Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^21 + 5*x^20 + 4*x^19 - 26*x^18 + 31*x^17 + 15*x^16 - 79*x^15 + 76*x^14 + 60*x^13 - 153*x^12 + 84*x^11 + 83*x^10 - 192*x^9 + 24*x^8 + 58*x^7 - 150*x^6 + 30*x^5 + 19*x^4 - 23*x^3 + 6*x^2 + 29*x + 1)
gp: K = bnfinit(y^22 - 4*y^21 + 5*y^20 + 4*y^19 - 26*y^18 + 31*y^17 + 15*y^16 - 79*y^15 + 76*y^14 + 60*y^13 - 153*y^12 + 84*y^11 + 83*y^10 - 192*y^9 + 24*y^8 + 58*y^7 - 150*y^6 + 30*y^5 + 19*y^4 - 23*y^3 + 6*y^2 + 29*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 4*x^21 + 5*x^20 + 4*x^19 - 26*x^18 + 31*x^17 + 15*x^16 - 79*x^15 + 76*x^14 + 60*x^13 - 153*x^12 + 84*x^11 + 83*x^10 - 192*x^9 + 24*x^8 + 58*x^7 - 150*x^6 + 30*x^5 + 19*x^4 - 23*x^3 + 6*x^2 + 29*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 4*x^21 + 5*x^20 + 4*x^19 - 26*x^18 + 31*x^17 + 15*x^16 - 79*x^15 + 76*x^14 + 60*x^13 - 153*x^12 + 84*x^11 + 83*x^10 - 192*x^9 + 24*x^8 + 58*x^7 - 150*x^6 + 30*x^5 + 19*x^4 - 23*x^3 + 6*x^2 + 29*x + 1)
\( x^{22} - 4 x^{21} + 5 x^{20} + 4 x^{19} - 26 x^{18} + 31 x^{17} + 15 x^{16} - 79 x^{15} + 76 x^{14} + \cdots + 1 \)
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: |
\(-931263224878297608080470016\)
\(\medspace = -\,2^{12}\cdot 11^{7}\cdot 19^{4}\cdot 547^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.82\) | 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: | $2^{2/3}11^{5/6}19^{1/2}547^{1/2}\approx 1193.6702363351474$ | ||
| Ramified primes: |
\(2\), \(11\), \(19\), \(547\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{11}a^{17}+\frac{3}{11}a^{15}+\frac{3}{11}a^{13}-\frac{4}{11}a^{12}-\frac{1}{11}a^{11}+\frac{1}{11}a^{10}-\frac{2}{11}a^{9}-\frac{5}{11}a^{8}-\frac{4}{11}a^{7}+\frac{1}{11}a^{6}-\frac{4}{11}a^{5}-\frac{4}{11}a^{4}-\frac{2}{11}a^{3}-\frac{1}{11}a^{2}-\frac{5}{11}a-\frac{1}{11}$, $\frac{1}{11}a^{18}+\frac{3}{11}a^{16}+\frac{3}{11}a^{14}-\frac{4}{11}a^{13}-\frac{1}{11}a^{12}+\frac{1}{11}a^{11}-\frac{2}{11}a^{10}-\frac{5}{11}a^{9}-\frac{4}{11}a^{8}+\frac{1}{11}a^{7}-\frac{4}{11}a^{6}-\frac{4}{11}a^{5}-\frac{2}{11}a^{4}-\frac{1}{11}a^{3}-\frac{5}{11}a^{2}-\frac{1}{11}a$, $\frac{1}{242}a^{19}+\frac{5}{242}a^{18}-\frac{5}{121}a^{17}-\frac{117}{242}a^{16}-\frac{51}{121}a^{15}+\frac{5}{22}a^{14}+\frac{83}{242}a^{13}-\frac{31}{121}a^{12}-\frac{105}{242}a^{11}-\frac{3}{121}a^{10}+\frac{63}{242}a^{9}+\frac{56}{121}a^{8}+\frac{53}{242}a^{7}+\frac{53}{121}a^{6}+\frac{48}{121}a^{5}+\frac{107}{242}a^{4}+\frac{93}{242}a^{3}-\frac{57}{242}a^{2}-\frac{39}{242}a-\frac{9}{242}$, $\frac{1}{242}a^{20}+\frac{9}{242}a^{18}-\frac{1}{242}a^{17}-\frac{111}{242}a^{16}+\frac{37}{242}a^{15}-\frac{30}{121}a^{14}+\frac{29}{242}a^{13}-\frac{103}{242}a^{12}+\frac{13}{242}a^{11}+\frac{71}{242}a^{10}-\frac{71}{242}a^{9}-\frac{45}{242}a^{8}+\frac{105}{242}a^{7}-\frac{30}{121}a^{6}-\frac{87}{242}a^{5}-\frac{34}{121}a^{4}+\frac{14}{121}a^{3}-\frac{20}{121}a^{2}+\frac{27}{121}a-\frac{21}{242}$, $\frac{1}{406526742211766}a^{21}-\frac{288595015055}{203263371105883}a^{20}+\frac{255666951553}{406526742211766}a^{19}-\frac{10643548821841}{406526742211766}a^{18}-\frac{10810179676633}{406526742211766}a^{17}+\frac{92462124524021}{406526742211766}a^{16}-\frac{5406253665840}{203263371105883}a^{15}-\frac{139971702813821}{406526742211766}a^{14}-\frac{14585103833331}{406526742211766}a^{13}+\frac{13764050961235}{36956976564706}a^{12}+\frac{163921142487191}{406526742211766}a^{11}-\frac{193045451521693}{406526742211766}a^{10}+\frac{27509531970517}{406526742211766}a^{9}-\frac{57330290235511}{406526742211766}a^{8}-\frac{31001972030638}{203263371105883}a^{7}+\frac{98468543453281}{406526742211766}a^{6}-\frac{36295292169886}{203263371105883}a^{5}+\frac{52852929139063}{203263371105883}a^{4}+\frac{75738032908635}{203263371105883}a^{3}-\frac{52209940478345}{203263371105883}a^{2}-\frac{295786960287}{36956976564706}a-\frac{92399281118454}{203263371105883}$
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: |
$\frac{4115115668275}{203263371105883}a^{21}-\frac{36217864001663}{406526742211766}a^{20}+\frac{42623957066059}{406526742211766}a^{19}+\frac{30352628641251}{203263371105883}a^{18}-\frac{265238532033837}{406526742211766}a^{17}+\frac{122180384755895}{203263371105883}a^{16}+\frac{348064011768861}{406526742211766}a^{15}-\frac{933952628700649}{406526742211766}a^{14}+\frac{203910682678299}{203263371105883}a^{13}+\frac{117345185580525}{36956976564706}a^{12}-\frac{935219496279468}{203263371105883}a^{11}-\frac{257815505528925}{406526742211766}a^{10}+\frac{11\!\cdots\!38}{203263371105883}a^{9}-\frac{16\!\cdots\!23}{406526742211766}a^{8}-\frac{525357975035906}{203263371105883}a^{7}+\frac{10\!\cdots\!84}{203263371105883}a^{6}-\frac{614923361666769}{406526742211766}a^{5}-\frac{470902278088683}{406526742211766}a^{4}+\frac{10\!\cdots\!21}{406526742211766}a^{3}-\frac{139444654007467}{406526742211766}a^{2}-\frac{2767222751515}{3359725142246}a+\frac{212136542349154}{203263371105883}$, $\frac{20237559936547}{406526742211766}a^{21}-\frac{43330777141568}{203263371105883}a^{20}+\frac{136259300898627}{406526742211766}a^{19}+\frac{13373028669289}{406526742211766}a^{18}-\frac{535954302110965}{406526742211766}a^{17}+\frac{893177466666807}{406526742211766}a^{16}-\frac{71618145619632}{203263371105883}a^{15}-\frac{16\!\cdots\!85}{406526742211766}a^{14}+\frac{25\!\cdots\!67}{406526742211766}a^{13}+\frac{3360983133265}{36956976564706}a^{12}-\frac{35\!\cdots\!31}{406526742211766}a^{11}+\frac{41\!\cdots\!55}{406526742211766}a^{10}+\frac{216517934003923}{406526742211766}a^{9}-\frac{55\!\cdots\!11}{406526742211766}a^{8}+\frac{15\!\cdots\!43}{203263371105883}a^{7}+\frac{162877126021101}{406526742211766}a^{6}-\frac{24\!\cdots\!74}{203263371105883}a^{5}+\frac{12\!\cdots\!45}{203263371105883}a^{4}+\frac{90850976191960}{203263371105883}a^{3}-\frac{380008083365270}{203263371105883}a^{2}+\frac{8583769093215}{3359725142246}a+\frac{559713515710311}{203263371105883}$, $\frac{2146482683207}{406526742211766}a^{21}-\frac{6113397812629}{203263371105883}a^{20}+\frac{28920784481841}{406526742211766}a^{19}-\frac{27662678848865}{406526742211766}a^{18}-\frac{39784824147279}{406526742211766}a^{17}+\frac{161344926785609}{406526742211766}a^{16}-\frac{88783581648204}{203263371105883}a^{15}-\frac{63799750311259}{406526742211766}a^{14}+\frac{443361481622179}{406526742211766}a^{13}-\frac{39842873864947}{36956976564706}a^{12}-\frac{189794453329839}{406526742211766}a^{11}+\frac{820918625859855}{406526742211766}a^{10}-\frac{735025945549199}{406526742211766}a^{9}-\frac{185221319315287}{406526742211766}a^{8}+\frac{374197392761029}{203263371105883}a^{7}-\frac{540296972242649}{406526742211766}a^{6}+\frac{89116735403169}{203263371105883}a^{5}+\frac{238581883961081}{203263371105883}a^{4}-\frac{421039483103645}{203263371105883}a^{3}+\frac{303729529910560}{203263371105883}a^{2}-\frac{26941151494127}{36956976564706}a-\frac{153290567465254}{203263371105883}$, $\frac{1563086874244}{203263371105883}a^{21}-\frac{15869702645895}{406526742211766}a^{20}+\frac{33209185878273}{406526742211766}a^{19}-\frac{6889507996321}{203263371105883}a^{18}-\frac{105962860444401}{406526742211766}a^{17}+\frac{122949937192814}{203263371105883}a^{16}-\frac{135474343672449}{406526742211766}a^{15}-\frac{384799365898575}{406526742211766}a^{14}+\frac{419730961460425}{203263371105883}a^{13}-\frac{2265171496323}{3359725142246}a^{12}-\frac{521508973567676}{203263371105883}a^{11}+\frac{16\!\cdots\!71}{406526742211766}a^{10}-\frac{53296123609824}{203263371105883}a^{9}-\frac{18\!\cdots\!15}{406526742211766}a^{8}+\frac{715699479503991}{203263371105883}a^{7}+\frac{104243482347988}{203263371105883}a^{6}-\frac{18\!\cdots\!13}{406526742211766}a^{5}+\frac{872711166596185}{406526742211766}a^{4}-\frac{50997770185193}{406526742211766}a^{3}-\frac{608194737341849}{406526742211766}a^{2}+\frac{43202030362505}{36956976564706}a+\frac{201826872297932}{203263371105883}$, $\frac{15266327456177}{406526742211766}a^{21}-\frac{61308269857247}{406526742211766}a^{20}+\frac{85497635203905}{406526742211766}a^{19}+\frac{10408871047935}{203263371105883}a^{18}-\frac{170490611689826}{203263371105883}a^{17}+\frac{254413488441186}{203263371105883}a^{16}-\frac{26328054702757}{406526742211766}a^{15}-\frac{874507660898437}{406526742211766}a^{14}+\frac{655498120818206}{203263371105883}a^{13}+\frac{3936379381447}{18478488282353}a^{12}-\frac{801533934984714}{203263371105883}a^{11}+\frac{947569978108333}{203263371105883}a^{10}-\frac{231024898964741}{203263371105883}a^{9}-\frac{11\!\cdots\!45}{203263371105883}a^{8}+\frac{13\!\cdots\!71}{406526742211766}a^{7}-\frac{858801485663379}{406526742211766}a^{6}-\frac{19\!\cdots\!07}{406526742211766}a^{5}+\frac{839531771558358}{203263371105883}a^{4}-\frac{219762449324364}{203263371105883}a^{3}+\frac{104909378060602}{203263371105883}a^{2}+\frac{5076300725349}{3359725142246}a+\frac{457820031523787}{406526742211766}$, $\frac{144582543784}{203263371105883}a^{21}-\frac{1994273802353}{203263371105883}a^{20}+\frac{1579238246481}{203263371105883}a^{19}+\frac{4695283362300}{203263371105883}a^{18}-\frac{1993351287005}{203263371105883}a^{17}-\frac{10547737196620}{203263371105883}a^{16}+\frac{34115665712884}{203263371105883}a^{15}+\frac{23524897165173}{203263371105883}a^{14}-\frac{130358363862029}{203263371105883}a^{13}+\frac{4195937616440}{18478488282353}a^{12}+\frac{135512758256983}{203263371105883}a^{11}-\frac{363124626815981}{203263371105883}a^{10}-\frac{145114952251905}{203263371105883}a^{9}+\frac{495894938761914}{203263371105883}a^{8}-\frac{54523508329079}{203263371105883}a^{7}-\frac{106174486030451}{203263371105883}a^{6}+\frac{799838155083276}{203263371105883}a^{5}+\frac{380426936502383}{203263371105883}a^{4}-\frac{48345226169767}{203263371105883}a^{3}+\frac{371025409732742}{203263371105883}a^{2}+\frac{10608837302533}{18478488282353}a-\frac{100152015582134}{203263371105883}$, $\frac{20721894289066}{203263371105883}a^{21}-\frac{169992439324791}{406526742211766}a^{20}+\frac{218609861014381}{406526742211766}a^{19}+\frac{82638640436756}{203263371105883}a^{18}-\frac{11\!\cdots\!65}{406526742211766}a^{17}+\frac{682281438385742}{203263371105883}a^{16}+\frac{631145442532479}{406526742211766}a^{15}-\frac{34\!\cdots\!79}{406526742211766}a^{14}+\frac{16\!\cdots\!03}{203263371105883}a^{13}+\frac{235544388000811}{36956976564706}a^{12}-\frac{34\!\cdots\!53}{203263371105883}a^{11}+\frac{37\!\cdots\!43}{406526742211766}a^{10}+\frac{19\!\cdots\!37}{203263371105883}a^{9}-\frac{85\!\cdots\!23}{406526742211766}a^{8}+\frac{626622029253779}{203263371105883}a^{7}+\frac{16\!\cdots\!67}{203263371105883}a^{6}-\frac{63\!\cdots\!03}{406526742211766}a^{5}+\frac{13\!\cdots\!97}{406526742211766}a^{4}+\frac{841635626037039}{406526742211766}a^{3}-\frac{13\!\cdots\!01}{406526742211766}a^{2}+\frac{17635461832739}{36956976564706}a+\frac{552664253418331}{203263371105883}$, $\frac{51352388734319}{406526742211766}a^{21}-\frac{180719931552745}{406526742211766}a^{20}+\frac{80798916028030}{203263371105883}a^{19}+\frac{307247252975881}{406526742211766}a^{18}-\frac{605113228622767}{203263371105883}a^{17}+\frac{10\!\cdots\!89}{406526742211766}a^{16}+\frac{13\!\cdots\!85}{406526742211766}a^{15}-\frac{17\!\cdots\!14}{203263371105883}a^{14}+\frac{22\!\cdots\!77}{406526742211766}a^{13}+\frac{192914300650253}{18478488282353}a^{12}-\frac{62\!\cdots\!89}{406526742211766}a^{11}+\frac{806756618077090}{203263371105883}a^{10}+\frac{49\!\cdots\!19}{406526742211766}a^{9}-\frac{42\!\cdots\!44}{203263371105883}a^{8}-\frac{746484021624008}{203263371105883}a^{7}+\frac{29\!\cdots\!97}{406526742211766}a^{6}-\frac{69\!\cdots\!95}{406526742211766}a^{5}+\frac{175191514238689}{406526742211766}a^{4}+\frac{20\!\cdots\!21}{406526742211766}a^{3}-\frac{11\!\cdots\!13}{406526742211766}a^{2}+\frac{26175687928846}{18478488282353}a+\frac{508126367028023}{203263371105883}$, $\frac{2023821221483}{406526742211766}a^{21}+\frac{5565190016273}{203263371105883}a^{20}-\frac{42309125836213}{203263371105883}a^{19}+\frac{74663948256015}{203263371105883}a^{18}+\frac{34668257828305}{406526742211766}a^{17}-\frac{300128417708089}{203263371105883}a^{16}+\frac{456823318900204}{203263371105883}a^{15}+\frac{109825234017238}{203263371105883}a^{14}-\frac{10\!\cdots\!26}{203263371105883}a^{13}+\frac{197222629260883}{36956976564706}a^{12}+\frac{665082984919147}{203263371105883}a^{11}-\frac{45\!\cdots\!35}{406526742211766}a^{10}+\frac{10\!\cdots\!04}{203263371105883}a^{9}+\frac{27\!\cdots\!37}{406526742211766}a^{8}-\frac{45\!\cdots\!63}{406526742211766}a^{7}+\frac{13\!\cdots\!15}{406526742211766}a^{6}+\frac{15\!\cdots\!03}{203263371105883}a^{5}-\frac{24\!\cdots\!47}{406526742211766}a^{4}+\frac{913994294006803}{406526742211766}a^{3}+\frac{442579574970873}{406526742211766}a^{2}-\frac{35479156652151}{18478488282353}a-\frac{325093544964663}{406526742211766}$, $\frac{2704001463122}{203263371105883}a^{21}-\frac{6405227232297}{203263371105883}a^{20}-\frac{5647504773371}{406526742211766}a^{19}+\frac{48119977667371}{406526742211766}a^{18}-\frac{38527773464509}{203263371105883}a^{17}-\frac{30174356942461}{406526742211766}a^{16}+\frac{101516203373863}{203263371105883}a^{15}-\frac{142047641725775}{406526742211766}a^{14}-\frac{72824935914983}{406526742211766}a^{13}+\frac{16342191285635}{18478488282353}a^{12}-\frac{56196788930221}{406526742211766}a^{11}-\frac{49258326624026}{203263371105883}a^{10}-\frac{91942308507031}{406526742211766}a^{9}-\frac{190990370983037}{203263371105883}a^{8}-\frac{142260453945603}{406526742211766}a^{7}-\frac{83169046410282}{203263371105883}a^{6}-\frac{101431630180189}{203263371105883}a^{5}-\frac{176680334769677}{406526742211766}a^{4}-\frac{274317453904859}{406526742211766}a^{3}+\frac{131506949109667}{406526742211766}a^{2}+\frac{8589363571301}{36956976564706}a-\frac{245144617662771}{406526742211766}$, $\frac{5793855784066}{203263371105883}a^{21}-\frac{37699991092243}{406526742211766}a^{20}+\frac{32477587372057}{406526742211766}a^{19}+\frac{28788684647802}{203263371105883}a^{18}-\frac{246065824866867}{406526742211766}a^{17}+\frac{106445469566053}{203263371105883}a^{16}+\frac{223251651478907}{406526742211766}a^{15}-\frac{676366843362049}{406526742211766}a^{14}+\frac{282000046752849}{203263371105883}a^{13}+\frac{6167539644577}{3359725142246}a^{12}-\frac{590676552361326}{203263371105883}a^{11}+\frac{618390590407297}{406526742211766}a^{10}+\frac{463397904145423}{203263371105883}a^{9}-\frac{20\!\cdots\!25}{406526742211766}a^{8}-\frac{230463547025487}{203263371105883}a^{7}+\frac{201264490864259}{203263371105883}a^{6}-\frac{20\!\cdots\!43}{406526742211766}a^{5}-\frac{491305417552263}{406526742211766}a^{4}+\frac{209417098891555}{406526742211766}a^{3}-\frac{525386407288537}{406526742211766}a^{2}-\frac{1647659842747}{3359725142246}a+\frac{212449209038723}{203263371105883}$, $\frac{56996998546933}{406526742211766}a^{21}-\frac{106965314402570}{203263371105883}a^{20}+\frac{110684655528177}{203263371105883}a^{19}+\frac{155984543750984}{203263371105883}a^{18}-\frac{14\!\cdots\!83}{406526742211766}a^{17}+\frac{673188771353915}{203263371105883}a^{16}+\frac{665634100818002}{203263371105883}a^{15}-\frac{20\!\cdots\!92}{203263371105883}a^{14}+\frac{14\!\cdots\!34}{203263371105883}a^{13}+\frac{35725003220601}{3359725142246}a^{12}-\frac{36\!\cdots\!35}{203263371105883}a^{11}+\frac{22\!\cdots\!31}{406526742211766}a^{10}+\frac{24\!\cdots\!87}{203263371105883}a^{9}-\frac{89\!\cdots\!23}{406526742211766}a^{8}-\frac{501037183884739}{406526742211766}a^{7}+\frac{28\!\cdots\!81}{406526742211766}a^{6}-\frac{34\!\cdots\!95}{203263371105883}a^{5}+\frac{11\!\cdots\!89}{406526742211766}a^{4}+\frac{11\!\cdots\!49}{406526742211766}a^{3}-\frac{11\!\cdots\!87}{406526742211766}a^{2}+\frac{2096064997681}{1679862571123}a+\frac{10\!\cdots\!81}{406526742211766}$
| 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: | \( 56735.8548792 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{9}\cdot 56735.8548792 \cdot 1}{2\cdot\sqrt{931263224878297608080470016}}\cr\approx \mathstrut & 0.227002258793 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^21 + 5*x^20 + 4*x^19 - 26*x^18 + 31*x^17 + 15*x^16 - 79*x^15 + 76*x^14 + 60*x^13 - 153*x^12 + 84*x^11 + 83*x^10 - 192*x^9 + 24*x^8 + 58*x^7 - 150*x^6 + 30*x^5 + 19*x^4 - 23*x^3 + 6*x^2 + 29*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 - 4*x^21 + 5*x^20 + 4*x^19 - 26*x^18 + 31*x^17 + 15*x^16 - 79*x^15 + 76*x^14 + 60*x^13 - 153*x^12 + 84*x^11 + 83*x^10 - 192*x^9 + 24*x^8 + 58*x^7 - 150*x^6 + 30*x^5 + 19*x^4 - 23*x^3 + 6*x^2 + 29*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 - 4*x^21 + 5*x^20 + 4*x^19 - 26*x^18 + 31*x^17 + 15*x^16 - 79*x^15 + 76*x^14 + 60*x^13 - 153*x^12 + 84*x^11 + 83*x^10 - 192*x^9 + 24*x^8 + 58*x^7 - 150*x^6 + 30*x^5 + 19*x^4 - 23*x^3 + 6*x^2 + 29*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 - 4*x^21 + 5*x^20 + 4*x^19 - 26*x^18 + 31*x^17 + 15*x^16 - 79*x^15 + 76*x^14 + 60*x^13 - 153*x^12 + 84*x^11 + 83*x^10 - 192*x^9 + 24*x^8 + 58*x^7 - 150*x^6 + 30*x^5 + 19*x^4 - 23*x^3 + 6*x^2 + 29*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
$C_2^{11}.A_{11}$ (as 22T52):
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 40874803200 |
| The 400 conjugacy class representatives for $C_2^{11}.A_{11}$ |
| Character table for $C_2^{11}.A_{11}$ |
Intermediate fields
| 11.3.836463893056.1 |
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 44 sibling: | 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 | R | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.4.0.1}{4} }$ | $22$ | R | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.9.0.1}{9} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.11.0.1}{11} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 2.9.6.1 | $x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ | |
| 2.9.6.1 | $x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ | |
|
\(11\)
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 11.6.3.1 | $x^{6} + 242 x^{2} - 11979$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 11.6.4.1 | $x^{6} + 21 x^{5} + 153 x^{4} + 449 x^{3} + 537 x^{2} + 1569 x + 3440$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(19\)
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.14.0.1 | $x^{14} + 11 x^{7} + 11 x^{6} + 11 x^{5} + x^{4} + 5 x^{3} + 16 x^{2} + 7 x + 2$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
|
\(547\)
| $\Q_{547}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{547}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |