Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 5*x^21 + 17*x^20 - 40*x^19 + 70*x^18 - 93*x^17 + 99*x^16 - 86*x^15 + 53*x^14 + 37*x^13 - 283*x^12 + 778*x^11 - 1493*x^10 + 2198*x^9 - 2575*x^8 + 2425*x^7 - 1840*x^6 + 1108*x^5 - 512*x^4 + 173*x^3 - 33*x^2 + x + 1)
gp: K = bnfinit(y^22 - 5*y^21 + 17*y^20 - 40*y^19 + 70*y^18 - 93*y^17 + 99*y^16 - 86*y^15 + 53*y^14 + 37*y^13 - 283*y^12 + 778*y^11 - 1493*y^10 + 2198*y^9 - 2575*y^8 + 2425*y^7 - 1840*y^6 + 1108*y^5 - 512*y^4 + 173*y^3 - 33*y^2 + y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 5*x^21 + 17*x^20 - 40*x^19 + 70*x^18 - 93*x^17 + 99*x^16 - 86*x^15 + 53*x^14 + 37*x^13 - 283*x^12 + 778*x^11 - 1493*x^10 + 2198*x^9 - 2575*x^8 + 2425*x^7 - 1840*x^6 + 1108*x^5 - 512*x^4 + 173*x^3 - 33*x^2 + x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 5*x^21 + 17*x^20 - 40*x^19 + 70*x^18 - 93*x^17 + 99*x^16 - 86*x^15 + 53*x^14 + 37*x^13 - 283*x^12 + 778*x^11 - 1493*x^10 + 2198*x^9 - 2575*x^8 + 2425*x^7 - 1840*x^6 + 1108*x^5 - 512*x^4 + 173*x^3 - 33*x^2 + x + 1)
\( x^{22} - 5 x^{21} + 17 x^{20} - 40 x^{19} + 70 x^{18} - 93 x^{17} + 99 x^{16} - 86 x^{15} + 53 x^{14} + \cdots + 1 \)
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: | $[2, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1767712543554828434373148672\)
\(\medspace = 2^{16}\cdot 37^{5}\cdot 4441^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.32\) | 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: | $2^{7/4}37^{1/2}4441^{2/3}\approx 5527.862135514047$ | ||
| Ramified primes: |
\(2\), \(37\), \(4441\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{37}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{202754029442297}a^{21}+\frac{79688049092785}{202754029442297}a^{20}-\frac{1478281960188}{202754029442297}a^{19}-\frac{55105790424430}{202754029442297}a^{18}-\frac{11264961496878}{202754029442297}a^{17}-\frac{4723551140204}{202754029442297}a^{16}-\frac{20916506265705}{202754029442297}a^{15}+\frac{86762012322381}{202754029442297}a^{14}-\frac{1557648614393}{202754029442297}a^{13}-\frac{29660620814315}{202754029442297}a^{12}+\frac{62720115835771}{202754029442297}a^{11}-\frac{82360132251769}{202754029442297}a^{10}+\frac{12109015057649}{202754029442297}a^{9}-\frac{36883678097667}{202754029442297}a^{8}-\frac{51530693902629}{202754029442297}a^{7}+\frac{80735861159545}{202754029442297}a^{6}+\frac{12215261598062}{202754029442297}a^{5}+\frac{60644651752149}{202754029442297}a^{4}-\frac{34062663527123}{202754029442297}a^{3}+\frac{88672585002187}{202754029442297}a^{2}+\frac{41315886664057}{202754029442297}a+\frac{57864770783523}{202754029442297}$
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: |
$\frac{1946041744636}{2278135162273}a^{21}-\frac{6638994035987}{2278135162273}a^{20}+\frac{21408620171262}{2278135162273}a^{19}-\frac{40131052244184}{2278135162273}a^{18}+\frac{60713495363720}{2278135162273}a^{17}-\frac{63167130188986}{2278135162273}a^{16}+\frac{61123879075396}{2278135162273}a^{15}-\frac{39457819172365}{2278135162273}a^{14}+\frac{11483357136605}{2278135162273}a^{13}+\frac{106942149640487}{2278135162273}a^{12}-\frac{382296458378708}{2278135162273}a^{11}+\frac{843392328476426}{2278135162273}a^{10}-\frac{13\!\cdots\!12}{2278135162273}a^{9}+\frac{16\!\cdots\!31}{2278135162273}a^{8}-\frac{16\!\cdots\!48}{2278135162273}a^{7}+\frac{12\!\cdots\!82}{2278135162273}a^{6}-\frac{784583114862857}{2278135162273}a^{5}+\frac{348325595893351}{2278135162273}a^{4}-\frac{129259084819211}{2278135162273}a^{3}+\frac{21404937614254}{2278135162273}a^{2}-\frac{2456269274825}{2278135162273}a+\frac{1863737548628}{2278135162273}$, $\frac{1998922649978}{2278135162273}a^{21}-\frac{6224917000407}{2278135162273}a^{20}+\frac{19791297332161}{2278135162273}a^{19}-\frac{34227639823382}{2278135162273}a^{18}+\frac{48539101650115}{2278135162273}a^{17}-\frac{43951230826279}{2278135162273}a^{16}+\frac{39959872249820}{2278135162273}a^{15}-\frac{19082919275724}{2278135162273}a^{14}-\frac{2491195936218}{2278135162273}a^{13}+\frac{114080557312161}{2278135162273}a^{12}-\frac{358500710196921}{2278135162273}a^{11}+\frac{740308391781517}{2278135162273}a^{10}-\frac{11\!\cdots\!95}{2278135162273}a^{9}+\frac{12\!\cdots\!04}{2278135162273}a^{8}-\frac{10\!\cdots\!37}{2278135162273}a^{7}+\frac{712391723444858}{2278135162273}a^{6}-\frac{353034908192362}{2278135162273}a^{5}+\frac{85936310240539}{2278135162273}a^{4}-\frac{23576504344760}{2278135162273}a^{3}-\frac{9873387353396}{2278135162273}a^{2}-\frac{3198499506727}{2278135162273}a+\frac{757757480455}{2278135162273}$, $\frac{839941390857819}{202754029442297}a^{21}-\frac{33\!\cdots\!20}{202754029442297}a^{20}+\frac{10\!\cdots\!86}{202754029442297}a^{19}-\frac{22\!\cdots\!68}{202754029442297}a^{18}+\frac{35\!\cdots\!82}{202754029442297}a^{17}-\frac{40\!\cdots\!70}{202754029442297}a^{16}+\frac{39\!\cdots\!79}{202754029442297}a^{15}-\frac{29\!\cdots\!58}{202754029442297}a^{14}+\frac{12\!\cdots\!80}{202754029442297}a^{13}+\frac{45\!\cdots\!41}{202754029442297}a^{12}-\frac{19\!\cdots\!71}{202754029442297}a^{11}+\frac{45\!\cdots\!73}{202754029442297}a^{10}-\frac{77\!\cdots\!95}{202754029442297}a^{9}+\frac{10\!\cdots\!19}{202754029442297}a^{8}-\frac{10\!\cdots\!99}{202754029442297}a^{7}+\frac{88\!\cdots\!53}{202754029442297}a^{6}-\frac{58\!\cdots\!02}{202754029442297}a^{5}+\frac{28\!\cdots\!83}{202754029442297}a^{4}-\frac{10\!\cdots\!90}{202754029442297}a^{3}+\frac{22\!\cdots\!39}{202754029442297}a^{2}-\frac{14\!\cdots\!95}{202754029442297}a-\frac{559434501483143}{202754029442297}$, $\frac{835234990282381}{202754029442297}a^{21}-\frac{33\!\cdots\!40}{202754029442297}a^{20}+\frac{10\!\cdots\!75}{202754029442297}a^{19}-\frac{22\!\cdots\!46}{202754029442297}a^{18}+\frac{36\!\cdots\!27}{202754029442297}a^{17}-\frac{42\!\cdots\!93}{202754029442297}a^{16}+\frac{41\!\cdots\!43}{202754029442297}a^{15}-\frac{30\!\cdots\!07}{202754029442297}a^{14}+\frac{13\!\cdots\!27}{202754029442297}a^{13}+\frac{44\!\cdots\!55}{202754029442297}a^{12}-\frac{19\!\cdots\!14}{202754029442297}a^{11}+\frac{46\!\cdots\!74}{202754029442297}a^{10}-\frac{79\!\cdots\!08}{202754029442297}a^{9}+\frac{10\!\cdots\!22}{202754029442297}a^{8}-\frac{11\!\cdots\!78}{202754029442297}a^{7}+\frac{93\!\cdots\!89}{202754029442297}a^{6}-\frac{61\!\cdots\!57}{202754029442297}a^{5}+\frac{30\!\cdots\!51}{202754029442297}a^{4}-\frac{11\!\cdots\!29}{202754029442297}a^{3}+\frac{25\!\cdots\!89}{202754029442297}a^{2}-\frac{13\!\cdots\!17}{202754029442297}a-\frac{663756304858043}{202754029442297}$, $\frac{135163805162867}{202754029442297}a^{21}-\frac{555119167692120}{202754029442297}a^{20}+\frac{17\!\cdots\!93}{202754029442297}a^{19}-\frac{37\!\cdots\!14}{202754029442297}a^{18}+\frac{59\!\cdots\!11}{202754029442297}a^{17}-\frac{69\!\cdots\!55}{202754029442297}a^{16}+\frac{69\!\cdots\!66}{202754029442297}a^{15}-\frac{53\!\cdots\!62}{202754029442297}a^{14}+\frac{25\!\cdots\!78}{202754029442297}a^{13}+\frac{70\!\cdots\!13}{202754029442297}a^{12}-\frac{31\!\cdots\!84}{202754029442297}a^{11}+\frac{76\!\cdots\!20}{202754029442297}a^{10}-\frac{13\!\cdots\!75}{202754029442297}a^{9}+\frac{17\!\cdots\!09}{202754029442297}a^{8}-\frac{18\!\cdots\!94}{202754029442297}a^{7}+\frac{15\!\cdots\!09}{202754029442297}a^{6}-\frac{10\!\cdots\!25}{202754029442297}a^{5}+\frac{58\!\cdots\!20}{202754029442297}a^{4}-\frac{25\!\cdots\!90}{202754029442297}a^{3}+\frac{73\!\cdots\!82}{202754029442297}a^{2}-\frac{872472462692759}{202754029442297}a-\frac{130139949066225}{202754029442297}$, $\frac{827105077984667}{202754029442297}a^{21}-\frac{31\!\cdots\!92}{202754029442297}a^{20}+\frac{10\!\cdots\!69}{202754029442297}a^{19}-\frac{20\!\cdots\!21}{202754029442297}a^{18}+\frac{32\!\cdots\!76}{202754029442297}a^{17}-\frac{36\!\cdots\!31}{202754029442297}a^{16}+\frac{36\!\cdots\!67}{202754029442297}a^{15}-\frac{26\!\cdots\!63}{202754029442297}a^{14}+\frac{10\!\cdots\!42}{202754029442297}a^{13}+\frac{44\!\cdots\!47}{202754029442297}a^{12}-\frac{18\!\cdots\!47}{202754029442297}a^{11}+\frac{42\!\cdots\!85}{202754029442297}a^{10}-\frac{71\!\cdots\!12}{202754029442297}a^{9}+\frac{93\!\cdots\!60}{202754029442297}a^{8}-\frac{97\!\cdots\!28}{202754029442297}a^{7}+\frac{80\!\cdots\!85}{202754029442297}a^{6}-\frac{52\!\cdots\!67}{202754029442297}a^{5}+\frac{25\!\cdots\!47}{202754029442297}a^{4}-\frac{96\!\cdots\!72}{202754029442297}a^{3}+\frac{18\!\cdots\!50}{202754029442297}a^{2}-\frac{463355823362006}{202754029442297}a-\frac{831412313350918}{202754029442297}$, $\frac{668971452431538}{202754029442297}a^{21}-\frac{24\!\cdots\!85}{202754029442297}a^{20}+\frac{79\!\cdots\!44}{202754029442297}a^{19}-\frac{15\!\cdots\!55}{202754029442297}a^{18}+\frac{24\!\cdots\!94}{202754029442297}a^{17}-\frac{27\!\cdots\!77}{202754029442297}a^{16}+\frac{26\!\cdots\!33}{202754029442297}a^{15}-\frac{19\!\cdots\!14}{202754029442297}a^{14}+\frac{74\!\cdots\!00}{202754029442297}a^{13}+\frac{36\!\cdots\!31}{202754029442297}a^{12}-\frac{14\!\cdots\!39}{202754029442297}a^{11}+\frac{32\!\cdots\!96}{202754029442297}a^{10}-\frac{54\!\cdots\!81}{202754029442297}a^{9}+\frac{69\!\cdots\!46}{202754029442297}a^{8}-\frac{71\!\cdots\!12}{202754029442297}a^{7}+\frac{58\!\cdots\!91}{202754029442297}a^{6}-\frac{38\!\cdots\!40}{202754029442297}a^{5}+\frac{18\!\cdots\!88}{202754029442297}a^{4}-\frac{71\!\cdots\!99}{202754029442297}a^{3}+\frac{15\!\cdots\!67}{202754029442297}a^{2}-\frac{15\!\cdots\!39}{202754029442297}a-\frac{181269701790142}{202754029442297}$, $\frac{403296665228262}{202754029442297}a^{21}-\frac{15\!\cdots\!39}{202754029442297}a^{20}+\frac{52\!\cdots\!72}{202754029442297}a^{19}-\frac{10\!\cdots\!82}{202754029442297}a^{18}+\frac{17\!\cdots\!21}{202754029442297}a^{17}-\frac{20\!\cdots\!50}{202754029442297}a^{16}+\frac{19\!\cdots\!17}{202754029442297}a^{15}-\frac{15\!\cdots\!35}{202754029442297}a^{14}+\frac{69\!\cdots\!91}{202754029442297}a^{13}+\frac{21\!\cdots\!13}{202754029442297}a^{12}-\frac{91\!\cdots\!52}{202754029442297}a^{11}+\frac{21\!\cdots\!66}{202754029442297}a^{10}-\frac{37\!\cdots\!18}{202754029442297}a^{9}+\frac{50\!\cdots\!20}{202754029442297}a^{8}-\frac{53\!\cdots\!33}{202754029442297}a^{7}+\frac{45\!\cdots\!97}{202754029442297}a^{6}-\frac{30\!\cdots\!42}{202754029442297}a^{5}+\frac{15\!\cdots\!47}{202754029442297}a^{4}-\frac{64\!\cdots\!92}{202754029442297}a^{3}+\frac{17\!\cdots\!32}{202754029442297}a^{2}-\frac{22\!\cdots\!09}{202754029442297}a-\frac{243313774016409}{202754029442297}$, $\frac{414397613645}{2278135162273}a^{21}-\frac{125946323589}{2278135162273}a^{20}+\frac{405765395978}{2278135162273}a^{19}+\frac{4832715625462}{2278135162273}a^{18}-\frac{11123219289034}{2278135162273}a^{17}+\frac{22174517294735}{2278135162273}a^{16}-\frac{22141766438131}{2278135162273}a^{15}+\frac{25485684301926}{2278135162273}a^{14}-\frac{17494745649180}{2278135162273}a^{13}+\frac{26816068841470}{2278135162273}a^{12}-\frac{10332375021048}{2278135162273}a^{11}-\frac{59895114962898}{2278135162273}a^{10}+\frac{224696691304441}{2278135162273}a^{9}-\frac{441836813574902}{2278135162273}a^{8}+\frac{608060814273356}{2278135162273}a^{7}-\frac{641194117314023}{2278135162273}a^{6}+\frac{501784279595082}{2278135162273}a^{5}-\frac{325430558944197}{2278135162273}a^{4}+\frac{136154017707111}{2278135162273}a^{3}-\frac{57568297658626}{2278135162273}a^{2}+\frac{7729816363969}{2278135162273}a-\frac{2041871661180}{2278135162273}$, $\frac{146193524101864}{202754029442297}a^{21}-\frac{510327304387893}{202754029442297}a^{20}+\frac{15\!\cdots\!06}{202754029442297}a^{19}-\frac{29\!\cdots\!13}{202754029442297}a^{18}+\frac{42\!\cdots\!83}{202754029442297}a^{17}-\frac{42\!\cdots\!21}{202754029442297}a^{16}+\frac{37\!\cdots\!49}{202754029442297}a^{15}-\frac{21\!\cdots\!07}{202754029442297}a^{14}+\frac{18724476982916}{202754029442297}a^{13}+\frac{86\!\cdots\!19}{202754029442297}a^{12}-\frac{29\!\cdots\!64}{202754029442297}a^{11}+\frac{62\!\cdots\!64}{202754029442297}a^{10}-\frac{97\!\cdots\!83}{202754029442297}a^{9}+\frac{11\!\cdots\!28}{202754029442297}a^{8}-\frac{10\!\cdots\!17}{202754029442297}a^{7}+\frac{72\!\cdots\!74}{202754029442297}a^{6}-\frac{36\!\cdots\!62}{202754029442297}a^{5}+\frac{10\!\cdots\!90}{202754029442297}a^{4}-\frac{98569279407055}{202754029442297}a^{3}-\frac{21\!\cdots\!07}{202754029442297}a^{2}+\frac{10\!\cdots\!11}{202754029442297}a-\frac{87215782465093}{202754029442297}$, $\frac{388585934915240}{202754029442297}a^{21}-\frac{15\!\cdots\!84}{202754029442297}a^{20}+\frac{50\!\cdots\!36}{202754029442297}a^{19}-\frac{10\!\cdots\!42}{202754029442297}a^{18}+\frac{16\!\cdots\!29}{202754029442297}a^{17}-\frac{19\!\cdots\!26}{202754029442297}a^{16}+\frac{19\!\cdots\!69}{202754029442297}a^{15}-\frac{14\!\cdots\!85}{202754029442297}a^{14}+\frac{67\!\cdots\!67}{202754029442297}a^{13}+\frac{20\!\cdots\!28}{202754029442297}a^{12}-\frac{87\!\cdots\!99}{202754029442297}a^{11}+\frac{21\!\cdots\!61}{202754029442297}a^{10}-\frac{36\!\cdots\!33}{202754029442297}a^{9}+\frac{48\!\cdots\!74}{202754029442297}a^{8}-\frac{51\!\cdots\!61}{202754029442297}a^{7}+\frac{43\!\cdots\!91}{202754029442297}a^{6}-\frac{29\!\cdots\!60}{202754029442297}a^{5}+\frac{15\!\cdots\!83}{202754029442297}a^{4}-\frac{60\!\cdots\!51}{202754029442297}a^{3}+\frac{14\!\cdots\!31}{202754029442297}a^{2}-\frac{15\!\cdots\!99}{202754029442297}a-\frac{412264122043307}{202754029442297}$
| 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: | \( 51691.3054088 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 51691.3054088 \cdot 1}{2\cdot\sqrt{1767712543554828434373148672}}\cr\approx \mathstrut & 0.235798130420 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 5*x^21 + 17*x^20 - 40*x^19 + 70*x^18 - 93*x^17 + 99*x^16 - 86*x^15 + 53*x^14 + 37*x^13 - 283*x^12 + 778*x^11 - 1493*x^10 + 2198*x^9 - 2575*x^8 + 2425*x^7 - 1840*x^6 + 1108*x^5 - 512*x^4 + 173*x^3 - 33*x^2 + 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 - 5*x^21 + 17*x^20 - 40*x^19 + 70*x^18 - 93*x^17 + 99*x^16 - 86*x^15 + 53*x^14 + 37*x^13 - 283*x^12 + 778*x^11 - 1493*x^10 + 2198*x^9 - 2575*x^8 + 2425*x^7 - 1840*x^6 + 1108*x^5 - 512*x^4 + 173*x^3 - 33*x^2 + 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 - 5*x^21 + 17*x^20 - 40*x^19 + 70*x^18 - 93*x^17 + 99*x^16 - 86*x^15 + 53*x^14 + 37*x^13 - 283*x^12 + 778*x^11 - 1493*x^10 + 2198*x^9 - 2575*x^8 + 2425*x^7 - 1840*x^6 + 1108*x^5 - 512*x^4 + 173*x^3 - 33*x^2 + 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 - 5*x^21 + 17*x^20 - 40*x^19 + 70*x^18 - 93*x^17 + 99*x^16 - 86*x^15 + 53*x^14 + 37*x^13 - 283*x^12 + 778*x^11 - 1493*x^10 + 2198*x^9 - 2575*x^8 + 2425*x^7 - 1840*x^6 + 1108*x^5 - 512*x^4 + 173*x^3 - 33*x^2 + 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
| 11.3.6912019581184.1 |
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 | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | $22$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | $18{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $22$ | ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | R | ${\href{/padicField/41.11.0.1}{11} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.11.0.1}{11} }^{2}$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | $22$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.0.1 | $x^{6} + x^{4} + x^{3} + x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 2.8.8.3 | $x^{8} + 2 x^{7} + 32 x^{6} + 116 x^{5} + 456 x^{4} + 696 x^{3} + 1152 x^{2} + 432 x + 1296$ | $2$ | $4$ | $8$ | $C_2^3: C_4$ | $[2, 2, 2]^{4}$ | |
| 2.8.8.3 | $x^{8} + 2 x^{7} + 32 x^{6} + 116 x^{5} + 456 x^{4} + 696 x^{3} + 1152 x^{2} + 432 x + 1296$ | $2$ | $4$ | $8$ | $C_2^3: C_4$ | $[2, 2, 2]^{4}$ | |
|
\(37\)
| 37.2.1.2 | $x^{2} + 74$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 37.4.0.1 | $x^{4} + 6 x^{2} + 24 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 37.4.0.1 | $x^{4} + 6 x^{2} + 24 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 37.4.0.1 | $x^{4} + 6 x^{2} + 24 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(4441\)
| $\Q_{4441}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{4441}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{4441}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{4441}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $6$ | $3$ | $2$ | $4$ | ||||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |