Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 3*x^18 + 6*x^17 + 33*x^16 + 34*x^15 + 65*x^14 + 69*x^13 + 127*x^12 + 132*x^11 + 144*x^10 + 143*x^9 + 101*x^8 + 141*x^7 + 116*x^6 - 32*x^5 + 110*x^4 - 20*x^3 + 20*x^2 + 6*x + 1)
gp: K = bnfinit(y^20 - y^19 + 3*y^18 + 6*y^17 + 33*y^16 + 34*y^15 + 65*y^14 + 69*y^13 + 127*y^12 + 132*y^11 + 144*y^10 + 143*y^9 + 101*y^8 + 141*y^7 + 116*y^6 - 32*y^5 + 110*y^4 - 20*y^3 + 20*y^2 + 6*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^19 + 3*x^18 + 6*x^17 + 33*x^16 + 34*x^15 + 65*x^14 + 69*x^13 + 127*x^12 + 132*x^11 + 144*x^10 + 143*x^9 + 101*x^8 + 141*x^7 + 116*x^6 - 32*x^5 + 110*x^4 - 20*x^3 + 20*x^2 + 6*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - x^19 + 3*x^18 + 6*x^17 + 33*x^16 + 34*x^15 + 65*x^14 + 69*x^13 + 127*x^12 + 132*x^11 + 144*x^10 + 143*x^9 + 101*x^8 + 141*x^7 + 116*x^6 - 32*x^5 + 110*x^4 - 20*x^3 + 20*x^2 + 6*x + 1)
\( x^{20} - x^{19} + 3 x^{18} + 6 x^{17} + 33 x^{16} + 34 x^{15} + 65 x^{14} + 69 x^{13} + 127 x^{12} + \cdots + 1 \)
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: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(823067302269314181883621609\)
\(\medspace = 11^{18}\cdot 23^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(22.17\) | 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: | $11^{9/10}23^{1/2}\approx 41.50661671665305$ | ||
| Ramified primes: |
\(11\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{15}-\frac{1}{3}a^{14}+\frac{1}{3}a^{12}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{9}a^{17}-\frac{1}{9}a^{16}-\frac{1}{3}a^{15}-\frac{4}{9}a^{14}+\frac{4}{9}a^{13}-\frac{2}{9}a^{12}-\frac{1}{3}a^{11}-\frac{1}{9}a^{10}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{2}{9}a^{7}+\frac{2}{9}a^{6}-\frac{1}{9}a^{5}-\frac{1}{9}a^{4}-\frac{1}{9}a^{2}+\frac{4}{9}$, $\frac{1}{621}a^{18}+\frac{11}{207}a^{17}+\frac{8}{621}a^{16}+\frac{56}{621}a^{15}-\frac{56}{207}a^{14}+\frac{80}{621}a^{13}-\frac{206}{621}a^{12}-\frac{40}{621}a^{11}-\frac{280}{621}a^{10}-\frac{35}{207}a^{9}+\frac{76}{621}a^{8}+\frac{53}{207}a^{7}+\frac{103}{621}a^{6}-\frac{80}{621}a^{5}-\frac{286}{621}a^{4}+\frac{116}{621}a^{3}-\frac{25}{621}a^{2}-\frac{32}{621}a+\frac{154}{621}$, $\frac{1}{98\!\cdots\!47}a^{19}+\frac{76218567797968}{98\!\cdots\!47}a^{18}+\frac{34\!\cdots\!63}{98\!\cdots\!47}a^{17}-\frac{130516148346665}{22\!\cdots\!29}a^{16}+\frac{26\!\cdots\!35}{98\!\cdots\!47}a^{15}+\frac{31\!\cdots\!02}{98\!\cdots\!47}a^{14}+\frac{25\!\cdots\!07}{10\!\cdots\!83}a^{13}+\frac{11\!\cdots\!52}{32\!\cdots\!49}a^{12}-\frac{50\!\cdots\!83}{98\!\cdots\!47}a^{11}-\frac{38\!\cdots\!09}{98\!\cdots\!47}a^{10}-\frac{16\!\cdots\!98}{98\!\cdots\!47}a^{9}+\frac{39\!\cdots\!61}{98\!\cdots\!47}a^{8}-\frac{820654432835378}{22\!\cdots\!29}a^{7}+\frac{16\!\cdots\!74}{98\!\cdots\!47}a^{6}-\frac{58\!\cdots\!87}{32\!\cdots\!49}a^{5}+\frac{42\!\cdots\!58}{98\!\cdots\!47}a^{4}+\frac{27\!\cdots\!96}{98\!\cdots\!47}a^{3}+\frac{72\!\cdots\!84}{32\!\cdots\!49}a^{2}+\frac{30\!\cdots\!96}{98\!\cdots\!47}a-\frac{10\!\cdots\!52}{98\!\cdots\!47}$
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
$C_{2}\times C_{2}$, which has order $4$
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: | $9$ | 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: |
$\frac{37\!\cdots\!72}{98\!\cdots\!47}a^{19}-\frac{31\!\cdots\!62}{98\!\cdots\!47}a^{18}+\frac{12\!\cdots\!89}{98\!\cdots\!47}a^{17}+\frac{574303985415520}{22\!\cdots\!29}a^{16}+\frac{13\!\cdots\!12}{98\!\cdots\!47}a^{15}+\frac{16\!\cdots\!09}{98\!\cdots\!47}a^{14}+\frac{11\!\cdots\!94}{32\!\cdots\!49}a^{13}+\frac{13\!\cdots\!14}{32\!\cdots\!49}a^{12}+\frac{65\!\cdots\!65}{98\!\cdots\!47}a^{11}+\frac{70\!\cdots\!22}{98\!\cdots\!47}a^{10}+\frac{80\!\cdots\!55}{98\!\cdots\!47}a^{9}+\frac{87\!\cdots\!69}{98\!\cdots\!47}a^{8}+\frac{16\!\cdots\!28}{22\!\cdots\!29}a^{7}+\frac{72\!\cdots\!41}{98\!\cdots\!47}a^{6}+\frac{60\!\cdots\!12}{10\!\cdots\!83}a^{5}-\frac{59\!\cdots\!59}{98\!\cdots\!47}a^{4}+\frac{51\!\cdots\!75}{98\!\cdots\!47}a^{3}-\frac{10\!\cdots\!40}{10\!\cdots\!83}a^{2}-\frac{42\!\cdots\!07}{98\!\cdots\!47}a+\frac{26\!\cdots\!73}{98\!\cdots\!47}$, $\frac{43\!\cdots\!25}{32\!\cdots\!49}a^{19}-\frac{24\!\cdots\!33}{10\!\cdots\!83}a^{18}+\frac{13\!\cdots\!00}{32\!\cdots\!49}a^{17}+\frac{458356856247599}{763833397232043}a^{16}+\frac{39\!\cdots\!13}{10\!\cdots\!83}a^{15}+\frac{35\!\cdots\!84}{32\!\cdots\!49}a^{14}+\frac{95\!\cdots\!18}{32\!\cdots\!49}a^{13}+\frac{12\!\cdots\!88}{32\!\cdots\!49}a^{12}+\frac{18\!\cdots\!94}{32\!\cdots\!49}a^{11}+\frac{71\!\cdots\!31}{10\!\cdots\!83}a^{10}-\frac{84\!\cdots\!33}{32\!\cdots\!49}a^{9}-\frac{50\!\cdots\!07}{10\!\cdots\!83}a^{8}-\frac{73\!\cdots\!16}{763833397232043}a^{7}+\frac{23\!\cdots\!63}{32\!\cdots\!49}a^{6}-\frac{93\!\cdots\!43}{32\!\cdots\!49}a^{5}-\frac{75\!\cdots\!79}{32\!\cdots\!49}a^{4}+\frac{32\!\cdots\!89}{32\!\cdots\!49}a^{3}-\frac{23\!\cdots\!35}{32\!\cdots\!49}a^{2}+\frac{20\!\cdots\!27}{32\!\cdots\!49}a+\frac{14\!\cdots\!55}{36\!\cdots\!61}$, $\frac{79\!\cdots\!53}{98\!\cdots\!47}a^{19}-\frac{91\!\cdots\!95}{98\!\cdots\!47}a^{18}+\frac{25\!\cdots\!82}{98\!\cdots\!47}a^{17}+\frac{10\!\cdots\!12}{22\!\cdots\!29}a^{16}+\frac{25\!\cdots\!41}{98\!\cdots\!47}a^{15}+\frac{23\!\cdots\!99}{98\!\cdots\!47}a^{14}+\frac{16\!\cdots\!86}{32\!\cdots\!49}a^{13}+\frac{17\!\cdots\!27}{32\!\cdots\!49}a^{12}+\frac{10\!\cdots\!58}{98\!\cdots\!47}a^{11}+\frac{98\!\cdots\!61}{98\!\cdots\!47}a^{10}+\frac{11\!\cdots\!31}{98\!\cdots\!47}a^{9}+\frac{11\!\cdots\!73}{98\!\cdots\!47}a^{8}+\frac{19\!\cdots\!34}{22\!\cdots\!29}a^{7}+\frac{11\!\cdots\!12}{98\!\cdots\!47}a^{6}+\frac{99\!\cdots\!04}{10\!\cdots\!83}a^{5}-\frac{32\!\cdots\!04}{98\!\cdots\!47}a^{4}+\frac{98\!\cdots\!61}{98\!\cdots\!47}a^{3}-\frac{89\!\cdots\!27}{36\!\cdots\!61}a^{2}+\frac{18\!\cdots\!55}{98\!\cdots\!47}a-\frac{17\!\cdots\!93}{98\!\cdots\!47}$, $\frac{14\!\cdots\!55}{98\!\cdots\!47}a^{19}-\frac{19\!\cdots\!34}{98\!\cdots\!47}a^{18}+\frac{51\!\cdots\!91}{98\!\cdots\!47}a^{17}+\frac{15\!\cdots\!15}{22\!\cdots\!29}a^{16}+\frac{45\!\cdots\!55}{98\!\cdots\!47}a^{15}+\frac{33\!\cdots\!93}{98\!\cdots\!47}a^{14}+\frac{95\!\cdots\!49}{10\!\cdots\!83}a^{13}+\frac{24\!\cdots\!72}{32\!\cdots\!49}a^{12}+\frac{16\!\cdots\!48}{98\!\cdots\!47}a^{11}+\frac{14\!\cdots\!72}{98\!\cdots\!47}a^{10}+\frac{17\!\cdots\!43}{98\!\cdots\!47}a^{9}+\frac{16\!\cdots\!77}{98\!\cdots\!47}a^{8}+\frac{24\!\cdots\!10}{22\!\cdots\!29}a^{7}+\frac{18\!\cdots\!77}{98\!\cdots\!47}a^{6}+\frac{39\!\cdots\!29}{32\!\cdots\!49}a^{5}-\frac{68\!\cdots\!47}{98\!\cdots\!47}a^{4}+\frac{20\!\cdots\!07}{98\!\cdots\!47}a^{3}-\frac{35\!\cdots\!10}{32\!\cdots\!49}a^{2}+\frac{64\!\cdots\!95}{98\!\cdots\!47}a+\frac{42\!\cdots\!94}{98\!\cdots\!47}$, $\frac{26\!\cdots\!73}{98\!\cdots\!47}a^{19}-\frac{30\!\cdots\!45}{98\!\cdots\!47}a^{18}+\frac{82\!\cdots\!81}{98\!\cdots\!47}a^{17}+\frac{33\!\cdots\!43}{22\!\cdots\!29}a^{16}+\frac{84\!\cdots\!49}{98\!\cdots\!47}a^{15}+\frac{76\!\cdots\!70}{98\!\cdots\!47}a^{14}+\frac{51\!\cdots\!12}{32\!\cdots\!49}a^{13}+\frac{49\!\cdots\!85}{32\!\cdots\!49}a^{12}+\frac{29\!\cdots\!29}{98\!\cdots\!47}a^{11}+\frac{28\!\cdots\!71}{98\!\cdots\!47}a^{10}+\frac{30\!\cdots\!90}{98\!\cdots\!47}a^{9}+\frac{29\!\cdots\!84}{98\!\cdots\!47}a^{8}+\frac{41\!\cdots\!28}{22\!\cdots\!29}a^{7}+\frac{30\!\cdots\!89}{98\!\cdots\!47}a^{6}+\frac{25\!\cdots\!03}{10\!\cdots\!83}a^{5}-\frac{13\!\cdots\!44}{98\!\cdots\!47}a^{4}+\frac{29\!\cdots\!89}{98\!\cdots\!47}a^{3}-\frac{11\!\cdots\!15}{10\!\cdots\!83}a^{2}+\frac{61\!\cdots\!20}{98\!\cdots\!47}a+\frac{10\!\cdots\!98}{98\!\cdots\!47}$, $\frac{219794153385214}{42\!\cdots\!89}a^{19}-\frac{543885287716874}{42\!\cdots\!89}a^{18}+\frac{619144145678966}{42\!\cdots\!89}a^{17}+\frac{14345109224095}{99630443117223}a^{16}+\frac{43\!\cdots\!12}{42\!\cdots\!89}a^{15}-\frac{56\!\cdots\!89}{42\!\cdots\!89}a^{14}-\frac{10\!\cdots\!72}{476012117115621}a^{13}-\frac{70\!\cdots\!93}{14\!\cdots\!63}a^{12}-\frac{19\!\cdots\!12}{42\!\cdots\!89}a^{11}-\frac{40\!\cdots\!34}{42\!\cdots\!89}a^{10}-\frac{59\!\cdots\!10}{42\!\cdots\!89}a^{9}-\frac{68\!\cdots\!32}{42\!\cdots\!89}a^{8}-\frac{18\!\cdots\!34}{99630443117223}a^{7}-\frac{22\!\cdots\!87}{186265611045243}a^{6}-\frac{16\!\cdots\!85}{14\!\cdots\!63}a^{5}-\frac{84\!\cdots\!71}{42\!\cdots\!89}a^{4}-\frac{504813416330078}{42\!\cdots\!89}a^{3}-\frac{66\!\cdots\!04}{14\!\cdots\!63}a^{2}-\frac{53\!\cdots\!38}{42\!\cdots\!89}a+\frac{24\!\cdots\!86}{42\!\cdots\!89}$, $\frac{210264943957801}{14\!\cdots\!63}a^{19}-\frac{71188940768299}{476012117115621}a^{18}+\frac{575634384388829}{14\!\cdots\!63}a^{17}+\frac{30945032508053}{33210147705741}a^{16}+\frac{22\!\cdots\!97}{476012117115621}a^{15}+\frac{67\!\cdots\!52}{14\!\cdots\!63}a^{14}+\frac{11\!\cdots\!83}{14\!\cdots\!63}a^{13}+\frac{13\!\cdots\!69}{14\!\cdots\!63}a^{12}+\frac{24\!\cdots\!00}{14\!\cdots\!63}a^{11}+\frac{82\!\cdots\!25}{476012117115621}a^{10}+\frac{23\!\cdots\!99}{14\!\cdots\!63}a^{9}+\frac{79\!\cdots\!08}{476012117115621}a^{8}+\frac{360913393825510}{33210147705741}a^{7}+\frac{10\!\cdots\!63}{62088537015081}a^{6}+\frac{21\!\cdots\!87}{14\!\cdots\!63}a^{5}-\frac{14\!\cdots\!15}{14\!\cdots\!63}a^{4}+\frac{17\!\cdots\!25}{14\!\cdots\!63}a^{3}+\frac{703893029303530}{14\!\cdots\!63}a^{2}-\frac{442328402996075}{14\!\cdots\!63}a+\frac{10605010992813}{52890235235069}$, $\frac{488221810721698}{32\!\cdots\!49}a^{19}+\frac{767793245079056}{10\!\cdots\!83}a^{18}+\frac{536511378767}{32\!\cdots\!49}a^{17}+\frac{253378891437620}{763833397232043}a^{16}+\frac{11\!\cdots\!18}{10\!\cdots\!83}a^{15}+\frac{11\!\cdots\!12}{32\!\cdots\!49}a^{14}+\frac{17\!\cdots\!91}{32\!\cdots\!49}a^{13}+\frac{28\!\cdots\!99}{32\!\cdots\!49}a^{12}+\frac{36\!\cdots\!06}{32\!\cdots\!49}a^{11}+\frac{18\!\cdots\!44}{10\!\cdots\!83}a^{10}+\frac{63\!\cdots\!10}{32\!\cdots\!49}a^{9}+\frac{23\!\cdots\!91}{10\!\cdots\!83}a^{8}+\frac{15\!\cdots\!46}{763833397232043}a^{7}+\frac{56\!\cdots\!64}{32\!\cdots\!49}a^{6}+\frac{57\!\cdots\!44}{32\!\cdots\!49}a^{5}+\frac{43\!\cdots\!34}{32\!\cdots\!49}a^{4}+\frac{85\!\cdots\!86}{32\!\cdots\!49}a^{3}+\frac{25\!\cdots\!25}{32\!\cdots\!49}a^{2}-\frac{30\!\cdots\!52}{32\!\cdots\!49}a+\frac{29\!\cdots\!03}{36\!\cdots\!61}$, $\frac{20\!\cdots\!05}{32\!\cdots\!49}a^{19}-\frac{496103371741916}{32\!\cdots\!49}a^{18}+\frac{48\!\cdots\!32}{32\!\cdots\!49}a^{17}+\frac{123787650816670}{254611132410681}a^{16}+\frac{75\!\cdots\!96}{32\!\cdots\!49}a^{15}+\frac{11\!\cdots\!40}{32\!\cdots\!49}a^{14}+\frac{18\!\cdots\!50}{32\!\cdots\!49}a^{13}+\frac{22\!\cdots\!89}{32\!\cdots\!49}a^{12}+\frac{11\!\cdots\!16}{10\!\cdots\!83}a^{11}+\frac{38\!\cdots\!53}{32\!\cdots\!49}a^{10}+\frac{38\!\cdots\!93}{32\!\cdots\!49}a^{9}+\frac{35\!\cdots\!09}{32\!\cdots\!49}a^{8}+\frac{57\!\cdots\!51}{763833397232043}a^{7}+\frac{84\!\cdots\!51}{10\!\cdots\!83}a^{6}+\frac{24\!\cdots\!45}{32\!\cdots\!49}a^{5}-\frac{20\!\cdots\!08}{12\!\cdots\!87}a^{4}+\frac{16\!\cdots\!38}{10\!\cdots\!83}a^{3}+\frac{60\!\cdots\!19}{32\!\cdots\!49}a^{2}-\frac{57\!\cdots\!68}{10\!\cdots\!83}a-\frac{14\!\cdots\!64}{32\!\cdots\!49}$
| 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: | \( 35594.0921154 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 35594.0921154 \cdot 4}{2\cdot\sqrt{823067302269314181883621609}}\cr\approx \mathstrut & 0.237951761872 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 3*x^18 + 6*x^17 + 33*x^16 + 34*x^15 + 65*x^14 + 69*x^13 + 127*x^12 + 132*x^11 + 144*x^10 + 143*x^9 + 101*x^8 + 141*x^7 + 116*x^6 - 32*x^5 + 110*x^4 - 20*x^3 + 20*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^20 - x^19 + 3*x^18 + 6*x^17 + 33*x^16 + 34*x^15 + 65*x^14 + 69*x^13 + 127*x^12 + 132*x^11 + 144*x^10 + 143*x^9 + 101*x^8 + 141*x^7 + 116*x^6 - 32*x^5 + 110*x^4 - 20*x^3 + 20*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^20 - x^19 + 3*x^18 + 6*x^17 + 33*x^16 + 34*x^15 + 65*x^14 + 69*x^13 + 127*x^12 + 132*x^11 + 144*x^10 + 143*x^9 + 101*x^8 + 141*x^7 + 116*x^6 - 32*x^5 + 110*x^4 - 20*x^3 + 20*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^20 - x^19 + 3*x^18 + 6*x^17 + 33*x^16 + 34*x^15 + 65*x^14 + 69*x^13 + 127*x^12 + 132*x^11 + 144*x^10 + 143*x^9 + 101*x^8 + 141*x^7 + 116*x^6 - 32*x^5 + 110*x^4 - 20*x^3 + 20*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
$C_2\wr C_5$ (as 20T40):
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 solvable group of order 160 |
| The 16 conjugacy class representatives for $C_2\wr C_5$ |
| Character table for $C_2\wr C_5$ |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.6.54232796893.1, 10.2.28689149556397.1, 10.2.113395848049.1 |
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 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
| Arithmetically equvalently siblings: | data not computed |
| Minimal sibling: | 10.6.54232796893.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$ | ${\href{/padicField/47.5.0.1}{5} }^{4}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.10.9.1 | $x^{10} + 110$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
| 11.10.9.1 | $x^{10} + 110$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |
|
\(23\)
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |