Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^18 - 2*x^17 + 36*x^16 - 91*x^14 + 5*x^13 + 221*x^12 + 163*x^11 - 156*x^10 - 264*x^9 - 62*x^8 + 22*x^7 - 69*x^6 - 26*x^5 + 47*x^4 + 2*x^3 - 7*x^2 + 10*x - 1)
gp: K = bnfinit(y^20 - 10*y^18 - 2*y^17 + 36*y^16 - 91*y^14 + 5*y^13 + 221*y^12 + 163*y^11 - 156*y^10 - 264*y^9 - 62*y^8 + 22*y^7 - 69*y^6 - 26*y^5 + 47*y^4 + 2*y^3 - 7*y^2 + 10*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 10*x^18 - 2*x^17 + 36*x^16 - 91*x^14 + 5*x^13 + 221*x^12 + 163*x^11 - 156*x^10 - 264*x^9 - 62*x^8 + 22*x^7 - 69*x^6 - 26*x^5 + 47*x^4 + 2*x^3 - 7*x^2 + 10*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 10*x^18 - 2*x^17 + 36*x^16 - 91*x^14 + 5*x^13 + 221*x^12 + 163*x^11 - 156*x^10 - 264*x^9 - 62*x^8 + 22*x^7 - 69*x^6 - 26*x^5 + 47*x^4 + 2*x^3 - 7*x^2 + 10*x - 1)
\( x^{20} - 10 x^{18} - 2 x^{17} + 36 x^{16} - 91 x^{14} + 5 x^{13} + 221 x^{12} + 163 x^{11} - 156 x^{10} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[8, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(4391182420642244613154093\)
\(\medspace = 13^{13}\cdot 347^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.07\) | 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: | $13^{3/4}347^{1/2}\approx 127.53290478781567$ | ||
| Ramified primes: |
\(13\), \(347\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{16164308853863}a^{19}+\frac{1669367367555}{16164308853863}a^{18}-\frac{6651558099593}{16164308853863}a^{17}+\frac{5475916295482}{16164308853863}a^{16}+\frac{4409873689881}{16164308853863}a^{15}-\frac{5815153775116}{16164308853863}a^{14}-\frac{1018610917361}{16164308853863}a^{13}+\frac{5510041340721}{16164308853863}a^{12}-\frac{6480410017993}{16164308853863}a^{11}+\frac{3537008899659}{16164308853863}a^{10}+\frac{5883594707282}{16164308853863}a^{9}-\frac{3114302082037}{16164308853863}a^{8}-\frac{6361090735756}{16164308853863}a^{7}+\frac{1890844133955}{16164308853863}a^{6}+\frac{2666336631673}{16164308853863}a^{5}+\frac{146807941364}{16164308853863}a^{4}+\frac{7130698383469}{16164308853863}a^{3}-\frac{5802159646925}{16164308853863}a^{2}+\frac{1329134576951}{16164308853863}a-\frac{4901564663028}{16164308853863}$
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: | $13$ | 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: |
$\frac{38901237515858}{16164308853863}a^{19}-\frac{51125810669806}{16164308853863}a^{18}-\frac{324894646146082}{16164308853863}a^{17}+\frac{352879864269951}{16164308853863}a^{16}+\frac{962340895899857}{16164308853863}a^{15}-\frac{12\!\cdots\!96}{16164308853863}a^{14}-\frac{19\!\cdots\!50}{16164308853863}a^{13}+\frac{28\!\cdots\!79}{16164308853863}a^{12}+\frac{50\!\cdots\!74}{16164308853863}a^{11}-\frac{493535882437094}{16164308853863}a^{10}-\frac{57\!\cdots\!31}{16164308853863}a^{9}-\frac{26\!\cdots\!42}{16164308853863}a^{8}+\frac{14\!\cdots\!09}{16164308853863}a^{7}-\frac{856702676534819}{16164308853863}a^{6}-\frac{15\!\cdots\!54}{16164308853863}a^{5}+\frac{12\!\cdots\!09}{16164308853863}a^{4}+\frac{336974507977199}{16164308853863}a^{3}-\frac{540126583364564}{16164308853863}a^{2}+\frac{375306177652393}{16164308853863}a-\frac{59505213230565}{16164308853863}$, $\frac{7842363665488}{16164308853863}a^{19}-\frac{2949138360940}{16164308853863}a^{18}-\frac{82466076443963}{16164308853863}a^{17}+\frac{25496833562746}{16164308853863}a^{16}+\frac{304883863668203}{16164308853863}a^{15}-\frac{171276828309887}{16164308853863}a^{14}-\frac{717493970084835}{16164308853863}a^{13}+\frac{453963443127581}{16164308853863}a^{12}+\frac{16\!\cdots\!60}{16164308853863}a^{11}+\frac{376818471695360}{16164308853863}a^{10}-\frac{18\!\cdots\!94}{16164308853863}a^{9}-\frac{13\!\cdots\!10}{16164308853863}a^{8}+\frac{600112644691685}{16164308853863}a^{7}+\frac{364755563050972}{16164308853863}a^{6}-\frac{672918254247246}{16164308853863}a^{5}-\frac{25980066026967}{16164308853863}a^{4}+\frac{319738153733262}{16164308853863}a^{3}-\frac{133702909557547}{16164308853863}a^{2}-\frac{3723066818105}{16164308853863}a+\frac{19665140658422}{16164308853863}$, $\frac{35533652044140}{16164308853863}a^{19}-\frac{39419098311435}{16164308853863}a^{18}-\frac{303498981760397}{16164308853863}a^{17}+\frac{254342296953299}{16164308853863}a^{16}+\frac{929023711850940}{16164308853863}a^{15}-\frac{952785307350795}{16164308853863}a^{14}-\frac{19\!\cdots\!83}{16164308853863}a^{13}+\frac{20\!\cdots\!69}{16164308853863}a^{12}+\frac{51\!\cdots\!12}{16164308853863}a^{11}+\frac{746144146433523}{16164308853863}a^{10}-\frac{52\!\cdots\!48}{16164308853863}a^{9}-\frac{37\!\cdots\!13}{16164308853863}a^{8}+\frac{489236439216214}{16164308853863}a^{7}-\frac{421044739846422}{16164308853863}a^{6}-\frac{13\!\cdots\!96}{16164308853863}a^{5}+\frac{764952476987725}{16164308853863}a^{4}+\frac{530299094649714}{16164308853863}a^{3}-\frac{307026246188930}{16164308853863}a^{2}+\frac{221374516505259}{16164308853863}a-\frac{6061148001293}{16164308853863}$, $\frac{76066665529395}{16164308853863}a^{19}-\frac{91297353976786}{16164308853863}a^{18}-\frac{645784635669514}{16164308853863}a^{17}+\frac{618241545559007}{16164308853863}a^{16}+\frac{19\!\cdots\!89}{16164308853863}a^{15}-\frac{23\!\cdots\!27}{16164308853863}a^{14}-\frac{39\!\cdots\!34}{16164308853863}a^{13}+\frac{50\!\cdots\!56}{16164308853863}a^{12}+\frac{10\!\cdots\!86}{16164308853863}a^{11}+\frac{170943239706893}{16164308853863}a^{10}-\frac{11\!\cdots\!67}{16164308853863}a^{9}-\frac{63\!\cdots\!27}{16164308853863}a^{8}+\frac{20\!\cdots\!46}{16164308853863}a^{7}-\frac{16\!\cdots\!84}{16164308853863}a^{6}-\frac{32\!\cdots\!27}{16164308853863}a^{5}+\frac{21\!\cdots\!50}{16164308853863}a^{4}+\frac{918675428357012}{16164308853863}a^{3}-\frac{907476096681973}{16164308853863}a^{2}+\frac{689009481131600}{16164308853863}a-\frac{72491523006444}{16164308853863}$, $\frac{20130342682970}{16164308853863}a^{19}-\frac{27277905147520}{16164308853863}a^{18}-\frac{164619244935746}{16164308853863}a^{17}+\frac{186584013491448}{16164308853863}a^{16}+\frac{473794275344972}{16164308853863}a^{15}-\frac{677175381348638}{16164308853863}a^{14}-\frac{926022832492593}{16164308853863}a^{13}+\frac{14\!\cdots\!55}{16164308853863}a^{12}+\frac{24\!\cdots\!68}{16164308853863}a^{11}-\frac{349362249238335}{16164308853863}a^{10}-\frac{27\!\cdots\!58}{16164308853863}a^{9}-\frac{957877732120126}{16164308853863}a^{8}+\frac{710784051465782}{16164308853863}a^{7}-\frac{897733779030476}{16164308853863}a^{6}-\frac{11\!\cdots\!49}{16164308853863}a^{5}+\frac{597134902536416}{16164308853863}a^{4}+\frac{210900254564245}{16164308853863}a^{3}-\frac{305905791599635}{16164308853863}a^{2}+\frac{200190923309760}{16164308853863}a+\frac{12058811636007}{16164308853863}$, $\frac{45641075165448}{16164308853863}a^{19}-\frac{47700536833056}{16164308853863}a^{18}-\frac{390806443414222}{16164308853863}a^{17}+\frac{303737214594782}{16164308853863}a^{16}+\frac{11\!\cdots\!66}{16164308853863}a^{15}-\frac{11\!\cdots\!77}{16164308853863}a^{14}-\frac{24\!\cdots\!76}{16164308853863}a^{13}+\frac{25\!\cdots\!23}{16164308853863}a^{12}+\frac{64\!\cdots\!28}{16164308853863}a^{11}+\frac{14\!\cdots\!56}{16164308853863}a^{10}-\frac{59\!\cdots\!09}{16164308853863}a^{9}-\frac{49\!\cdots\!96}{16164308853863}a^{8}-\frac{357189312785492}{16164308853863}a^{7}-\frac{12\!\cdots\!46}{16164308853863}a^{6}-\frac{16\!\cdots\!93}{16164308853863}a^{5}+\frac{10\!\cdots\!53}{16164308853863}a^{4}+\frac{481895007445156}{16164308853863}a^{3}-\frac{407079196553059}{16164308853863}a^{2}+\frac{419213162033895}{16164308853863}a-\frac{29903866135791}{16164308853863}$, $\frac{34927344960489}{16164308853863}a^{19}-\frac{49149569712753}{16164308853863}a^{18}-\frac{288484083950434}{16164308853863}a^{17}+\frac{342812679950379}{16164308853863}a^{16}+\frac{848582179707978}{16164308853863}a^{15}-\frac{12\!\cdots\!86}{16164308853863}a^{14}-\frac{16\!\cdots\!95}{16164308853863}a^{13}+\frac{26\!\cdots\!86}{16164308853863}a^{12}+\frac{44\!\cdots\!71}{16164308853863}a^{11}-\frac{879535894648715}{16164308853863}a^{10}-\frac{54\!\cdots\!16}{16164308853863}a^{9}-\frac{21\!\cdots\!48}{16164308853863}a^{8}+\frac{17\!\cdots\!47}{16164308853863}a^{7}-\frac{527743026732446}{16164308853863}a^{6}-\frac{12\!\cdots\!60}{16164308853863}a^{5}+\frac{11\!\cdots\!76}{16164308853863}a^{4}+\frac{284706901883248}{16164308853863}a^{3}-\frac{484641282031585}{16164308853863}a^{2}+\frac{336947924988737}{16164308853863}a-\frac{73144504676642}{16164308853863}$, $\frac{1136684249928}{16164308853863}a^{19}-\frac{370310324928}{16164308853863}a^{18}-\frac{18670148813080}{16164308853863}a^{17}+\frac{16535251838487}{16164308853863}a^{16}+\frac{85945909305856}{16164308853863}a^{15}-\frac{100949948472435}{16164308853863}a^{14}-\frac{187465623005612}{16164308853863}a^{13}+\frac{273766207672024}{16164308853863}a^{12}+\frac{381723210684809}{16164308853863}a^{11}-\frac{334670181578253}{16164308853863}a^{10}-\frac{788139316717586}{16164308853863}a^{9}-\frac{91795221588801}{16164308853863}a^{8}+\frac{660906448980130}{16164308853863}a^{7}+\frac{296960878534490}{16164308853863}a^{6}-\frac{92641997517772}{16164308853863}a^{5}+\frac{212362827896336}{16164308853863}a^{4}+\frac{118471772639360}{16164308853863}a^{3}-\frac{110210468943212}{16164308853863}a^{2}+\frac{39409419654486}{16164308853863}a-\frac{21398275121532}{16164308853863}$, $\frac{25410642625222}{16164308853863}a^{19}-\frac{24790642067519}{16164308853863}a^{18}-\frac{228516851937562}{16164308853863}a^{17}+\frac{165375460742374}{16164308853863}a^{16}+\frac{745702284385681}{16164308853863}a^{15}-\frac{669761545235185}{16164308853863}a^{14}-\frac{16\!\cdots\!89}{16164308853863}a^{13}+\frac{15\!\cdots\!23}{16164308853863}a^{12}+\frac{41\!\cdots\!47}{16164308853863}a^{11}+\frac{553478584844783}{16164308853863}a^{10}-\frac{45\!\cdots\!05}{16164308853863}a^{9}-\frac{31\!\cdots\!89}{16164308853863}a^{8}+\frac{10\!\cdots\!22}{16164308853863}a^{7}+\frac{392530264799693}{16164308853863}a^{6}-\frac{12\!\cdots\!01}{16164308853863}a^{5}+\frac{326210979403609}{16164308853863}a^{4}+\frac{562028533696752}{16164308853863}a^{3}-\frac{240436143700110}{16164308853863}a^{2}+\frac{101824325572270}{16164308853863}a+\frac{643486875136}{16164308853863}$, $\frac{33916698649}{16164308853863}a^{19}+\frac{3958895375053}{16164308853863}a^{18}-\frac{9609629623968}{16164308853863}a^{17}-\frac{19743159136946}{16164308853863}a^{16}+\frac{50428705263323}{16164308853863}a^{15}+\frac{19930832339196}{16164308853863}a^{14}-\frac{111907403982555}{16164308853863}a^{13}-\frac{12788589754907}{16164308853863}a^{12}+\frac{176057295652662}{16164308853863}a^{11}+\frac{206925252826763}{16164308853863}a^{10}+\frac{7476898835578}{16164308853863}a^{9}-\frac{187825517588346}{16164308853863}a^{8}-\frac{206841678196686}{16164308853863}a^{7}-\frac{205047551062076}{16164308853863}a^{6}-\frac{67885957897147}{16164308853863}a^{5}+\frac{132597680833992}{16164308853863}a^{4}+\frac{7720130485879}{16164308853863}a^{3}-\frac{72013601647600}{16164308853863}a^{2}+\frac{62332839579466}{16164308853863}a-\frac{21220353239982}{16164308853863}$, $\frac{106748518776358}{16164308853863}a^{19}-\frac{123129445315192}{16164308853863}a^{18}-\frac{924659070169027}{16164308853863}a^{17}+\frac{852996801621996}{16164308853863}a^{16}+\frac{28\!\cdots\!19}{16164308853863}a^{15}-\frac{32\!\cdots\!21}{16164308853863}a^{14}-\frac{58\!\cdots\!65}{16164308853863}a^{13}+\frac{72\!\cdots\!67}{16164308853863}a^{12}+\frac{15\!\cdots\!24}{16164308853863}a^{11}+\frac{73524572425533}{16164308853863}a^{10}-\frac{16\!\cdots\!56}{16164308853863}a^{9}-\frac{90\!\cdots\!34}{16164308853863}a^{8}+\frac{36\!\cdots\!49}{16164308853863}a^{7}-\frac{19\!\cdots\!53}{16164308853863}a^{6}-\frac{49\!\cdots\!59}{16164308853863}a^{5}+\frac{30\!\cdots\!90}{16164308853863}a^{4}+\frac{15\!\cdots\!85}{16164308853863}a^{3}-\frac{14\!\cdots\!84}{16164308853863}a^{2}+\frac{972649310046569}{16164308853863}a-\frac{83275645403157}{16164308853863}$, $\frac{8103130127930}{16164308853863}a^{19}-\frac{3000643060642}{16164308853863}a^{18}-\frac{87670796402783}{16164308853863}a^{17}+\frac{32823432304911}{16164308853863}a^{16}+\frac{324519634341657}{16164308853863}a^{15}-\frac{214168391687697}{16164308853863}a^{14}-\frac{735276655329449}{16164308853863}a^{13}+\frac{553277624451402}{16164308853863}a^{12}+\frac{16\!\cdots\!65}{16164308853863}a^{11}+\frac{257257153427872}{16164308853863}a^{10}-\frac{19\!\cdots\!99}{16164308853863}a^{9}-\frac{13\!\cdots\!83}{16164308853863}a^{8}+\frac{585473837554057}{16164308853863}a^{7}+\frac{255158985573929}{16164308853863}a^{6}-\frac{568236807032922}{16164308853863}a^{5}+\frac{204954762726284}{16164308853863}a^{4}+\frac{300051739468187}{16164308853863}a^{3}-\frac{185549976374026}{16164308853863}a^{2}+\frac{81526784903608}{16164308853863}a-\frac{1979252313860}{16164308853863}$, $\frac{26642979986017}{16164308853863}a^{19}-\frac{22562036296791}{16164308853863}a^{18}-\frac{236844794019851}{16164308853863}a^{17}+\frac{135807458581593}{16164308853863}a^{16}+\frac{751469457749589}{16164308853863}a^{15}-\frac{558559660605261}{16164308853863}a^{14}-\frac{16\!\cdots\!45}{16164308853863}a^{13}+\frac{12\!\cdots\!77}{16164308853863}a^{12}+\frac{41\!\cdots\!32}{16164308853863}a^{11}+\frac{14\!\cdots\!67}{16164308853863}a^{10}-\frac{37\!\cdots\!89}{16164308853863}a^{9}-\frac{37\!\cdots\!35}{16164308853863}a^{8}-\frac{401255486019551}{16164308853863}a^{7}-\frac{251272426870019}{16164308853863}a^{6}-\frac{980382018695202}{16164308853863}a^{5}+\frac{391637194994689}{16164308853863}a^{4}+\frac{400558219012837}{16164308853863}a^{3}-\frac{198252072986059}{16164308853863}a^{2}+\frac{152458009434872}{16164308853863}a+\frac{2752437993924}{16164308853863}$
| 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: | \( 45341.4653783 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{6}\cdot 45341.4653783 \cdot 1}{2\cdot\sqrt{4391182420642244613154093}}\cr\approx \mathstrut & 0.170409574513 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^18 - 2*x^17 + 36*x^16 - 91*x^14 + 5*x^13 + 221*x^12 + 163*x^11 - 156*x^10 - 264*x^9 - 62*x^8 + 22*x^7 - 69*x^6 - 26*x^5 + 47*x^4 + 2*x^3 - 7*x^2 + 10*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^20 - 10*x^18 - 2*x^17 + 36*x^16 - 91*x^14 + 5*x^13 + 221*x^12 + 163*x^11 - 156*x^10 - 264*x^9 - 62*x^8 + 22*x^7 - 69*x^6 - 26*x^5 + 47*x^4 + 2*x^3 - 7*x^2 + 10*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^20 - 10*x^18 - 2*x^17 + 36*x^16 - 91*x^14 + 5*x^13 + 221*x^12 + 163*x^11 - 156*x^10 - 264*x^9 - 62*x^8 + 22*x^7 - 69*x^6 - 26*x^5 + 47*x^4 + 2*x^3 - 7*x^2 + 10*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^20 - 10*x^18 - 2*x^17 + 36*x^16 - 91*x^14 + 5*x^13 + 221*x^12 + 163*x^11 - 156*x^10 - 264*x^9 - 62*x^8 + 22*x^7 - 69*x^6 - 26*x^5 + 47*x^4 + 2*x^3 - 7*x^2 + 10*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
$C_4\times C_2^4:S_5$ (as 20T369):
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 7680 |
| The 72 conjugacy class representatives for $C_4\times C_2^4:S_5$ |
| Character table for $C_4\times C_2^4:S_5$ |
Intermediate fields
| \(\Q(\sqrt{13}) \), 5.3.4511.1, 10.6.44707018837.1 |
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 20 sibling: | data not computed |
| Degree 40 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 | $20$ | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | $20$ | $20$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.4.0.1}{4} }^{5}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.10.0.1}{10} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 13.8.6.2 | $x^{8} + 130 x^{4} - 1521$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 13.8.4.1 | $x^{8} + 520 x^{7} + 101458 x^{6} + 8810644 x^{5} + 288610205 x^{4} + 142111548 x^{3} + 982314112 x^{2} + 3617879976 x + 920156436$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(347\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |