2174675}{59233963} a^{7} - \frac{25803668}{59233963} a^{6} - \frac{15365265}{59233963} a^{5} - \frac{6994309}{59233963} a^{4} + \frac{284974}{1910773} a^{3} + \frac{29286817}{59233963} a^{2} + \frac{558377}{3117577} a + \frac{12158757}{59233963}$, $\frac{1}{59233963} a^{19} - \frac{12868}{59233963} a^{17} - \frac{885057}{59233963} a^{16} + \frac{651732}{59233963} a^{15} - \frac{1311286}{59233963} a^{14} + \frac{13874}{884089} a^{13} + \frac{24158}{1910773} a^{12} - \frac{17151559}{59233963} a^{11} - \frac{23436960}{59233963} a^{10} - \frac{28474077}{59233963} a^{9} - \frac{2799355}{59233963} a^{8} + \frac{3129306}{59233963} a^{7} - \frac{3264090}{59233963} a^{6} - \frac{10531087}{59233963} a^{5} - \frac{8101235}{59233963} a^{4} - \frac{857707}{59233963} a^{3} - \frac{10625505}{59233963} a^{2} + \frac{25983499}{59233963} a + \frac{7255342}{59233963}$, $\frac{1}{1987734965291817008071595293662999007543825264434402458579708711580804091273120404047842354740889999} a^{20} + \frac{13550996264483041296380091837378974979649913014200601121463481998199720963551090147551624460}{1987734965291817008071595293662999007543825264434402458579708711580804091273120404047842354740889999} a^{19} - \frac{849998397603630861526928260697396062897510786429620398287661062632240923710014465211207669}{1987734965291817008071595293662999007543825264434402458579708711580804091273120404047842354740889999} a^{18} - \frac{2672883406522206588184537354024520502437925779547542000509142151381574042157905269189860903873}{8756541697320779771240507901599114570677644336715429332950258641325128155388195612545561034100837} a^{17} + \frac{19727146458362809299399350528434371442617735183309370626467585582641503910080086060856231204709158}{1987734965291817008071595293662999007543825264434402458579708711580804091273120404047842354740889999} a^{16} - \frac{47482353681270009124971773556428074642428159157297965345352778955458238108809503089967533925781595}{1987734965291817008071595293662999007543825264434402458579708711580804091273120404047842354740889999} a^{15} + \frac{259943911892438853476604414938274553815034590614859567140339069237972967354621739718162934905381}{1987734965291817008071595293662999007543825264434402458579708711580804091273120404047842354740889999} a^{14} - \frac{36450473918590816058602499814462467651029729646043507189273619622929915038955569535721066876867246}{1987734965291817008071595293662999007543825264434402458579708711580804091273120404047842354740889999} a^{13} - \frac{18221981547330364879524171579931909964116212151531037245698263673714361250830947140108896976157962}{1987734965291817008071595293662999007543825264434402458579708711580804091273120404047842354740889999} a^{12} - \frac{656918332755962547858896074550447940803085997377410906952152470050676862521325759747603613859702905}{1987734965291817008071595293662999007543825264434402458579708711580804091273120404047842354740889999} a^{11} + \frac{250937619134510739452834307709418286673761749313254126584545559763342804815676462159806933167134948}{1987734965291817008071595293662999007543825264434402458579708711580804091273120404047842354740889999} a^{10} + \frac{699735968903000146133870397159090462347535912417823228275447166058050726086850797241025563202357581}{1987734965291817008071595293662999007543825264434402458579708711580804091273120404047842354740889999} a^{9} - \frac{339273370388024029758677483403116437782907439086902311757939315616831420469440127490209412394418772}{1987734965291817008071595293662999007543825264434402458579708711580804091273120404047842354740889999} a^{8} - \frac{632001524465262835638899902714077718598127466062069657070521607515444924028601216830048257529241968}{1987734965291817008071595293662999007543825264434402458579708711580804091273120404047842354740889999} a^{7} + \frac{495774114292016033089892028904679737867291489148732293710322449949291503085476954842089919505500912}{1987734965291817008071595293662999007543825264434402458579708711580804091273120404047842354740889999} a^{6} - \frac{576730510113933298344351783555030393701558216643322365210828974959221955360459771532657591269886533}{1987734965291817008071595293662999007543825264434402458579708711580804091273120404047842354740889999} a^{5} - \frac{468783350396186537846128581295502485237009166296323300970846857089682140905422577514560294551622680}{1987734965291817008071595293662999007543825264434402458579708711580804091273120404047842354740889999} a^{4} + \frac{120156875098832248721966159324185926052205516393953143944283598652827615693684933660197087354404710}{1987734965291817008071595293662999007543825264434402458579708711580804091273120404047842354740889999} a^{3} - \frac{773681351073696149864796147047748535635471082873391719943695352157874388281707983590260014726783395}{1987734965291817008071595293662999007543825264434402458579708711580804091273120404047842354740889999} a^{2} - \frac{815935786228302480457654844261625011507768815120999884965225167316734437787749450261540812903951959}{1987734965291817008071595293662999007543825264434402458579708711580804091273120404047842354740889999} a + \frac{449875512697738594260195425564333192456119525275399795711990705241648568774249030544174402977723670}{1987734965291817008071595293662999007543825264434402458579708711580804091273120404047842354740889999}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $20$
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:  \( 97078377381360090000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{21}$ (as 21T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 21
The 21 conjugacy class representatives for $C_{21}$
Character table for $C_{21}$ is not computed

Intermediate fields

3.3.47089.1, 7.7.13841287201.1

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 $21$ $21$ $21$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ $21$ $21$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{21}$ $21$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ R ${\href{/LocalNumberField/37.7.0.1}{7} }^{3}$ $21$ $21$ $21$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ $21$

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
7Data not computed
$31$31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$