Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 36*x^18 - 82*x^17 + 590*x^16 - 1196*x^15 + 6360*x^14 - 11542*x^13 + 33069*x^12 - 73808*x^11 + 108170*x^10 - 178938*x^9 + 336647*x^8 - 235632*x^7 + 398012*x^6 - 428398*x^5 - 72149*x^4 - 133254*x^3 + 269144*x^2 + 611294*x + 805751)
gp: K = bnfinit(y^20 - 4*y^19 + 36*y^18 - 82*y^17 + 590*y^16 - 1196*y^15 + 6360*y^14 - 11542*y^13 + 33069*y^12 - 73808*y^11 + 108170*y^10 - 178938*y^9 + 336647*y^8 - 235632*y^7 + 398012*y^6 - 428398*y^5 - 72149*y^4 - 133254*y^3 + 269144*y^2 + 611294*y + 805751, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^19 + 36*x^18 - 82*x^17 + 590*x^16 - 1196*x^15 + 6360*x^14 - 11542*x^13 + 33069*x^12 - 73808*x^11 + 108170*x^10 - 178938*x^9 + 336647*x^8 - 235632*x^7 + 398012*x^6 - 428398*x^5 - 72149*x^4 - 133254*x^3 + 269144*x^2 + 611294*x + 805751);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 4*x^19 + 36*x^18 - 82*x^17 + 590*x^16 - 1196*x^15 + 6360*x^14 - 11542*x^13 + 33069*x^12 - 73808*x^11 + 108170*x^10 - 178938*x^9 + 336647*x^8 - 235632*x^7 + 398012*x^6 - 428398*x^5 - 72149*x^4 - 133254*x^3 + 269144*x^2 + 611294*x + 805751)
\( x^{20} - 4 x^{19} + 36 x^{18} - 82 x^{17} + 590 x^{16} - 1196 x^{15} + 6360 x^{14} - 11542 x^{13} + \cdots + 805751 \)
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: |
\(7303816416639774784171798530359296\)
\(\medspace = 2^{30}\cdot 11^{16}\cdot 23^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(49.34\) | 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: | $2^{15/8}11^{4/5}23^{1/2}\approx 119.78689508462926$ | ||
| Ramified primes: |
\(2\), \(11\), \(23\)
| 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 a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23}a^{17}+\frac{1}{23}a^{16}-\frac{3}{23}a^{15}-\frac{2}{23}a^{14}-\frac{7}{23}a^{12}-\frac{2}{23}a^{11}+\frac{3}{23}a^{10}-\frac{1}{23}a^{9}-\frac{2}{23}a^{8}-\frac{8}{23}a^{7}+\frac{3}{23}a^{6}-\frac{7}{23}a^{5}-\frac{11}{23}a^{4}-\frac{6}{23}a^{2}+\frac{7}{23}a+\frac{8}{23}$, $\frac{1}{23}a^{18}-\frac{4}{23}a^{16}+\frac{1}{23}a^{15}+\frac{2}{23}a^{14}-\frac{7}{23}a^{13}+\frac{5}{23}a^{12}+\frac{5}{23}a^{11}-\frac{4}{23}a^{10}-\frac{1}{23}a^{9}-\frac{6}{23}a^{8}+\frac{11}{23}a^{7}-\frac{10}{23}a^{6}-\frac{4}{23}a^{5}+\frac{11}{23}a^{4}-\frac{6}{23}a^{3}-\frac{10}{23}a^{2}+\frac{1}{23}a-\frac{8}{23}$, $\frac{1}{48\!\cdots\!93}a^{19}+\frac{15\!\cdots\!13}{20\!\cdots\!91}a^{18}-\frac{59\!\cdots\!57}{48\!\cdots\!93}a^{17}-\frac{38\!\cdots\!90}{48\!\cdots\!93}a^{16}-\frac{22\!\cdots\!09}{48\!\cdots\!93}a^{15}+\frac{13\!\cdots\!12}{48\!\cdots\!93}a^{14}+\frac{16\!\cdots\!63}{48\!\cdots\!93}a^{13}+\frac{23\!\cdots\!16}{48\!\cdots\!93}a^{12}-\frac{23\!