↑

Properties

Label 21.3.48900911821...1927.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 71^{18}$
Root discriminant $186.60$
Ramified primes $7, 71$
Class number $49$ (GRH)
Class group $[7, 7]$ (GRH)
Galois group $C_3\times D_7$ (as 21T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![176307053, 1206871419, 2567288464, 2465480719, 867279238, -214511808, -162134041, 40153179, 26343578, 1794625, -18483, 556052, -195159, -62573, -262, 6276, -2633, -326, -34, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 21*x^19 - 34*x^18 - 326*x^17 - 2633*x^16 + 6276*x^15 - 262*x^14 - 62573*x^13 - 195159*x^12 + 556052*x^11 - 18483*x^10 + 1794625*x^9 + 26343578*x^8 + 40153179*x^7 - 162134041*x^6 - 214511808*x^5 + 867279238*x^4 + 2465480719*x^3 + 2567288464*x^2 + 1206871419*x + 176307053)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 34*x^18 - 326*x^17 - 2633*x^16 + 6276*x^15 - 262*x^14 - 62573*x^13 - 195159*x^12 + 556052*x^11 - 18483*x^10 + 1794625*x^9 + 26343578*x^8 + 40153179*x^7 - 162134041*x^6 - 214511808*x^5 + 867279238*x^4 + 2465480719*x^3 + 2567288464*x^2 + 1206871419*x + 176307053, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} - 326 x^{17} - 2633 x^{16} + 6276 x^{15} - 262 x^{14} - 62573 x^{13} - 195159 x^{12} + 556052 x^{11} - 18483 x^{10} + 1794625 x^{9} + 26343578 x^{8} + 40153179 x^{7} - 162134041 x^{6} - 214511808 x^{5} + 867279238 x^{4} + 2465480719 x^{3} + 2567288464 x^{2} + 1206871419 x + 176307053 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-489009118212301150746875247421551446873029171927=-\,7^{17}\cdot 71^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $186.60$
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:  $7, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{13} a^{17} + \frac{2}{13} a^{16} + \frac{5}{13} a^{14} + \frac{5}{13} a^{13} - \frac{1}{13} a^{12} + \frac{1}{13} a^{11} - \frac{6}{13} a^{10} - \frac{6}{13} a^{9} - \frac{5}{13} a^{8} - \frac{3}{13} a^{7} + \frac{2}{13} a^{6} + \frac{6}{13} a^{5} + \frac{3}{13} a^{4} + \frac{2}{13} a^{3} + \frac{1}{13} a^{2} + \frac{5}{13} a$, $\frac{1}{923} a^{18} + \frac{14}{923} a^{17} + \frac{310}{923} a^{16} - \frac{138}{923} a^{15} + \frac{24}{71} a^{14} - \frac{318}{923} a^{13} + \frac{262}{923} a^{12} + \frac{175}{923} a^{11} + \frac{15}{71} a^{10} - \frac{259}{923} a^{9} - \frac{440}{923} a^{8} + \frac{200}{923} a^{7} - \frac{48}{923} a^{6} - \frac{185}{923} a^{5} - \frac{313}{923} a^{4} - \frac{105}{923} a^{3} + \frac{368}{923} a^{2} + \frac{190}{923} a + \frac{25}{71}$, $\frac{1}{2363803} a^{19} + \frac{799}{2363803} a^{18} - \frac{77095}{2363803} a^{17} + \frac{748519}{2363803} a^{16} + \frac{718067}{2363803} a^{15} + \frac{284433}{2363803} a^{14} - \frac{785489}{2363803} a^{13} - \frac{557689}{2363803} a^{12} + \frac{209777}{2363803} a^{11} + \frac{895476}{2363803} a^{10} - \frac{437629}{2363803} a^{9} + \frac{302604}{2363803} a^{8} + \frac{210770}{2363803} a^{7} + \frac{151776}{2363803} a^{6} - \frac{1010957}{2363803} a^{5} + \frac{470154}{2363803} a^{4} - \frac{608664}{2363803} a^{3} - \frac{168525}{2363803} a^{2} - \frac{74827}{181831} a + \frac{4366}{13987}$, $\frac{1}{420549673518180616372549218435551567393891926858537510456267300366073181207344936613827} a^{20} + \frac{52662440073945439378933364757900290549683451453714151110348879084522737551152278}{420549673518180616372549218435551567393891926858537510456267300366073181207344936613827} a^{19} + \frac{33756168830222092995173029339511826767187316388917515565814683840878791747953597686}{420549673518180616372549218435551567393891926858537510456267300366073181207344936613827} a^{18} + \frac{4998828584383300709742120447487400224015509971573065156550902558236765615376101304258}{420549673518180616372549218435551567393891926858537510456267300366073181207344936613827} a^{17} + \frac{155361059653556588734069289165138542758638775664121403170483175377793913524069627005874}{420549673518180616372549218435551567393891926858537510456267300366073181207344936613827} a^{16} + \frac{129529983513706195694912159263393279356318654619365396194374725438611386261166282802152}{420549673518180616372549218435551567393891926858537510456267300366073181207344936613827} a^{15} + \frac{12139497910148077394937151174047817810595669356608883216769134519623538629025215819829}{420549673518180616372549218435551567393891926858537510456267300366073181207344936613827} a^{14} - \frac{45056840296258548407426919032440466163949785544679250626437710534278739472740679182603}{420549673518180616372549218435551567393891926858537510456267300366073181207344936613827} a^{13} - \frac{129396004056804881772868689420417511869854307207986575320965553747728287655271858128301}{420549673518180616372549218435551567393891926858537510456267300366073181207344936613827} a^{12} + \frac{26788907554683775494067653751646654641394773400840471536040860676801088635515239287167}{420549673518180616372549218435551567393891926858537510456267300366073181207344936613827} a^{11} + \frac{179424184741239799935547463684579950126377274363748531864745871326609624271860336738448}{420549673518180616372549218435551567393891926858537510456267300366073181207344936613827} a^{10} + \frac{171432285762479166156254588466996239754572217233675344614691885473173869349892484745913}{420549673518180616372549218435551567393891926858537510456267300366073181207344936613827} a^{9} + \frac{177043090646635698272783590610427589710592312469848240617923583288459573703602900310564}{420549673518180616372549218435551567393891926858537510456267300366073181207344936613827} a^{8} + \frac{59130740742587260379379347987020909074645505151234394462377366053573616575856692376184}{420549673518180616372549218435551567393891926858537510456267300366073181207344936613827} a^{7} - \frac{155132107362333117043869555167411908284308110616543259047320552130708653188746313454222}{420549673518180616372549218435551567393891926858537510456267300366073181207344936613827} a^{6} + \frac{43549144025478367846716874101976911408391828868294395722209857664720421896348583108351}{420549673518180616372549218435551567393891926858537510456267300366073181207344936613827} a^{5} - \frac{51651766174175884221279729614156439195652884421762340438442064136265255716431634933625}{420549673518180616372549218435551567393891926858537510456267300366073181207344936613827} a^{4} + \frac{120978493117184904186956401632252563623703076058792280238587618875639508051008258302194}{420549673518180616372549218435551567393891926858537510456267300366073181207344936613827} a^{3} + \frac{12201987028853473907140270949585288191365124382685856673671631269874686141592747245176}{32349974886013893567119170648888582107222455912195193112020561566621013939026533585679} a^{2} - \frac{1111383551125810960778920143678213714062971592907053729329029352728668103485703466257}{2488459606616453351316859280683737085170958147091937931693889351278539533771271814283} a - \frac{48616846297768101266599266455564790745666455550162615153284537319125464934945012342}{191419969739727180870527636975672083474689088237841379361068411636810733367020908791}$

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

Class group and class number

$C_{7}\times C_{7}$, which has order $49$ (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:  \( 85935385452559.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times D_7$ (as 21T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 15 conjugacy class representatives for $C_3\times D_7$
Character table for $C_3\times D_7$

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.1.43938397384903.1

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 $21$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R $21$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$71$71.7.6.7$x^{7} - 4544$$7$$1$$6$$C_7$$[\ ]_{7}$
71.7.6.7$x^{7} - 4544$$7$$1$$6$$C_7$$[\ ]_{7}$
71.7.6.7$x^{7} - 4544$$7$$1$$6$$C_7$$[\ ]_{7}$