Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^19 - 2*x^18 + 15*x^17 + 6*x^16 - 23*x^15 - 30*x^14 + 23*x^13 + 44*x^12 + 12*x^11 - 43*x^10 - 61*x^9 + 51*x^8 + 34*x^7 - 5*x^6 - 30*x^5 - 9*x^4 + 13*x^3 + 7*x^2 - 1)
gp: K = bnfinit(y^21 - 3*y^19 - 2*y^18 + 15*y^17 + 6*y^16 - 23*y^15 - 30*y^14 + 23*y^13 + 44*y^12 + 12*y^11 - 43*y^10 - 61*y^9 + 51*y^8 + 34*y^7 - 5*y^6 - 30*y^5 - 9*y^4 + 13*y^3 + 7*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 3*x^19 - 2*x^18 + 15*x^17 + 6*x^16 - 23*x^15 - 30*x^14 + 23*x^13 + 44*x^12 + 12*x^11 - 43*x^10 - 61*x^9 + 51*x^8 + 34*x^7 - 5*x^6 - 30*x^5 - 9*x^4 + 13*x^3 + 7*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 3*x^19 - 2*x^18 + 15*x^17 + 6*x^16 - 23*x^15 - 30*x^14 + 23*x^13 + 44*x^12 + 12*x^11 - 43*x^10 - 61*x^9 + 51*x^8 + 34*x^7 - 5*x^6 - 30*x^5 - 9*x^4 + 13*x^3 + 7*x^2 - 1)
\( x^{21} - 3 x^{19} - 2 x^{18} + 15 x^{17} + 6 x^{16} - 23 x^{15} - 30 x^{14} + 23 x^{13} + 44 x^{12} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[5, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(41784109442335295334261365153\)
\(\medspace = 23^{7}\cdot 107^{3}\cdot 21557^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(23.06\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $23^{1/2}107^{1/2}21557^{1/2}\approx 7283.665080163969$ | ||
| Ramified primes: |
\(23\), \(107\), \(21557\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{53051777}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3}a^{18}+\frac{1}{3}a^{16}-\frac{1}{3}a^{15}+\frac{1}{3}a^{14}+\frac{1}{3}a^{12}+\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{19}+\frac{1}{3}a^{17}-\frac{1}{3}a^{16}+\frac{1}{3}a^{15}+\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{34045695873057}a^{20}+\frac{458051679451}{34045695873057}a^{19}+\frac{845742090404}{11348565291019}a^{18}-\frac{5669586646814}{11348565291019}a^{17}-\frac{12785298772183}{34045695873057}a^{16}+\frac{5873170515638}{34045695873057}a^{15}-\frac{4918559842192}{11348565291019}a^{14}+\frac{11949733820195}{34045695873057}a^{13}-\frac{23332136950}{11348565291019}a^{12}-\frac{2655751095264}{11348565291019}a^{11}+\frac{3136665381098}{34045695873057}a^{10}+\frac{314533635275}{1031687753729}a^{9}-\frac{4784078147466}{11348565291019}a^{8}-\frac{6925659998}{3095063261187}a^{7}-\frac{8915169087736}{34045695873057}a^{6}+\frac{9397387192675}{34045695873057}a^{5}-\frac{9598898515352}{34045695873057}a^{4}+\frac{9021023073407}{34045695873057}a^{3}-\frac{3905340730367}{11348565291019}a^{2}-\frac{893528499562}{34045695873057}a+\frac{285466716407}{34045695873057}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $12$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{3036942159536}{34045695873057}a^{20}-\frac{7270984132547}{34045695873057}a^{19}-\frac{4103957412913}{34045695873057}a^{18}+\frac{14871246617816}{34045695873057}a^{17}+\frac{46024033380625}{34045695873057}a^{16}-\frac{97100951078678}{34045695873057}a^{15}-\frac{39125579753140}{34045695873057}a^{14}+\frac{29436591046421}{11348565291019}a^{13}+\frac{179587060219801}{34045695873057}a^{12}-\frac{142230956761258}{34045695873057}a^{11}-\frac{52010418380234}{11348565291019}a^{10}-\frac{6070518307129}{3095063261187}a^{9}+\frac{144907771266275}{34045695873057}a^{8}+\frac{13159801451617}{1031687753729}a^{7}-\frac{500025579886999}{34045695873057}a^{6}+\frac{7652314574555}{34045695873057}a^{5}-\frac{6083965610381}{11348565291019}a^{4}+\frac{132519201866047}{34045695873057}a^{3}+\frac{31885279195261}{34045695873057}a^{2}-\frac{68714398711436}{34045695873057}a-\frac{3555764056773}{11348565291019}$, $\frac{2839594961648}{34045695873057}a^{20}-\frac{4848676375484}{34045695873057}a^{19}-\frac{7411692737881}{34045695873057}a^{18}+\frac{6181727032616}{34045695873057}a^{17}+\frac{47056907376343}{34045695873057}a^{16}-\frac{50992199749352}{34045695873057}a^{15}-\frac{67049124002218}{34045695873057}a^{14}-\frac{82573220071}{11348565291019}a^{13}+\frac{144535077578503}{34045695873057}a^{12}+\frac{16111212406178}{34045695873057}a^{11}-\frac{13906136976702}{11348565291019}a^{10}-\frac{10741061973541}{3095063261187}a^{9}-\frac{67918115624851}{34045695873057}a^{8}+\frac{8615012105097}{1031687753729}a^{7}-\frac{171160149343585}{34045695873057}a^{6}+\frac{70793825827055}{34045695873057}a^{5}-\frac{13584923052365}{11348565291019}a^{4}+\frac{24168699816193}{34045695873057}a^{3}+\frac{2785749741361}{34045695873057}a^{2}+\frac{8471477018425}{34045695873057}a+\frac{3716933507950}{11348565291019}$, $\frac{5006869065695}{34045695873057}a^{20}-\frac{289273820251}{11348565291019}a^{19}-\frac{4690689092503}{11348565291019}a^{18}-\frac{6257842047689}{34045695873057}a^{17}+\frac{24783331570490}{11348565291019}a^{16}+\frac{12185402876762}{34045695873057}a^{15}-\frac{36130711141979}{11348565291019}a^{14}-\frac{108623776732261}{34045695873057}a^{13}+\frac{127448740776934}{34045695873057}a^{12}+\frac{50354873690731}{11348565291019}a^{11}+\frac{17508925298450}{34045695873057}a^{10}-\frac{14921875847122}{3095063261187}a^{9}-\frac{77487140850442}{11348565291019}a^{8}+\frac{24550044170803}{3095063261187}a^{7}+\frac{36501447292202}{34045695873057}a^{6}-\frac{58277842674539}{34045695873057}a^{5}-\frac{73033826071630}{34045695873057}a^{4}+\frac{4549799894923}{34045695873057}a^{3}+\frac{6183900884453}{34045695873057}a^{2}+\frac{3036942159536}{34045695873057}a+\frac{26774711740510}{34045695873057}$, $\frac{8928073896221}{34045695873057}a^{20}+\frac{294359636061}{11348565291019}a^{19}-\frac{5757642801908}{11348565291019}a^{18}-\frac{20433047340110}{34045695873057}a^{17}+\frac{35428649848497}{11348565291019}a^{16}+\frac{50177648600894}{34045695873057}a^{15}-\frac{21170539065197}{11348565291019}a^{14}-\frac{240521064763927}{34045695873057}a^{13}-\frac{8052660401048}{34045695873057}a^{12}+\frac{57582166695857}{11348565291019}a^{11}+\frac{338135674535477}{34045695873057}a^{10}-\frac{4877661785824}{3095063261187}a^{9}-\frac{161875130424417}{11348565291019}a^{8}+\frac{9275772346501}{3095063261187}a^{7}-\frac{117134326309708}{34045695873057}a^{6}+\frac{436021653832828}{34045695873057}a^{5}-\frac{130417163043430}{34045695873057}a^{4}-\frac{169215974248601}{34045695873057}a^{3}-\frac{62579218899715}{34045695873057}a^{2}+\frac{26828351540960}{34045695873057}a+\frac{85105940064217}{34045695873057}$, $\frac{1451992039996}{11348565291019}a^{20}+\frac{7895319488611}{34045695873057}a^{19}-\frac{7683912729307}{34045695873057}a^{18}-\frac{28096125292322}{34045695873057}a^{17}+\frac{39525753161395}{34045695873057}a^{16}+\frac{125582620652693}{34045695873057}a^{15}+\frac{4350733004924}{34045695873057}a^{14}-\frac{237337034713703}{34045695873057}a^{13}-\frac{53191535168880}{11348565291019}a^{12}+\frac{202659093422936}{34045695873057}a^{11}+\frac{318376589285554}{34045695873057}a^{10}+\frac{798663069499}{1031687753729}a^{9}-\frac{370213042913482}{34045695873057}a^{8}-\frac{19700955065740}{3095063261187}a^{7}+\frac{81791890064129}{11348565291019}a^{6}+\frac{198698765385223}{34045695873057}a^{5}-\frac{126674696868659}{34045695873057}a^{4}-\frac{43960190946583}{11348565291019}a^{3}-\frac{23059183230199}{34045695873057}a^{2}+\frac{12724992465758}{34045695873057}a+\frac{23909966767976}{34045695873057}$, $\frac{488792662023}{11348565291019}a^{20}-\frac{305109951711}{11348565291019}a^{19}-\frac{914290020993}{11348565291019}a^{18}-\frac{1048718896151}{11348565291019}a^{17}+\frac{5967225614574}{11348565291019}a^{16}-\frac{260807521124}{11348565291019}a^{15}-\frac{2767412424870}{11348565291019}a^{14}-\frac{17662252605995}{11348565291019}a^{13}-\frac{241995364046}{11348565291019}a^{12}+\frac{14365399081485}{11348565291019}a^{11}+\frac{28367390523932}{11348565291019}a^{10}-\frac{1192628213608}{1031687753729}a^{9}-\frac{39438332472696}{11348565291019}a^{8}+\frac{847235790731}{1031687753729}a^{7}-\frac{14876227327384}{11348565291019}a^{6}+\frac{65734317964237}{11348565291019}a^{5}-\frac{32009222564524}{11348565291019}a^{4}-\frac{10603299876563}{11348565291019}a^{3}+\frac{105221672475}{11348565291019}a^{2}+\frac{5096693433356}{11348565291019}a+\frac{16830855640546}{11348565291019}$, $\frac{28195573066732}{34045695873057}a^{20}+\frac{3947365352242}{34045695873057}a^{19}-\frac{76006023742762}{34045695873057}a^{18}-\frac{21144283525529}{11348565291019}a^{17}+\frac{391530889757146}{34045695873057}a^{16}+\frac{67046044583656}{11348565291019}a^{15}-\frac{508384195778476}{34045695873057}a^{14}-\frac{825082525838959}{34045695873057}a^{13}+\frac{380139025713014}{34045695873057}a^{12}+\frac{10\!\cdots\!05}{34045695873057}a^{11}+\frac{575766352380023}{34045695873057}a^{10}-\frac{69314831414615}{3095063261187}a^{9}-\frac{16\!\cdots\!07}{34045695873057}a^{8}+\frac{85082451498508}{3095063261187}a^{7}+\frac{523063689826078}{34045695873057}a^{6}+\frac{51217004639431}{11348565291019}a^{5}-\frac{460246242142702}{34045695873057}a^{4}-\frac{312109698288511}{34045695873057}a^{3}+\frac{48292032592512}{11348565291019}a^{2}+\frac{16832664986963}{11348565291019}a+\frac{30816183343372}{34045695873057}$, $\frac{3012549234943}{11348565291019}a^{20}-\frac{22082975104598}{34045695873057}a^{19}-\frac{4948398015492}{11348565291019}a^{18}+\frac{39181322398693}{34045695873057}a^{17}+\frac{143346132096476}{34045695873057}a^{16}-\frac{275568360153569}{34045695873057}a^{15}-\frac{47804040838262}{11348565291019}a^{14}+\frac{176306081791090}{34045695873057}a^{13}+\frac{552172392897379}{34045695873057}a^{12}-\frac{91672721805092}{11348565291019}a^{11}-\frac{377562368304281}{34045695873057}a^{10}-\frac{31337232640684}{3095063261187}a^{9}+\frac{83959567645962}{11348565291019}a^{8}+\frac{115462744312868}{3095063261187}a^{7}-\frac{12\!\cdots\!02}{34045695873057}a^{6}+\frac{221035702307384}{34045695873057}a^{5}-\frac{78147787340619}{11348565291019}a^{4}+\frac{122178854324760}{11348565291019}a^{3}+\frac{123414024317066}{34045695873057}a^{2}-\frac{203844359725202}{34045695873057}a+\frac{8377888615560}{11348565291019}$, $\frac{23362699391839}{34045695873057}a^{20}+\frac{12217376711962}{34045695873057}a^{19}-\frac{21488653189851}{11348565291019}a^{18}-\frac{27790376003125}{11348565291019}a^{17}+\frac{314132950968305}{34045695873057}a^{16}+\frac{313795075486262}{34045695873057}a^{15}-\frac{131300353050913}{11348565291019}a^{14}-\frac{960771883696342}{34045695873057}a^{13}+\frac{37448707504137}{11348565291019}a^{12}+\frac{395517857959605}{11348565291019}a^{11}+\frac{847334401389269}{34045695873057}a^{10}-\frac{23152413923123}{1031687753729}a^{9}-\frac{601544427754445}{11348565291019}a^{8}+\frac{36325527352195}{3095063261187}a^{7}+\frac{11\!\cdots\!69}{34045695873057}a^{6}+\frac{438621074131450}{34045695873057}a^{5}-\frac{837567634153661}{34045695873057}a^{4}-\frac{424424867841262}{34045695873057}a^{3}+\frac{55992773709108}{11348565291019}a^{2}+\frac{210514507995971}{34045695873057}a+\frac{66668868172355}{34045695873057}$, $\frac{47161537524}{1031687753729}a^{20}+\frac{213365540647}{3095063261187}a^{19}-\frac{358177738553}{3095063261187}a^{18}-\frac{806486395751}{3095063261187}a^{17}+\frac{446293564961}{1031687753729}a^{16}+\frac{1164445337882}{1031687753729}a^{15}-\frac{925591773932}{3095063261187}a^{14}-\frac{6411805869644}{3095063261187}a^{13}-\frac{5426092380523}{3095063261187}a^{12}+\frac{5133997659220}{3095063261187}a^{11}+\frac{9672439080619}{3095063261187}a^{10}+\frac{7386726826691}{3095063261187}a^{9}-\frac{7893649713167}{3095063261187}a^{8}-\frac{6358920371606}{3095063261187}a^{7}+\frac{1815043640309}{3095063261187}a^{6}-\frac{70950352469}{1031687753729}a^{5}+\frac{6293985293747}{3095063261187}a^{4}-\frac{1950752202296}{1031687753729}a^{3}+\frac{141302212724}{3095063261187}a^{2}-\frac{2062784559642}{1031687753729}a+\frac{251434551232}{3095063261187}$, $\frac{1491236218349}{3095063261187}a^{20}-\frac{1561561495322}{3095063261187}a^{19}-\frac{3474195079793}{3095063261187}a^{18}+\frac{1606462979018}{3095063261187}a^{17}+\frac{7556537219724}{1031687753729}a^{16}-\frac{15940484752564}{3095063261187}a^{15}-\frac{28700106629483}{3095063261187}a^{14}-\frac{1691468705348}{1031687753729}a^{13}+\frac{19666054203611}{1031687753729}a^{12}+\frac{5322757561561}{3095063261187}a^{11}-\frac{7998313500369}{1031687753729}a^{10}-\frac{15628964440467}{1031687753729}a^{9}-\frac{22185612325067}{3095063261187}a^{8}+\frac{42625610586328}{1031687753729}a^{7}-\frac{77349438969875}{3095063261187}a^{6}+\frac{1530200846962}{3095063261187}a^{5}-\frac{13753623202793}{3095063261187}a^{4}+\frac{20075065197766}{3095063261187}a^{3}+\frac{4469291017768}{3095063261187}a^{2}-\frac{14101004718346}{3095063261187}a+\frac{2988144439403}{3095063261187}$, $\frac{2143495226266}{34045695873057}a^{20}-\frac{2123646712321}{11348565291019}a^{19}-\frac{15448512936847}{34045695873057}a^{18}+\frac{11535334043564}{34045695873057}a^{17}+\frac{64855338214013}{34045695873057}a^{16}-\frac{59468101346890}{34045695873057}a^{15}-\frac{198947671364521}{34045695873057}a^{14}+\frac{3545781256060}{34045695873057}a^{13}+\frac{342016314629941}{34045695873057}a^{12}+\frac{198375487567649}{34045695873057}a^{11}-\frac{278817713161859}{34045695873057}a^{10}-\frac{35535050671174}{3095063261187}a^{9}-\frac{75835527435040}{34045695873057}a^{8}+\frac{54523171564562}{3095063261187}a^{7}+\frac{58057772392865}{11348565291019}a^{6}-\frac{444321348908213}{34045695873057}a^{5}-\frac{64827566222809}{34045695873057}a^{4}+\frac{64990116257795}{34045695873057}a^{3}+\frac{209882131597231}{34045695873057}a^{2}+\frac{2081285736542}{34045695873057}a-\frac{34416987482072}{34045695873057}$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | \( 895675.424859 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{8}\cdot 895675.424859 \cdot 1}{2\cdot\sqrt{41784109442335295334261365153}}\cr\approx \mathstrut & 0.170295829661 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^19 - 2*x^18 + 15*x^17 + 6*x^16 - 23*x^15 - 30*x^14 + 23*x^13 + 44*x^12 + 12*x^11 - 43*x^10 - 61*x^9 + 51*x^8 + 34*x^7 - 5*x^6 - 30*x^5 - 9*x^4 + 13*x^3 + 7*x^2 - 1)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^21 - 3*x^19 - 2*x^18 + 15*x^17 + 6*x^16 - 23*x^15 - 30*x^14 + 23*x^13 + 44*x^12 + 12*x^11 - 43*x^10 - 61*x^9 + 51*x^8 + 34*x^7 - 5*x^6 - 30*x^5 - 9*x^4 + 13*x^3 + 7*x^2 - 1, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^21 - 3*x^19 - 2*x^18 + 15*x^17 + 6*x^16 - 23*x^15 - 30*x^14 + 23*x^13 + 44*x^12 + 12*x^11 - 43*x^10 - 61*x^9 + 51*x^8 + 34*x^7 - 5*x^6 - 30*x^5 - 9*x^4 + 13*x^3 + 7*x^2 - 1);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 3*x^19 - 2*x^18 + 15*x^17 + 6*x^16 - 23*x^15 - 30*x^14 + 23*x^13 + 44*x^12 + 12*x^11 - 43*x^10 - 61*x^9 + 51*x^8 + 34*x^7 - 5*x^6 - 30*x^5 - 9*x^4 + 13*x^3 + 7*x^2 - 1);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$S_3\times S_7$ (as 21T74):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A non-solvable group of order 30240 |
| The 45 conjugacy class representatives for $S_3\times S_7$ |
| Character table for $S_3\times S_7$ is not computed |
Intermediate fields
| 3.1.23.1, 7.5.2306599.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 42 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $21$ | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{3}$ | ${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | $21$ | ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.7.0.1}{7} }$ | R | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{5}$ | $21$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{5}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{10}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(23\)
| 23.7.0.1 | $x^{7} + 21 x + 18$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |
| 23.14.7.2 | $x^{14} - 21574 x^{13} + 234933293 x^{12} - 1215548742590 x^{11} + 4112603302919993 x^{10} - 2725136947640868418 x^{9} + 363970304488058959670 x^{8} + 412439955621146008597774 x^{7} + 8371317003225356072410 x^{6} - 1441597445302019393122 x^{5} + 50038044386627554831 x^{4} - 340160375675128190 x^{3} + 1512111255867499 x^{2} - 3193726269286 x + 3404825447$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ | |
|
\(107\)
| $\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 107.2.0.1 | $x^{2} + 103 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 107.2.0.1 | $x^{2} + 103 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 107.2.0.1 | $x^{2} + 103 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 107.2.0.1 | $x^{2} + 103 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 107.2.0.1 | $x^{2} + 103 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 107.2.1.1 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 107.2.0.1 | $x^{2} + 103 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 107.2.0.1 | $x^{2} + 103 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 107.4.2.1 | $x^{4} + 206 x^{3} + 10827 x^{2} + 22454 x + 1146188$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(21557\)
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $6$ | $2$ | $3$ | $3$ |