\cdots\!20}{48\!\cdots\!93}a^{11}-\frac{18\!\cdots\!33}{48\!\cdots\!93}a^{10}+\frac{14\!\cdots\!73}{48\!\cdots\!93}a^{9}-\frac{19\!\cdots\!74}{48\!\cdots\!93}a^{8}-\frac{10\!\cdots\!71}{48\!\cdots\!93}a^{7}+\frac{78\!\cdots\!01}{48\!\cdots\!93}a^{6}-\frac{14\!\cdots\!03}{48\!\cdots\!93}a^{5}+\frac{10\!\cdots\!11}{48\!\cdots\!93}a^{4}-\frac{89\!\cdots\!58}{48\!\cdots\!93}a^{3}+\frac{19\!\cdots\!48}{48\!\cdots\!93}a^{2}+\frac{59\!\cdots\!81}{48\!\cdots\!93}a+\frac{20\!\cdots\!31}{48\!\cdots\!93}$
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
$C_{2}\times C_{4}\times C_{20}$, which has order $160$ (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: | $9$ | 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: |
$\frac{17\!\cdots\!60}{28\!\cdots\!83}a^{19}-\frac{12\!\cdots\!48}{28\!\cdots\!83}a^{18}+\frac{92\!\cdots\!88}{28\!\cdots\!83}a^{17}-\frac{36\!\cdots\!69}{28\!\cdots\!83}a^{16}+\frac{17\!\cdots\!94}{28\!\cdots\!83}a^{15}-\frac{56\!\cdots\!66}{28\!\cdots\!83}a^{14}+\frac{21\!\cdots\!00}{28\!\cdots\!83}a^{13}-\frac{58\!\cdots\!82}{28\!\cdots\!83}a^{12}+\frac{15\!\cdots\!20}{28\!\cdots\!83}a^{11}-\frac{33\!\cdots\!08}{28\!\cdots\!83}a^{10}+\frac{68\!\cdots\!98}{28\!\cdots\!83}a^{9}-\frac{11\!\cdots\!52}{28\!\cdots\!83}a^{8}+\frac{15\!\cdots\!64}{28\!\cdots\!83}a^{7}-\frac{21\!\cdots\!62}{28\!\cdots\!83}a^{6}+\frac{25\!\cdots\!52}{28\!\cdots\!83}a^{5}-\frac{14\!\cdots\!71}{28\!\cdots\!83}a^{4}+\frac{46\!\cdots\!46}{28\!\cdots\!83}a^{3}+\frac{27\!\cdots\!30}{12\!\cdots\!21}a^{2}-\frac{31\!\cdots\!42}{28\!\cdots\!83}a-\frac{73\!\cdots\!25}{28\!\cdots\!83}$, $\frac{65\!\cdots\!28}{28\!\cdots\!83}a^{19}-\frac{31\!\cdots\!16}{28\!\cdots\!83}a^{18}+\frac{32\!\cdots\!78}{28\!\cdots\!83}a^{17}-\frac{96\!\cdots\!95}{28\!\cdots\!83}a^{16}+\frac{64\!\cdots\!08}{28\!\cdots\!83}a^{15}-\frac{15\!\cdots\!78}{28\!\cdots\!83}a^{14}+\frac{82\!\cdots\!12}{28\!\cdots\!83}a^{13}-\frac{17\!\cdots\!65}{28\!\cdots\!83}a^{12}+\frac{63\!\cdots\!88}{28\!\cdots\!83}a^{11}-\frac{12\!\cdots\!00}{28\!\cdots\!83}a^{10}+\frac{27\!\cdots\!80}{28\!\cdots\!83}a^{9}-\frac{55\!\cdots\!60}{28\!\cdots\!83}a^{8}+\frac{69\!\cdots\!94}{28\!\cdots\!83}a^{7}-\frac{10\!\cdots\!48}{28\!\cdots\!83}a^{6}+\frac{16\!\cdots\!88}{28\!\cdots\!83}a^{5}-\frac{50\!\cdots\!49}{28\!\cdots\!83}a^{4}+\frac{10\!\cdots\!92}{28\!\cdots\!83}a^{3}-\frac{27\!\cdots\!64}{12\!\cdots\!21}a^{2}-\frac{28\!\cdots\!94}{28\!\cdots\!83}a+\frac{20\!\cdots\!16}{28\!\cdots\!83}$, $\frac{43\!\cdots\!64}{28\!\cdots\!83}a^{19}-\frac{11\!\cdots\!28}{28\!\cdots\!83}a^{18}+\frac{13\!\cdots\!00}{28\!\cdots\!83}a^{17}-\frac{14\!\cdots\!81}{28\!\cdots\!83}a^{16}+\frac{21\!\cdots\!16}{28\!\cdots\!83}a^{15}-\frac{19\!\cdots\!16}{28\!\cdots\!83}a^{14}+\frac{21\!\cdots\!28}{28\!\cdots\!83}a^{13}-\frac{16\!\cdots\!50}{28\!\cdots\!83}a^{12}+\frac{91\!\cdots\!16}{28\!\cdots\!83}a^{11}-\frac{18\!\cdots\!56}{28\!\cdots\!83}a^{10}+\frac{18\!\cdots\!16}{28\!\cdots\!83}a^{9}-\frac{50\!\cdots\!86}{28\!\cdots\!83}a^{8}+\frac{11\!\cdots\!24}{28\!\cdots\!83}a^{7}-\frac{35\!\cdots\!68}{28\!\cdots\!83}a^{6}+\frac{15\!\cdots\!04}{28\!\cdots\!83}a^{5}-\frac{11\!\cdots\!08}{28\!\cdots\!83}a^{4}-\frac{13\!\cdots\!68}{28\!\cdots\!83}a^{3}-\frac{45\!\cdots\!36}{12\!\cdots\!21}a^{2}-\frac{17\!\cdots\!88}{28\!\cdots\!83}a+\frac{96\!\cdots\!67}{28\!\cdots\!83}$, $\frac{65\!\cdots\!28}{28\!\cdots\!83}a^{19}-\frac{31\!\cdots\!16}{28\!\cdots\!83}a^{18}+\frac{32\!\cdots\!78}{28\!\cdots\!83}a^{17}-\frac{96\!\cdots\!95}{28\!\cdots\!83}a^{16}+\frac{64\!\cdots\!08}{28\!\cdots\!83}a^{15}-\frac{15\!\cdots\!78}{28\!\cdots\!83}a^{14}+\frac{82\!\cdots\!12}{28\!\cdots\!83}a^{13}-\frac{17\!\cdots\!65}{28\!\cdots\!83}a^{12}+\frac{63\!\cdots\!88}{28\!\cdots\!83}a^{11}-\frac{12\!\cdots\!00}{28\!\cdots\!83}a^{10}+\frac{27\!\cdots\!80}{28\!\cdots\!83}a^{9}-\frac{55\!\cdots\!60}{28\!\cdots\!83}a^{8}+\frac{69\!\cdots\!94}{28\!\cdots\!83}a^{7}-\frac{10\!\cdots\!48}{28\!\cdots\!83}a^{6}+\frac{16\!\cdots\!88}{28\!\cdots\!83}a^{5}-\frac{50\!\cdots\!49}{28\!\cdots\!83}a^{4}+\frac{10\!\cdots\!92}{28\!\cdots\!83}a^{3}-\frac{27\!\cdots\!64}{12\!\cdots\!21}a^{2}-\frac{28\!\cdots\!94}{28\!\cdots\!83}a-\frac{77\!\cdots\!67}{28\!\cdots\!83}$, $\frac{86\!\cdots\!33}{11\!\cdots\!51}a^{19}+\frac{95\!\cdots\!88}{11\!\cdots\!51}a^{18}+\frac{93\!\cdots\!23}{11\!\cdots\!51}a^{17}+\frac{10\!\cdots\!21}{11\!\cdots\!51}a^{16}+\frac{27\!\cdots\!97}{11\!\cdots\!51}a^{15}+\frac{17\!\cdots\!73}{11\!\cdots\!51}a^{14}-\frac{15\!\cdots\!54}{11\!\cdots\!51}a^{13}+\frac{19\!\cdots\!84}{11\!\cdots\!51}a^{12}-\frac{38\!\cdots\!26}{11\!\cdots\!51}a^{11}+\frac{92\!\cdots\!01}{11\!\cdots\!51}a^{10}-\frac{26\!\cdots\!34}{11\!\cdots\!51}a^{9}+\frac{40\!\cdots\!92}{11\!\cdots\!51}a^{8}-\frac{43\!\cdots\!67}{11\!\cdots\!51}a^{7}+\frac{11\!\cdots\!80}{11\!\cdots\!51}a^{6}-\frac{86\!\cdots\!60}{11\!\cdots\!51}a^{5}+\frac{47\!\cdots\!15}{11\!\cdots\!51}a^{4}-\frac{78\!\cdots\!40}{11\!\cdots\!51}a^{3}-\frac{10\!\cdots\!95}{11\!\cdots\!51}a^{2}+\frac{91\!\cdots\!26}{11\!\cdots\!51}a+\frac{33\!\cdots\!73}{11\!\cdots\!51}$, $\frac{11\!\cdots\!84}{11\!\cdots\!51}a^{19}-\frac{28\!\cdots\!85}{11\!\cdots\!51}a^{18}+\frac{11\!\cdots\!94}{11\!\cdots\!51}a^{17}-\frac{84\!\cdots\!38}{11\!\cdots\!51}a^{16}+\frac{17\!\cdots\!24}{11\!\cdots\!51}a^{15}-\frac{57\!\cdots\!26}{48\!\cdots\!37}a^{14}+\frac{20\!\cdots\!85}{11\!\cdots\!51}a^{13}-\frac{13\!\cdots\!27}{11\!\cdots\!51}a^{12}+\frac{14\!\cdots\!82}{11\!\cdots\!51}a^{11}-\frac{53\!\cdots\!53}{11\!\cdots\!51}a^{10}+\frac{94\!\cdots\!11}{11\!\cdots\!51}a^{9}-\frac{22\!\cdots\!77}{48\!\cdots\!37}a^{8}+\frac{20\!\cdots\!21}{11\!\cdots\!51}a^{7}-\frac{10\!\cdots\!13}{11\!\cdots\!51}a^{6}-\frac{16\!\cdots\!27}{11\!\cdots\!51}a^{5}-\frac{32\!\cdots\!01}{11\!\cdots\!51}a^{4}-\frac{16\!\cdots\!61}{11\!\cdots\!51}a^{3}+\frac{12\!\cdots\!28}{11\!\cdots\!51}a^{2}+\frac{16\!\cdots\!74}{11\!\cdots\!51}a+\frac{82\!\cdots\!46}{11\!\cdots\!51}$, $\frac{40\!\cdots\!24}{11\!\cdots\!51}a^{19}-\frac{44\!\cdots\!01}{11\!\cdots\!51}a^{18}+\frac{25\!\cdots\!71}{11\!\cdots\!51}a^{17}-\frac{11\!\cdots\!51}{11\!\cdots\!51}a^{16}+\frac{37\!\cdots\!29}{11\!\cdots\!51}a^{15}-\frac{15\!\cdots\!38}{11\!\cdots\!51}a^{14}+\frac{39\!\cdots\!29}{11\!\cdots\!51}a^{13}-\frac{58\!\cdots\!19}{48\!\cdots\!37}a^{12}+\frac{18\!\cdots\!29}{11\!\cdots\!51}a^{11}-\frac{24\!\cdots\!28}{11\!\cdots\!51}a^{10}+\frac{14\!\cdots\!36}{11\!\cdots\!51}a^{9}+\frac{15\!\cdots\!19}{11\!\cdots\!51}a^{8}-\frac{47\!\cdots\!93}{11\!\cdots\!51}a^{7}+\frac{62\!\cdots\!12}{11\!\cdots\!51}a^{6}-\frac{93\!\cdots\!44}{11\!\cdots\!51}a^{5}+\frac{86\!\cdots\!98}{11\!\cdots\!51}a^{4}+\frac{15\!\cdots\!77}{11\!\cdots\!51}a^{3}-\frac{69\!\cdots\!78}{11\!\cdots\!51}a^{2}+\frac{72\!\cdots\!11}{11\!\cdots\!51}a-\frac{12\!\cdots\!36}{11\!\cdots\!51}$, $\frac{77\!\cdots\!83}{11\!\cdots\!51}a^{19}-\frac{19\!\cdots\!10}{11\!\cdots\!51}a^{18}+\frac{17\!\cdots\!49}{11\!\cdots\!51}a^{17}-\frac{36\!\cdots\!24}{11\!\cdots\!51}a^{16}+\frac{18\!\cdots\!22}{11\!\cdots\!51}a^{15}-\frac{21\!\cdots\!09}{11\!\cdots\!51}a^{14}+\frac{10\!\cdots\!59}{11\!\cdots\!51}a^{13}+\frac{13\!\cdots\!46}{11\!\cdots\!51}a^{12}-\frac{11\!\cdots\!72}{11\!\cdots\!51}a^{11}+\frac{28\!\cdots\!02}{11\!\cdots\!51}a^{10}-\frac{67\!\cdots\!02}{11\!\cdots\!51}a^{9}+\frac{17\!\cdots\!61}{11\!\cdots\!51}a^{8}+\frac{26\!\cdots\!84}{11\!\cdots\!51}a^{7}+\frac{30\!\cdots\!64}{11\!\cdots\!51}a^{6}-\frac{67\!\cdots\!63}{11\!\cdots\!51}a^{5}-\frac{33\!\cdots\!47}{11\!\cdots\!51}a^{4}-\frac{25\!\cdots\!00}{11\!\cdots\!51}a^{3}+\frac{10\!\cdots\!78}{11\!\cdots\!51}a^{2}+\frac{21\!\cdots\!13}{11\!\cdots\!51}a+\frac{64\!\cdots\!35}{11\!\cdots\!51}$, $\frac{18\!\cdots\!72}{11\!\cdots\!51}a^{19}-\frac{62\!\cdots\!24}{11\!\cdots\!51}a^{18}+\frac{57\!\cdots\!24}{11\!\cdots\!51}a^{17}-\frac{93\!\cdots\!91}{11\!\cdots\!51}a^{16}+\frac{84\!\cdots\!09}{11\!\cdots\!51}a^{15}-\frac{12\!\cdots\!10}{11\!\cdots\!51}a^{14}+\frac{80\!\cdots\!81}{11\!\cdots\!51}a^{13}-\frac{10\!\cdots\!41}{11\!\cdots\!51}a^{12}+\frac{24\!\cdots\!18}{11\!\cdots\!51}a^{11}-\frac{30\!\cdots\!19}{48\!\cdots\!37}a^{10}+\frac{35\!\cdots\!38}{11\!\cdots\!51}a^{9}-\frac{95\!\cdots\!32}{11\!\cdots\!51}a^{8}+\frac{13\!\cdots\!71}{48\!\cdots\!37}a^{7}+\frac{99\!\cdots\!58}{11\!\cdots\!51}a^{6}-\frac{33\!\cdots\!27}{11\!\cdots\!51}a^{5}-\frac{61\!\cdots\!61}{11\!\cdots\!51}a^{4}-\frac{40\!\cdots\!46}{11\!\cdots\!51}a^{3}+\frac{11\!\cdots\!43}{11\!\cdots\!51}a^{2}+\frac{19\!\cdots\!69}{11\!\cdots\!51}a+\frac{71\!\cdots\!61}{11\!\cdots\!51}$
| 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: | \( 2417625.83378 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 2417625.83378 \cdot 160}{2\cdot\sqrt{7303816416639774784171798530359296}}\cr\approx \mathstrut & 0.217021536538 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 36*x^18 - 82*x^17 + 590*x^16 - 1196*x^15 + 6360*x^14 - 11542*x^13 + 33069*x^12 - 73808*x^11 + 108170*x^10 - 178938*x^9 + 336647*x^8 - 235632*x^7 + 398012*x^6 - 428398*x^5 - 72149*x^4 - 133254*x^3 + 269144*x^2 + 611294*x + 805751)
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 - 4*x^19 + 36*x^18 - 82*x^17 + 590*x^16 - 1196*x^15 + 6360*x^14 - 11542*x^13 + 33069*x^12 - 73808*x^11 + 108170*x^10 - 178938*x^9 + 336647*x^8 - 235632*x^7 + 398012*x^6 - 428398*x^5 - 72149*x^4 - 133254*x^3 + 269144*x^2 + 611294*x + 805751, 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 - 4*x^19 + 36*x^18 - 82*x^17 + 590*x^16 - 1196*x^15 + 6360*x^14 - 11542*x^13 + 33069*x^12 - 73808*x^11 + 108170*x^10 - 178938*x^9 + 336647*x^8 - 235632*x^7 + 398012*x^6 - 428398*x^5 - 72149*x^4 - 133254*x^3 + 269144*x^2 + 611294*x + 805751);
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 - 4*x^19 + 36*x^18 - 82*x^17 + 590*x^16 - 1196*x^15 + 6360*x^14 - 11542*x^13 + 33069*x^12 - 73808*x^11 + 108170*x^10 - 178938*x^9 + 336647*x^8 - 235632*x^7 + 398012*x^6 - 428398*x^5 - 72149*x^4 - 133254*x^3 + 269144*x^2 + 611294*x + 805751);
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
| \(\Q(\zeta_{11})^+\), 10.0.2670699013250048.1, 10.0.5048580365312.1, 10.10.116117348402176.1 |
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: | 10.0.5048580365312.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/13.5.0.1}{5} }^{4}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$ | ${\href{/padicField/47.5.0.1}{5} }^{4}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $20$ | $4$ | $5$ | $30$ | |||
|
\(11\)
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |
|
\(23\)
| 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 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 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 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |