Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 + 2*x^19 - 4*x^16 - 6*x^15 + 8*x^14 - 16*x^13 + 30*x^12 - 24*x^11 + 22*x^10 - 48*x^9 - 4*x^8 - 30*x^7 + 8*x^6 - 4*x^5 - 12*x^4 - 10*x^3 + 12*x^2 - 4*x + 2)
gp: K = bnfinit(y^21 - 2*y^20 + 2*y^19 - 4*y^16 - 6*y^15 + 8*y^14 - 16*y^13 + 30*y^12 - 24*y^11 + 22*y^10 - 48*y^9 - 4*y^8 - 30*y^7 + 8*y^6 - 4*y^5 - 12*y^4 - 10*y^3 + 12*y^2 - 4*y + 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 2*x^20 + 2*x^19 - 4*x^16 - 6*x^15 + 8*x^14 - 16*x^13 + 30*x^12 - 24*x^11 + 22*x^10 - 48*x^9 - 4*x^8 - 30*x^7 + 8*x^6 - 4*x^5 - 12*x^4 - 10*x^3 + 12*x^2 - 4*x + 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 2*x^20 + 2*x^19 - 4*x^16 - 6*x^15 + 8*x^14 - 16*x^13 + 30*x^12 - 24*x^11 + 22*x^10 - 48*x^9 - 4*x^8 - 30*x^7 + 8*x^6 - 4*x^5 - 12*x^4 - 10*x^3 + 12*x^2 - 4*x + 2)
\( x^{21} - 2 x^{20} + 2 x^{19} - 4 x^{16} - 6 x^{15} + 8 x^{14} - 16 x^{13} + 30 x^{12} - 24 x^{11} + \cdots + 2 \)
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: | $[3, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-20735011601517857380131405824\)
\(\medspace = -\,2^{20}\cdot 11^{7}\cdot 317^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(22.31\) | 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: | $2^{20/21}11^{1/2}317^{1/2}\approx 114.26710056592353$ | ||
| Ramified primes: |
\(2\), \(11\), \(317\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23}a^{18}+\frac{9}{23}a^{17}+\frac{6}{23}a^{16}-\frac{2}{23}a^{15}+\frac{5}{23}a^{14}-\frac{5}{23}a^{13}+\frac{6}{23}a^{12}-\frac{1}{23}a^{11}-\frac{1}{23}a^{10}+\frac{6}{23}a^{9}-\frac{6}{23}a^{8}+\frac{2}{23}a^{7}-\frac{1}{23}a^{6}-\frac{5}{23}a^{5}-\frac{3}{23}a^{4}+\frac{8}{23}a^{3}-\frac{1}{23}a^{2}+\frac{7}{23}a-\frac{5}{23}$, $\frac{1}{23}a^{19}-\frac{6}{23}a^{17}-\frac{10}{23}a^{16}-\frac{4}{23}a^{14}+\frac{5}{23}a^{13}-\frac{9}{23}a^{12}+\frac{8}{23}a^{11}-\frac{8}{23}a^{10}+\frac{9}{23}a^{9}+\frac{10}{23}a^{8}+\frac{4}{23}a^{7}+\frac{4}{23}a^{6}-\frac{4}{23}a^{5}-\frac{11}{23}a^{4}-\frac{4}{23}a^{3}-\frac{7}{23}a^{2}+\frac{1}{23}a-\frac{1}{23}$, $\frac{1}{10\!\cdots\!21}a^{20}-\frac{12054179123160}{10\!\cdots\!21}a^{19}-\frac{4346269375880}{10\!\cdots\!21}a^{18}+\frac{527625497675}{46257401324027}a^{17}-\frac{363890337254282}{10\!\cdots\!21}a^{16}+\frac{146260499158340}{10\!\cdots\!21}a^{15}+\frac{236050473918702}{10\!\cdots\!21}a^{14}+\frac{100564808288743}{10\!\cdots\!21}a^{13}+\frac{77401986278483}{10\!\cdots\!21}a^{12}-\frac{198856096870300}{10\!\cdots\!21}a^{11}-\frac{20187872653725}{10\!\cdots\!21}a^{10}+\frac{251968074462890}{10\!\cdots\!21}a^{9}+\frac{144136428842320}{10\!\cdots\!21}a^{8}-\frac{23086822132333}{46257401324027}a^{7}+\frac{379213950108131}{10\!\cdots\!21}a^{6}+\frac{359202937962588}{10\!\cdots\!21}a^{5}+\frac{221842076581147}{10\!\cdots\!21}a^{4}-\frac{263169622699011}{10\!\cdots\!21}a^{3}-\frac{299802845248077}{10\!\cdots\!21}a^{2}+\frac{441664910091037}{10\!\cdots\!21}a+\frac{266251492481954}{10\!\cdots\!21}$
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: | $11$ | 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: |
$\frac{4822929489489}{46257401324027}a^{20}-\frac{14022264516234}{46257401324027}a^{19}+\frac{14520922531886}{46257401324027}a^{18}-\frac{1393895688721}{46257401324027}a^{17}-\frac{7255989729408}{46257401324027}a^{16}-\frac{18811894827872}{46257401324027}a^{15}-\frac{13086471864662}{46257401324027}a^{14}+\frac{81131459865558}{46257401324027}a^{13}-\frac{86715395148406}{46257401324027}a^{12}+\frac{186232257716704}{46257401324027}a^{11}-\frac{188304896903121}{46257401324027}a^{10}+\frac{94961741838240}{46257401324027}a^{9}-\frac{242846344921392}{46257401324027}a^{8}+\frac{97078382100379}{46257401324027}a^{7}+\frac{85699829969506}{46257401324027}a^{6}+\frac{193311436989800}{46257401324027}a^{5}+\frac{80835834483656}{46257401324027}a^{4}-\frac{76223197701576}{46257401324027}a^{3}+\frac{11675200072752}{46257401324027}a^{2}+\frac{117036601172184}{46257401324027}a-\frac{49414021910429}{46257401324027}$, $\frac{13193494957663}{10\!\cdots\!21}a^{20}-\frac{25229990656184}{10\!\cdots\!21}a^{19}+\frac{38946058021463}{10\!\cdots\!21}a^{18}-\frac{8785813142964}{10\!\cdots\!21}a^{17}-\frac{17664146952858}{10\!\cdots\!21}a^{16}-\frac{2688283716579}{10\!\cdots\!21}a^{15}-\frac{87832047899974}{10\!\cdots\!21}a^{14}+\frac{38541427391791}{10\!\cdots\!21}a^{13}-\frac{410310591904359}{10\!\cdots\!21}a^{12}+\frac{458773416693152}{10\!\cdots\!21}a^{11}-\frac{217461125338295}{10\!\cdots\!21}a^{10}+\frac{8856996992208}{46257401324027}a^{9}-\frac{321273444101830}{10\!\cdots\!21}a^{8}-\frac{336469835905226}{10\!\cdots\!21}a^{7}-\frac{598577675412166}{10\!\cdots\!21}a^{6}-\frac{11\!\cdots\!38}{10\!\cdots\!21}a^{5}+\frac{316741169507592}{10\!\cdots\!21}a^{4}-\frac{37732204033787}{10\!\cdots\!21}a^{3}+\frac{115194354855866}{10\!\cdots\!21}a^{2}+\frac{14865669823930}{10\!\cdots\!21}a-\frac{477958103939093}{10\!\cdots\!21}$, $\frac{26760491048508}{10\!\cdots\!21}a^{20}+\frac{27427543275057}{10\!\cdots\!21}a^{19}-\frac{137514892132704}{10\!\cdots\!21}a^{18}+\frac{169292948479484}{10\!\cdots\!21}a^{17}+\frac{32167428611339}{10\!\cdots\!21}a^{16}-\frac{174349173220375}{10\!\cdots\!21}a^{15}-\frac{542565901009682}{10\!\cdots\!21}a^{14}-\frac{107339775891445}{10\!\cdots\!21}a^{13}+\frac{584673298865413}{10\!\cdots\!21}a^{12}-\frac{384839136775470}{10\!\cdots\!21}a^{11}+\frac{18\!\cdots\!35}{10\!\cdots\!21}a^{10}-\frac{66636596232347}{46257401324027}a^{9}+\frac{58264769383822}{10\!\cdots\!21}a^{8}-\frac{41\!\cdots\!46}{10\!\cdots\!21}a^{7}-\frac{146270231887793}{10\!\cdots\!21}a^{6}-\frac{450712329526768}{10\!\cdots\!21}a^{5}+\frac{22\!\cdots\!36}{10\!\cdots\!21}a^{4}+\frac{317061751010923}{10\!\cdots\!21}a^{3}-\frac{557953399025612}{10\!\cdots\!21}a^{2}+\frac{251927062332438}{10\!\cdots\!21}a+\frac{14\!\cdots\!05}{10\!\cdots\!21}$, $\frac{477724828819801}{10\!\cdots\!21}a^{20}-\frac{885624984168294}{10\!\cdots\!21}a^{19}+\frac{829637021086445}{10\!\cdots\!21}a^{18}+\frac{119592502959892}{10\!\cdots\!21}a^{17}+\frac{44434521205160}{10\!\cdots\!21}a^{16}-\frac{84971580158373}{46257401324027}a^{15}-\frac{31\!\cdots\!74}{10\!\cdots\!21}a^{14}+\frac{34\!\cdots\!96}{10\!\cdots\!21}a^{13}-\frac{72\!\cdots\!89}{10\!\cdots\!21}a^{12}+\frac{13\!\cdots\!60}{10\!\cdots\!21}a^{11}-\frac{96\!\cdots\!19}{10\!\cdots\!21}a^{10}+\frac{93\!\cdots\!91}{10\!\cdots\!21}a^{9}-\frac{21\!\cdots\!71}{10\!\cdots\!21}a^{8}-\frac{45\!\cdots\!17}{10\!\cdots\!21}a^{7}-\frac{677249400773162}{46257401324027}a^{6}+\frac{11\!\cdots\!43}{10\!\cdots\!21}a^{5}-\frac{23\!\cdots\!16}{10\!\cdots\!21}a^{4}-\frac{71\!\cdots\!40}{10\!\cdots\!21}a^{3}-\frac{60\!\cdots\!69}{10\!\cdots\!21}a^{2}+\frac{43\!\cdots\!32}{10\!\cdots\!21}a-\frac{13\!\cdots\!01}{10\!\cdots\!21}$, $\frac{21225665923150}{10\!\cdots\!21}a^{20}+\frac{15444062391033}{10\!\cdots\!21}a^{19}-\frac{134920837167559}{10\!\cdots\!21}a^{18}+\frac{189538800218830}{10\!\cdots\!21}a^{17}-\frac{34435261669767}{10\!\cdots\!21}a^{16}-\frac{173203399807838}{10\!\cdots\!21}a^{15}-\frac{344976653094324}{10\!\cdots\!21}a^{14}-\frac{279683844706}{10\!\cdots\!21}a^{13}+\frac{741690830008846}{10\!\cdots\!21}a^{12}-\frac{442018222072826}{10\!\cdots\!21}a^{11}+\frac{18\!\cdots\!50}{10\!\cdots\!21}a^{10}-\frac{86560696643417}{46257401324027}a^{9}+\frac{260654061543102}{10\!\cdots\!21}a^{8}-\frac{28\!\cdots\!44}{10\!\cdots\!21}a^{7}+\frac{759341064576292}{10\!\cdots\!21}a^{6}+\frac{14\!\cdots\!08}{10\!\cdots\!21}a^{5}+\frac{22\!\cdots\!24}{10\!\cdots\!21}a^{4}+\frac{835746724927450}{10\!\cdots\!21}a^{3}-\frac{12\!\cdots\!66}{10\!\cdots\!21}a^{2}+\frac{396411742165112}{10\!\cdots\!21}a+\frac{20\!\cdots\!91}{10\!\cdots\!21}$, $\frac{310742332893564}{10\!\cdots\!21}a^{20}-\frac{741987863296601}{10\!\cdots\!21}a^{19}+\frac{785492353242023}{10\!\cdots\!21}a^{18}-\frac{46302737589739}{10\!\cdots\!21}a^{17}-\frac{251628787964230}{10\!\cdots\!21}a^{16}-\frac{11\!\cdots\!57}{10\!\cdots\!21}a^{15}-\frac{13\!\cdots\!08}{10\!\cdots\!21}a^{14}+\frac{34\!\cdots\!52}{10\!\cdots\!21}a^{13}-\frac{239597072093082}{46257401324027}a^{12}+\frac{10\!\cdots\!79}{10\!\cdots\!21}a^{11}-\frac{93\!\cdots\!86}{10\!\cdots\!21}a^{10}+\frac{66\!\cdots\!95}{10\!\cdots\!21}a^{9}-\frac{14\!\cdots\!64}{10\!\cdots\!21}a^{8}+\frac{19\!\cdots\!28}{10\!\cdots\!21}a^{7}-\frac{46\!\cdots\!02}{10\!\cdots\!21}a^{6}+\frac{54\!\cdots\!65}{10\!\cdots\!21}a^{5}-\frac{10\!\cdots\!64}{10\!\cdots\!21}a^{4}-\frac{33\!\cdots\!29}{10\!\cdots\!21}a^{3}-\frac{14\!\cdots\!37}{10\!\cdots\!21}a^{2}+\frac{52\!\cdots\!99}{10\!\cdots\!21}a-\frac{12\!\cdots\!65}{10\!\cdots\!21}$, $\frac{57593144934088}{10\!\cdots\!21}a^{20}+\frac{41113718111330}{10\!\cdots\!21}a^{19}-\frac{225047794252013}{10\!\cdots\!21}a^{18}+\frac{291746191050708}{10\!\cdots\!21}a^{17}+\frac{64578524982782}{10\!\cdots\!21}a^{16}-\frac{300873205564963}{10\!\cdots\!21}a^{15}-\frac{11\!\cdots\!28}{10\!\cdots\!21}a^{14}-\frac{252184942208586}{10\!\cdots\!21}a^{13}+\frac{762623679766001}{10\!\cdots\!21}a^{12}-\frac{497912063727820}{10\!\cdots\!21}a^{11}+\frac{33\!\cdots\!80}{10\!\cdots\!21}a^{10}-\frac{97665984004326}{46257401324027}a^{9}-\frac{22472473328970}{10\!\cdots\!21}a^{8}-\frac{82\!\cdots\!98}{10\!\cdots\!21}a^{7}-\frac{10\!\cdots\!52}{10\!\cdots\!21}a^{6}-\frac{22\!\cdots\!32}{10\!\cdots\!21}a^{5}+\frac{41\!\cdots\!48}{10\!\cdots\!21}a^{4}+\frac{219584305394916}{10\!\cdots\!21}a^{3}-\frac{554938193176700}{10\!\cdots\!21}a^{2}+\frac{363281677215014}{10\!\cdots\!21}a+\frac{26\!\cdots\!01}{10\!\cdots\!21}$, $\frac{527120946910521}{10\!\cdots\!21}a^{20}-\frac{965782630736355}{10\!\cdots\!21}a^{19}+\frac{849736961670200}{10\!\cdots\!21}a^{18}+\frac{193589820856949}{10\!\cdots\!21}a^{17}-\frac{42446697731031}{10\!\cdots\!21}a^{16}-\frac{20\!\cdots\!30}{10\!\cdots\!21}a^{15}-\frac{36\!\cdots\!53}{10\!\cdots\!21}a^{14}+\frac{37\!\cdots\!63}{10\!\cdots\!21}a^{13}-\frac{73\!\cdots\!48}{10\!\cdots\!21}a^{12}+\frac{14\!\cdots\!66}{10\!\cdots\!21}a^{11}-\frac{94\!\cdots\!23}{10\!\cdots\!21}a^{10}+\frac{89\!\cdots\!64}{10\!\cdots\!21}a^{9}-\frac{22\!\cdots\!34}{10\!\cdots\!21}a^{8}-\frac{79\!\cdots\!32}{10\!\cdots\!21}a^{7}-\frac{14\!\cdots\!84}{10\!\cdots\!21}a^{6}+\frac{24\!\cdots\!52}{10\!\cdots\!21}a^{5}+\frac{16\!\cdots\!45}{10\!\cdots\!21}a^{4}-\frac{36\!\cdots\!03}{10\!\cdots\!21}a^{3}-\frac{50\!\cdots\!65}{10\!\cdots\!21}a^{2}+\frac{60\!\cdots\!37}{10\!\cdots\!21}a-\frac{278716594784867}{10\!\cdots\!21}$, $\frac{100229997793131}{10\!\cdots\!21}a^{20}-\frac{169881699651564}{10\!\cdots\!21}a^{19}+\frac{138873381678560}{10\!\cdots\!21}a^{18}+\frac{48475813693415}{10\!\cdots\!21}a^{17}+\frac{8934204027029}{10\!\cdots\!21}a^{16}-\frac{425041685588382}{10\!\cdots\!21}a^{15}-\frac{716400597902967}{10\!\cdots\!21}a^{14}+\frac{612774569663750}{10\!\cdots\!21}a^{13}-\frac{14\!\cdots\!16}{10\!\cdots\!21}a^{12}+\frac{27\!\cdots\!69}{10\!\cdots\!21}a^{11}-\frac{12\!\cdots\!27}{10\!\cdots\!21}a^{10}+\frac{16\!\cdots\!37}{10\!\cdots\!21}a^{9}-\frac{43\!\cdots\!54}{10\!\cdots\!21}a^{8}-\frac{15\!\cdots\!67}{10\!\cdots\!21}a^{7}-\frac{32\!\cdots\!98}{10\!\cdots\!21}a^{6}-\frac{413351780700988}{10\!\cdots\!21}a^{5}+\frac{854332860095097}{10\!\cdots\!21}a^{4}+\frac{2807464408686}{10\!\cdots\!21}a^{3}+\frac{187315371560482}{10\!\cdots\!21}a^{2}+\frac{23\!\cdots\!45}{10\!\cdots\!21}a+\frac{487394601091003}{10\!\cdots\!21}$, $\frac{380313749367025}{10\!\cdots\!21}a^{20}-\frac{743141052927562}{10\!\cdots\!21}a^{19}+\frac{667597820395522}{10\!\cdots\!21}a^{18}+\frac{123731590254986}{10\!\cdots\!21}a^{17}-\frac{45530492935844}{10\!\cdots\!21}a^{16}-\frac{16\!\cdots\!94}{10\!\cdots\!21}a^{15}-\frac{22\!\cdots\!70}{10\!\cdots\!21}a^{14}+\frac{31\!\cdots\!97}{10\!\cdots\!21}a^{13}-\frac{55\!\cdots\!48}{10\!\cdots\!21}a^{12}+\frac{10\!\cdots\!45}{10\!\cdots\!21}a^{11}-\frac{80\!\cdots\!28}{10\!\cdots\!21}a^{10}+\frac{67\!\cdots\!42}{10\!\cdots\!21}a^{9}-\frac{17\!\cdots\!64}{10\!\cdots\!21}a^{8}-\frac{23\!\cdots\!27}{10\!\cdots\!21}a^{7}-\frac{98\!\cdots\!28}{10\!\cdots\!21}a^{6}+\frac{45\!\cdots\!52}{10\!\cdots\!21}a^{5}+\frac{396641482550001}{10\!\cdots\!21}a^{4}-\frac{53\!\cdots\!26}{10\!\cdots\!21}a^{3}-\frac{39\!\cdots\!16}{10\!\cdots\!21}a^{2}+\frac{41\!\cdots\!56}{10\!\cdots\!21}a-\frac{550909807784673}{10\!\cdots\!21}$, $\frac{169027951625789}{10\!\cdots\!21}a^{20}-\frac{285933407235628}{10\!\cdots\!21}a^{19}+\frac{283789148268025}{10\!\cdots\!21}a^{18}-\frac{3807501357574}{10\!\cdots\!21}a^{17}+\frac{129166059692125}{10\!\cdots\!21}a^{16}-\frac{714801041297595}{10\!\cdots\!21}a^{15}-\frac{11\!\cdots\!03}{10\!\cdots\!21}a^{14}+\frac{797638751651822}{10\!\cdots\!21}a^{13}-\frac{24\!\cdots\!72}{10\!\cdots\!21}a^{12}+\frac{45\!\cdots\!50}{10\!\cdots\!21}a^{11}-\frac{35\!\cdots\!59}{10\!\cdots\!21}a^{10}+\frac{43\!\cdots\!68}{10\!\cdots\!21}a^{9}-\frac{87\!\cdots\!67}{10\!\cdots\!21}a^{8}-\frac{12\!\cdots\!22}{10\!\cdots\!21}a^{7}-\frac{374918196230603}{46257401324027}a^{6}+\frac{37590350742395}{46257401324027}a^{5}-\frac{31\!\cdots\!62}{10\!\cdots\!21}a^{4}-\frac{13\!\cdots\!92}{10\!\cdots\!21}a^{3}-\frac{25\!\cdots\!55}{10\!\cdots\!21}a^{2}+\frac{546861938898325}{10\!\cdots\!21}a+\frac{91608356423949}{10\!\cdots\!21}$
| 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: | \( 999740.077738 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 999740.077738 \cdot 1}{2\cdot\sqrt{20735011601517857380131405824}}\cr\approx \mathstrut & 0.423851542233 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 + 2*x^19 - 4*x^16 - 6*x^15 + 8*x^14 - 16*x^13 + 30*x^12 - 24*x^11 + 22*x^10 - 48*x^9 - 4*x^8 - 30*x^7 + 8*x^6 - 4*x^5 - 12*x^4 - 10*x^3 + 12*x^2 - 4*x + 2)
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 - 2*x^20 + 2*x^19 - 4*x^16 - 6*x^15 + 8*x^14 - 16*x^13 + 30*x^12 - 24*x^11 + 22*x^10 - 48*x^9 - 4*x^8 - 30*x^7 + 8*x^6 - 4*x^5 - 12*x^4 - 10*x^3 + 12*x^2 - 4*x + 2, 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 - 2*x^20 + 2*x^19 - 4*x^16 - 6*x^15 + 8*x^14 - 16*x^13 + 30*x^12 - 24*x^11 + 22*x^10 - 48*x^9 - 4*x^8 - 30*x^7 + 8*x^6 - 4*x^5 - 12*x^4 - 10*x^3 + 12*x^2 - 4*x + 2);
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 - 2*x^20 + 2*x^19 - 4*x^16 - 6*x^15 + 8*x^14 - 16*x^13 + 30*x^12 - 24*x^11 + 22*x^10 - 48*x^9 - 4*x^8 - 30*x^7 + 8*x^6 - 4*x^5 - 12*x^4 - 10*x^3 + 12*x^2 - 4*x + 2);
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
$S_3\times \GL(3,2)$ (as 21T27):
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 1008 |
| The 18 conjugacy class representatives for $S_3\times \GL(3,2)$ |
| Character table for $S_3\times \GL(3,2)$ |
Intermediate fields
| 3.1.44.1, 7.3.6431296.1 |
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 24 sibling: | data not computed |
| Degree 42 siblings: | data not computed |
| Arithmetically equvalently 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 | R | $21$ | $21$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.7.0.1}{7} }$ | R | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $21$ | $21$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.7.0.1}{7} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{7}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.21.20.1 | $x^{21} + 2$ | $21$ | $1$ | $20$ | 21T11 | $[\ ]_{21}^{6}$ |
|
\(11\)
| $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(317\)
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $2$ | $3$ | $3$ | ||||
| Deg $6$ | $2$ | $3$ | $3$ |