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Properties

Label 21.3.60558130520...9727.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 379^{18}$
Root discriminant $784.13$
Ramified primes $7, 379$
Class number $16807$ (GRH)
Class group $[7, 7, 7, 7, 7]$ (GRH)
Galois group $F_7$ (as 21T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![122588921675584, -276127419440096, 193183257790096, -12644866212328, -48713312542188, 28259771083054, -7765632654107, 1235524808890, -124794841242, 16406088995, -5119024426, 1254295567, -167147434, 9992983, -976238, 370487, -68226, 5335, -34, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 5335*x^17 - 68226*x^16 + 370487*x^15 - 976238*x^14 + 9992983*x^13 - 167147434*x^12 + 1254295567*x^11 - 5119024426*x^10 + 16406088995*x^9 - 124794841242*x^8 + 1235524808890*x^7 - 7765632654107*x^6 + 28259771083054*x^5 - 48713312542188*x^4 - 12644866212328*x^3 + 193183257790096*x^2 - 276127419440096*x + 122588921675584)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 5335*x^17 - 68226*x^16 + 370487*x^15 - 976238*x^14 + 9992983*x^13 - 167147434*x^12 + 1254295567*x^11 - 5119024426*x^10 + 16406088995*x^9 - 124794841242*x^8 + 1235524808890*x^7 - 7765632654107*x^6 + 28259771083054*x^5 - 48713312542188*x^4 - 12644866212328*x^3 + 193183257790096*x^2 - 276127419440096*x + 122588921675584, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 5335 x^{17} - 68226 x^{16} + 370487 x^{15} - 976238 x^{14} + 9992983 x^{13} - 167147434 x^{12} + 1254295567 x^{11} - 5119024426 x^{10} + 16406088995 x^{9} - 124794841242 x^{8} + 1235524808890 x^{7} - 7765632654107 x^{6} + 28259771083054 x^{5} - 48713312542188 x^{4} - 12644866212328 x^{3} + 193183257790096 x^{2} - 276127419440096 x + 122588921675584 \)

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:  \(-6055813052022442741267437386720431302520208823407199686319727=-\,7^{17}\cdot 379^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $784.13$
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:  $7, 379$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{3}$, $\frac{1}{3032} a^{18} + \frac{67}{3032} a^{17} - \frac{131}{3032} a^{16} + \frac{27}{758} a^{15} + \frac{283}{3032} a^{14} - \frac{157}{758} a^{13} - \frac{741}{3032} a^{12} + \frac{151}{758} a^{11} - \frac{425}{3032} a^{10} - \frac{34}{379} a^{9} + \frac{515}{3032} a^{8} - \frac{39}{379} a^{7} + \frac{243}{3032} a^{6} - \frac{4}{379} a^{5} - \frac{253}{1516} a^{4} + \frac{857}{3032} a^{3} + \frac{553}{1516} a^{2} - \frac{89}{379} a - \frac{58}{379}$, $\frac{1}{685232} a^{19} - \frac{29}{685232} a^{18} - \frac{66445}{685232} a^{17} - \frac{10473}{171308} a^{16} - \frac{41921}{685232} a^{15} - \frac{6317}{85654} a^{14} + \frac{161119}{685232} a^{13} - \frac{20723}{171308} a^{12} + \frac{56807}{685232} a^{11} - \frac{2773}{42827} a^{10} + \frac{88783}{685232} a^{9} - \frac{4815}{42827} a^{8} - \frac{157789}{685232} a^{7} - \frac{20410}{42827} a^{6} - \frac{13119}{342616} a^{5} - \frac{215867}{685232} a^{4} - \frac{2562}{42827} a^{3} + \frac{37425}{85654} a^{2} - \frac{42679}{85654} a - \frac{6312}{42827}$, $\frac{1}{76587880848914252043473383372180725617604275683047703779988735959343026828176686359223598554679618690415581091085754063057632} a^{20} + \frac{41834719139910887533473398765626364522754885680461014673957571359142923466640795714924600365799650305738117591336218005}{76587880848914252043473383372180725617604275683047703779988735959343026828176686359223598554679618690415581091085754063057632} a^{19} + \frac{12332917164757393777707874646968953373895800648252981153777919208828240211024481010435253380843244137611142323738003031025}{76587880848914252043473383372180725617604275683047703779988735959343026828176686359223598554679618690415581091085754063057632} a^{18} - \frac{350778443589697332028113158822588304658293819012655490568215626422865671345997666752694983108157323492858393739089123360003}{38293940424457126021736691686090362808802137841523851889994367979671513414088343179611799277339809345207790545542877031528816} a^{17} - \frac{2092274473939845220322010295148971566058811791635695663633757502359651910264409098348774453930348475502929680214351871672297}{76587880848914252043473383372180725617604275683047703779988735959343026828176686359223598554679618690415581091085754063057632} a^{16} + \frac{510322030807815334000125058072550620638433397715895822643928824180800167366918943039625236283728313572125260593037559731117}{38293940424457126021736691686090362808802137841523851889994367979671513414088343179611799277339809345207790545542877031528816} a^{15} - \frac{4146272585819854582092756418475170552401193634913776298906068026483173803927785675510980224457963426183253330442313368543537}{76587880848914252043473383372180725617604275683047703779988735959343026828176686359223598554679618690415581091085754063057632} a^{14} + \frac{8572162571501534461183617976445985675861040081223025933310439314602827020892789669259319757644069749395410054147378845009855}{38293940424457126021736691686090362808802137841523851889994367979671513414088343179611799277339809345207790545542877031528816} a^{13} - \frac{17367631158327534995658589760761247917909633252351344939190194255840392025919088141141514171297874931308787461800657065384129}{76587880848914252043473383372180725617604275683047703779988735959343026828176686359223598554679618690415581091085754063057632} a^{12} - \frac{5053634638606737629750570271094137030197840023992138682005731920266022961217350084247930465538397081538418928536472017625815}{38293940424457126021736691686090362808802137841523851889994367979671513414088343179611799277339809345207790545542877031528816} a^{11} - \frac{874073616914634169807552885187706084432805022920386464314430194533304743096899636841458043428321177091238028189399458301401}{76587880848914252043473383372180725617604275683047703779988735959343026828176686359223598554679618690415581091085754063057632} a^{10} + \frac{5763618786974069777196409428301220408916697166690295508576431928914631651088185098504740290477512500765624343953480726234057}{38293940424457126021736691686090362808802137841523851889994367979671513414088343179611799277339809345207790545542877031528816} a^{9} - \frac{17236259606107189528617006575002375766440186292872150739968275238182956232896362606987854161034545188734704865707316827297749}{76587880848914252043473383372180725617604275683047703779988735959343026828176686359223598554679618690415581091085754063057632} a^{8} + \frac{708039310922877921514325088480519218818420804238828302311115677934697366666094579017366898537824620769692858863417878025417}{38293940424457126021736691686090362808802137841523851889994367979671513414088343179611799277339809345207790545542877031528816} a^{7} + \frac{8639603696478506248418167000999024784764364909104270175537588651770102600916962731716243486152667612188080196723739474885825}{38293940424457126021736691686090362808802137841523851889994367979671513414088343179611799277339809345207790545542877031528816} a^{6} - \frac{27243214869314230618898712503596591038958810473406440189775582176540719484530559005954400049644641235434323282719000587415659}{76587880848914252043473383372180725617604275683047703779988735959343026828176686359223598554679618690415581091085754063057632} a^{5} - \frac{16389369813377821191889458016165616132111949846888195839348851264919429771446868854901589101206213638200340031540496025988779}{38293940424457126021736691686090362808802137841523851889994367979671513414088343179611799277339809345207790545542877031528816} a^{4} - \frac{7685111371277497937513019005619723291271097669241729659336672629592889854066335289313710797150313413052629896220261900947855}{19146970212228563010868345843045181404401068920761925944997183989835756707044171589805899638669904672603895272771438515764408} a^{3} + \frac{150411313829731244290148206943013514360328216349369817743277838094348244483338671433264382596126241065799477313679658982516}{2393371276528570376358543230380647675550133615095240743124647998729469588380521448725737454833738084075486909096429814470551} a^{2} - \frac{1168683273431578021705561138775431904700278793935203067863495252688909192671610707568418153940827085858493014192686912519969}{2393371276528570376358543230380647675550133615095240743124647998729469588380521448725737454833738084075486909096429814470551} a - \frac{866062367692539935974412325099210099405450000198764426870265538650265337676977942988595037852735870833208047737685118543155}{2393371276528570376358543230380647675550133615095240743124647998729469588380521448725737454833738084075486909096429814470551}$

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

Class group and class number

$C_{7}\times C_{7}\times C_{7}\times C_{7}\times C_{7}$, which has order $16807$ (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:  \( 2564983668887593500 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_7$ (as 21T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 7 conjugacy class representatives for $F_7$
Character table for $F_7$

Intermediate fields

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

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

Sibling fields

Galois closure: data not computed
Degree 7 sibling: data not computed
Degree 14 sibling: 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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}$
379Data not computed