Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + x^20 + x^19 + x^18 + 5*x^17 - 11*x^16 + 6*x^15 + 19*x^14 - 22*x^13 - 4*x^12 + 6*x^11 + 13*x^10 + 26*x^9 - 75*x^8 - 11*x^7 + 101*x^6 - 32*x^5 - 59*x^4 + 28*x^3 + 15*x^2 - 6*x - 1)
gp: K = bnfinit(y^22 - y^21 + y^20 + y^19 + y^18 + 5*y^17 - 11*y^16 + 6*y^15 + 19*y^14 - 22*y^13 - 4*y^12 + 6*y^11 + 13*y^10 + 26*y^9 - 75*y^8 - 11*y^7 + 101*y^6 - 32*y^5 - 59*y^4 + 28*y^3 + 15*y^2 - 6*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - x^21 + x^20 + x^19 + x^18 + 5*x^17 - 11*x^16 + 6*x^15 + 19*x^14 - 22*x^13 - 4*x^12 + 6*x^11 + 13*x^10 + 26*x^9 - 75*x^8 - 11*x^7 + 101*x^6 - 32*x^5 - 59*x^4 + 28*x^3 + 15*x^2 - 6*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - x^21 + x^20 + x^19 + x^18 + 5*x^17 - 11*x^16 + 6*x^15 + 19*x^14 - 22*x^13 - 4*x^12 + 6*x^11 + 13*x^10 + 26*x^9 - 75*x^8 - 11*x^7 + 101*x^6 - 32*x^5 - 59*x^4 + 28*x^3 + 15*x^2 - 6*x - 1)
\( x^{22} - x^{21} + x^{20} + x^{19} + x^{18} + 5 x^{17} - 11 x^{16} + 6 x^{15} + 19 x^{14} - 22 x^{13} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[4, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-1897242698910438481935564683\)
\(\medspace = -\,47147\cdot 200601609583^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.37\) | 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: | $47147^{1/2}200601609583^{1/2}\approx 97251036.43154505$ | ||
| Ramified primes: |
\(47147\), \(200601609583\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-47147}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{23}a^{20}+\frac{5}{23}a^{19}-\frac{9}{23}a^{18}-\frac{7}{23}a^{16}+\frac{9}{23}a^{15}+\frac{1}{23}a^{14}-\frac{3}{23}a^{13}+\frac{7}{23}a^{12}+\frac{2}{23}a^{11}+\frac{4}{23}a^{10}-\frac{4}{23}a^{9}-\frac{10}{23}a^{8}+\frac{11}{23}a^{7}+\frac{9}{23}a^{5}-\frac{6}{23}a^{4}+\frac{9}{23}a^{3}+\frac{5}{23}a^{2}-\frac{3}{23}a+\frac{4}{23}$, $\frac{1}{140492380027}a^{21}-\frac{489918049}{140492380027}a^{20}+\frac{1670174114}{140492380027}a^{19}+\frac{64023947760}{140492380027}a^{18}-\frac{15049179365}{140492380027}a^{17}-\frac{50331626752}{140492380027}a^{16}+\frac{60766032187}{140492380027}a^{15}-\frac{59800005798}{140492380027}a^{14}+\frac{70000161927}{140492380027}a^{13}-\frac{66127248310}{140492380027}a^{12}+\frac{4594787493}{140492380027}a^{11}-\frac{2545358882}{6108364349}a^{10}+\frac{63508459473}{140492380027}a^{9}-\frac{43533025151}{140492380027}a^{8}+\frac{40914747276}{140492380027}a^{7}+\frac{51904774829}{140492380027}a^{6}-\frac{23326460387}{140492380027}a^{5}-\frac{2514555658}{140492380027}a^{4}-\frac{30149135848}{140492380027}a^{3}-\frac{13950411235}{140492380027}a^{2}-\frac{48112699384}{140492380027}a-\frac{46026854520}{140492380027}$
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: |
$\frac{146135252591}{140492380027}a^{21}-\frac{11393413462}{6108364349}a^{20}+\frac{334836102676}{140492380027}a^{19}-\frac{94416785039}{140492380027}a^{18}+\frac{196299204765}{140492380027}a^{17}+\frac{581743704310}{140492380027}a^{16}-\frac{2094687085931}{140492380027}a^{15}+\frac{2470019229450}{140492380027}a^{14}+\frac{1063458664580}{140492380027}a^{13}-\frac{4203772481453}{140492380027}a^{12}+\frac{2508545834385}{140492380027}a^{11}-\frac{774801190886}{140492380027}a^{10}+\frac{2491424982795}{140492380027}a^{9}+\frac{1906041534055}{140492380027}a^{8}-\frac{12934767090746}{140492380027}a^{7}+\frac{8095753339226}{140492380027}a^{6}+\frac{9711136917935}{140492380027}a^{5}-\frac{12338521454989}{140492380027}a^{4}-\frac{190574776542}{140492380027}a^{3}+\frac{4525666868883}{140492380027}a^{2}-\frac{874800692251}{140492380027}a-\frac{78910713704}{140492380027}$, $\frac{230951054654}{140492380027}a^{21}-\frac{404591326234}{140492380027}a^{20}+\frac{507250128116}{140492380027}a^{19}-\frac{129028618879}{140492380027}a^{18}+\frac{302793330696}{140492380027}a^{17}+\frac{902781980471}{140492380027}a^{16}-\frac{3259333439797}{140492380027}a^{15}+\frac{3688034440814}{140492380027}a^{14}+\frac{1887507297694}{140492380027}a^{13}-\frac{6615677453483}{140492380027}a^{12}+\frac{3536977370947}{140492380027}a^{11}-\frac{889775313959}{140492380027}a^{10}+\frac{3784004731889}{140492380027}a^{9}+\frac{3126404131565}{140492380027}a^{8}-\frac{20199867968825}{140492380027}a^{7}+\frac{11616971544225}{140492380027}a^{6}+\frac{16322045549870}{140492380027}a^{5}-\frac{18984861473906}{140492380027}a^{4}-\frac{69341726537}{6108364349}a^{3}+\frac{7671181616855}{140492380027}a^{2}-\frac{1148479940504}{140492380027}a-\frac{390955532472}{140492380027}$, $\frac{253322986066}{140492380027}a^{21}-\frac{421521813042}{140492380027}a^{20}+\frac{544056074067}{140492380027}a^{19}-\frac{142146003082}{140492380027}a^{18}+\frac{356021265579}{140492380027}a^{17}+\frac{1021012363762}{140492380027}a^{16}-\frac{151593109582}{6108364349}a^{15}+\frac{3842717461563}{140492380027}a^{14}+\frac{1984118636031}{140492380027}a^{13}-\frac{6696383822857}{140492380027}a^{12}+\frac{3701296755771}{140492380027}a^{11}-\frac{1478649070415}{140492380027}a^{10}+\frac{4356522933730}{140492380027}a^{9}+\frac{3981041044404}{140492380027}a^{8}-\frac{21171970793051}{140492380027}a^{7}+\frac{11518388941437}{140492380027}a^{6}+\frac{16237543369752}{140492380027}a^{5}-\frac{18323119633099}{140492380027}a^{4}-\frac{457544273055}{140492380027}a^{3}+\frac{6481567988379}{140492380027}a^{2}-\frac{1600445705051}{140492380027}a-\frac{41987988774}{140492380027}$, $\frac{96050121604}{140492380027}a^{21}-\frac{182195008315}{140492380027}a^{20}+\frac{258156429974}{140492380027}a^{19}-\frac{83424094300}{140492380027}a^{18}+\frac{148021576110}{140492380027}a^{17}+\frac{423854431909}{140492380027}a^{16}-\frac{1380681174936}{140492380027}a^{15}+\frac{1950564067007}{140492380027}a^{14}+\frac{437123620867}{140492380027}a^{13}-\frac{2797941536255}{140492380027}a^{12}+\frac{2441940070816}{140492380027}a^{11}-\frac{822173795463}{140492380027}a^{10}+\frac{1626957796089}{140492380027}a^{9}+\frac{1102930949519}{140492380027}a^{8}-\frac{8470449738822}{140492380027}a^{7}+\frac{7108838428016}{140492380027}a^{6}+\frac{4867219195812}{140492380027}a^{5}-\frac{9769627012335}{140492380027}a^{4}+\frac{2096554077807}{140492380027}a^{3}+\frac{3036702889372}{140492380027}a^{2}-\frac{1614520802367}{140492380027}a-\frac{84295409530}{140492380027}$, $\frac{168198826976}{140492380027}a^{21}-\frac{290733088001}{140492380027}a^{20}+\frac{17194303876}{6108364349}a^{19}-\frac{102698279513}{140492380027}a^{18}+\frac{245602566568}{140492380027}a^{17}+\frac{700088673660}{140492380027}a^{16}-\frac{2322779545167}{140492380027}a^{15}+\frac{2829018099223}{140492380027}a^{14}+\frac{1123278129405}{140492380027}a^{13}-\frac{4714588700035}{140492380027}a^{12}+\frac{2998586986811}{140492380027}a^{11}-\frac{1063324114872}{140492380027}a^{10}+\frac{2605356593312}{140492380027}a^{9}+\frac{2172746838101}{140492380027}a^{8}-\frac{14304941788163}{140492380027}a^{7}+\frac{9348078222914}{140492380027}a^{6}+\frac{10216784078987}{140492380027}a^{5}-\frac{14488511904839}{140492380027}a^{4}+\frac{611475621469}{140492380027}a^{3}+\frac{5400290496041}{140492380027}a^{2}-\frac{1477949927622}{140492380027}a-\frac{253322986066}{140492380027}$, $\frac{295755863567}{140492380027}a^{21}-\frac{496991090350}{140492380027}a^{20}+\frac{29012103801}{6108364349}a^{19}-\frac{197404018731}{140492380027}a^{18}+\frac{460768312882}{140492380027}a^{17}+\frac{1168819930294}{140492380027}a^{16}-\frac{4027669804489}{140492380027}a^{15}+\frac{4632798535071}{140492380027}a^{14}+\frac{2019699925667}{140492380027}a^{13}-\frac{7743563077266}{140492380027}a^{12}+\frac{4563186089844}{140492380027}a^{11}-\frac{2067684607623}{140492380027}a^{10}+\frac{5082486758129}{140492380027}a^{9}+\frac{4158082693677}{140492380027}a^{8}-\frac{24415319659781}{140492380027}a^{7}+\frac{14355410286977}{140492380027}a^{6}+\frac{17656589125400}{140492380027}a^{5}-\frac{21855714056487}{140492380027}a^{4}+\frac{348975054337}{140492380027}a^{3}+\frac{7584426106497}{140492380027}a^{2}-\frac{1974114174799}{140492380027}a-\frac{115815896405}{140492380027}$, $\frac{29715827871}{140492380027}a^{21}-\frac{33068802378}{140492380027}a^{20}+\frac{14090245818}{140492380027}a^{19}+\frac{35983830436}{140492380027}a^{18}-\frac{7166056845}{140492380027}a^{17}+\frac{128426675219}{140492380027}a^{16}-\frac{401070086128}{140492380027}a^{15}+\frac{88372958708}{140492380027}a^{14}+\frac{650573218902}{140492380027}a^{13}-\frac{869467909371}{140492380027}a^{12}-\frac{305242418078}{140492380027}a^{11}+\frac{347885217799}{140492380027}a^{10}+\frac{252434972824}{140492380027}a^{9}+\frac{892774239309}{140492380027}a^{8}-\frac{2591143182611}{140492380027}a^{7}-\frac{597781550642}{140492380027}a^{6}+\frac{3930519127527}{140492380027}a^{5}-\frac{1186290469116}{140492380027}a^{4}-\frac{2291597783730}{140492380027}a^{3}+\frac{1260729488551}{140492380027}a^{2}+\frac{510239344493}{140492380027}a-\frac{235692048932}{140492380027}$, $\frac{174712343959}{140492380027}a^{21}-\frac{14294168946}{6108364349}a^{20}+\frac{451934288065}{140492380027}a^{19}-\frac{167025786394}{140492380027}a^{18}+\frac{295129589015}{140492380027}a^{17}+\frac{669912129669}{140492380027}a^{16}-\frac{2468016432864}{140492380027}a^{15}+\frac{3284304871626}{140492380027}a^{14}+\frac{886028845721}{140492380027}a^{13}-\frac{4924228734910}{140492380027}a^{12}+\frac{3675938978603}{140492380027}a^{11}-\frac{1273841422796}{140492380027}a^{10}+\frac{3117682928521}{140492380027}a^{9}+\frac{1821059076403}{140492380027}a^{8}-\frac{15190851520130}{140492380027}a^{7}+\frac{11425924244242}{140492380027}a^{6}+\frac{9867444912462}{140492380027}a^{5}-\frac{16007057150293}{140492380027}a^{4}+\frac{1607812880934}{140492380027}a^{3}+\frac{5521323823307}{140492380027}a^{2}-\frac{1883971736999}{140492380027}a-\frac{191694414035}{140492380027}$, $\frac{24618261837}{140492380027}a^{21}+\frac{12587359949}{140492380027}a^{20}-\frac{2069250585}{140492380027}a^{19}+\frac{2483884034}{6108364349}a^{18}+\frac{52644844161}{140492380027}a^{17}+\frac{184148065438}{140492380027}a^{16}-\frac{101548585072}{140492380027}a^{15}-\frac{194996178604}{140492380027}a^{14}+\frac{530090031528}{140492380027}a^{13}+\frac{109772802802}{140492380027}a^{12}-\frac{655451564985}{140492380027}a^{11}-\frac{214261459840}{140492380027}a^{10}+\frac{281713647878}{140492380027}a^{9}+\frac{1114059665580}{140492380027}a^{8}-\frac{29873074685}{6108364349}a^{7}-\frac{2564014455608}{140492380027}a^{6}+\frac{1242583656813}{140492380027}a^{5}+\frac{2226966474423}{140492380027}a^{4}-\frac{1357278200904}{140492380027}a^{3}-\frac{972078361453}{140492380027}a^{2}+\frac{550781047316}{140492380027}a+\frac{9254241688}{6108364349}$, $\frac{72305931941}{140492380027}a^{21}-\frac{69907344545}{140492380027}a^{20}+\frac{139915306325}{140492380027}a^{19}-\frac{6317318067}{140492380027}a^{18}+\frac{197934307754}{140492380027}a^{17}+\frac{383821678612}{140492380027}a^{16}-\frac{642934719160}{140492380027}a^{15}+\frac{766281657593}{140492380027}a^{14}+\frac{670785355344}{140492380027}a^{13}-\frac{765098761633}{140492380027}a^{12}+\frac{581226592735}{140492380027}a^{11}-\frac{870234678847}{140492380027}a^{10}+\frac{1436187681405}{140492380027}a^{9}+\frac{1742147449037}{140492380027}a^{8}-\frac{4207646920700}{140492380027}a^{7}+\frac{534846211740}{140492380027}a^{6}+\frac{2261069273510}{140492380027}a^{5}-\frac{1158206643328}{140492380027}a^{4}+\frac{314009645570}{140492380027}a^{3}-\frac{1007357492846}{140492380027}a^{2}-\frac{476691132635}{140492380027}a+\frac{572550671643}{140492380027}$, $\frac{17917311977}{140492380027}a^{21}-\frac{112944410992}{140492380027}a^{20}+\frac{97549914726}{140492380027}a^{19}-\frac{132426189623}{140492380027}a^{18}-\frac{45344402929}{140492380027}a^{17}-\frac{115252057008}{140492380027}a^{16}-\frac{731323615583}{140492380027}a^{15}+\frac{920731130909}{140492380027}a^{14}-\frac{439799762575}{140492380027}a^{13}-\frac{1721911777770}{140492380027}a^{12}+\frac{1373905189082}{140492380027}a^{11}-\frac{170101535590}{140492380027}a^{10}+\frac{544267715529}{140492380027}a^{9}-\frac{758008807825}{140492380027}a^{8}-\frac{3636958046940}{140492380027}a^{7}+\frac{5566368758194}{140492380027}a^{6}+\frac{1961647738095}{140492380027}a^{5}-\frac{6220096341425}{140492380027}a^{4}+\frac{1684135466231}{140492380027}a^{3}+\frac{2783686384896}{140492380027}a^{2}-\frac{862882292643}{140492380027}a-\frac{361587529455}{140492380027}$, $\frac{14612920145}{140492380027}a^{21}+\frac{72433832300}{140492380027}a^{20}-\frac{80052840804}{140492380027}a^{19}+\frac{150348683122}{140492380027}a^{18}+\frac{55168520937}{140492380027}a^{17}+\frac{223400966939}{140492380027}a^{16}+\frac{296249092980}{140492380027}a^{15}-\frac{818622884119}{140492380027}a^{14}+\frac{1129367190735}{140492380027}a^{13}+\frac{931508661026}{140492380027}a^{12}-\frac{1692447932899}{140492380027}a^{11}+\frac{526784828277}{140492380027}a^{10}+\frac{36430473564}{140492380027}a^{9}+\frac{1780247713655}{140492380027}a^{8}+\frac{973557839297}{140492380027}a^{7}-\frac{6344705182353}{140492380027}a^{6}+\frac{2094670186930}{140492380027}a^{5}+\frac{5630863101020}{140492380027}a^{4}-\frac{4133775817844}{140492380027}a^{3}-\frac{1831140271822}{140492380027}a^{2}+\frac{1409744550197}{140492380027}a+\frac{16238064135}{140492380027}$
| 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: | \( 64048.1745685 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{9}\cdot 64048.1745685 \cdot 1}{2\cdot\sqrt{1897242698910438481935564683}}\cr\approx \mathstrut & 0.179537061833 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + x^20 + x^19 + x^18 + 5*x^17 - 11*x^16 + 6*x^15 + 19*x^14 - 22*x^13 - 4*x^12 + 6*x^11 + 13*x^10 + 26*x^9 - 75*x^8 - 11*x^7 + 101*x^6 - 32*x^5 - 59*x^4 + 28*x^3 + 15*x^2 - 6*x - 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^22 - x^21 + x^20 + x^19 + x^18 + 5*x^17 - 11*x^16 + 6*x^15 + 19*x^14 - 22*x^13 - 4*x^12 + 6*x^11 + 13*x^10 + 26*x^9 - 75*x^8 - 11*x^7 + 101*x^6 - 32*x^5 - 59*x^4 + 28*x^3 + 15*x^2 - 6*x - 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^22 - x^21 + x^20 + x^19 + x^18 + 5*x^17 - 11*x^16 + 6*x^15 + 19*x^14 - 22*x^13 - 4*x^12 + 6*x^11 + 13*x^10 + 26*x^9 - 75*x^8 - 11*x^7 + 101*x^6 - 32*x^5 - 59*x^4 + 28*x^3 + 15*x^2 - 6*x - 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^22 - x^21 + x^20 + x^19 + x^18 + 5*x^17 - 11*x^16 + 6*x^15 + 19*x^14 - 22*x^13 - 4*x^12 + 6*x^11 + 13*x^10 + 26*x^9 - 75*x^8 - 11*x^7 + 101*x^6 - 32*x^5 - 59*x^4 + 28*x^3 + 15*x^2 - 6*x - 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
$C_2^{10}.(C_2\times S_{11})$ (as 22T53):
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 81749606400 |
| The 752 conjugacy class representatives for $C_2^{10}.(C_2\times S_{11})$ |
| Character table for $C_2^{10}.(C_2\times S_{11})$ |
Intermediate fields
| 11.5.200601609583.1 |
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 22 sibling: | data not computed |
| Degree 44 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 | $22$ | ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.6.0.1}{6} }^{2}$ | $16{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.9.0.1}{9} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}$ | $18{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.10.0.1}{10} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(47147\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
|
\(200601609583\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ |