Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 34*x^18 - 132*x^17 + 502*x^16 - 1450*x^15 + 3398*x^14 - 7814*x^13 + 13793*x^12 - 30364*x^11 + 73744*x^10 - 254376*x^9 + 822868*x^8 - 2139784*x^7 + 4734462*x^6 - 7831492*x^5 + 11001528*x^4 - 10686038*x^3 + 8803206*x^2 - 3761584*x + 1294987)
gp: K = bnfinit(y^20 - 6*y^19 + 34*y^18 - 132*y^17 + 502*y^16 - 1450*y^15 + 3398*y^14 - 7814*y^13 + 13793*y^12 - 30364*y^11 + 73744*y^10 - 254376*y^9 + 822868*y^8 - 2139784*y^7 + 4734462*y^6 - 7831492*y^5 + 11001528*y^4 - 10686038*y^3 + 8803206*y^2 - 3761584*y + 1294987, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 6*x^19 + 34*x^18 - 132*x^17 + 502*x^16 - 1450*x^15 + 3398*x^14 - 7814*x^13 + 13793*x^12 - 30364*x^11 + 73744*x^10 - 254376*x^9 + 822868*x^8 - 2139784*x^7 + 4734462*x^6 - 7831492*x^5 + 11001528*x^4 - 10686038*x^3 + 8803206*x^2 - 3761584*x + 1294987);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 6*x^19 + 34*x^18 - 132*x^17 + 502*x^16 - 1450*x^15 + 3398*x^14 - 7814*x^13 + 13793*x^12 - 30364*x^11 + 73744*x^10 - 254376*x^9 + 822868*x^8 - 2139784*x^7 + 4734462*x^6 - 7831492*x^5 + 11001528*x^4 - 10686038*x^3 + 8803206*x^2 - 3761584*x + 1294987)
\( x^{20} - 6 x^{19} + 34 x^{18} - 132 x^{17} + 502 x^{16} - 1450 x^{15} + 3398 x^{14} - 7814 x^{13} + \cdots + 1294987 \)
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23}a^{17}-\frac{1}{23}a^{16}-\frac{8}{23}a^{15}-\frac{7}{23}a^{14}-\frac{9}{23}a^{13}-\frac{6}{23}a^{12}-\frac{1}{23}a^{11}-\frac{9}{23}a^{10}-\frac{1}{23}a^{9}-\frac{11}{23}a^{8}+\frac{9}{23}a^{7}+\frac{6}{23}a^{6}+\frac{11}{23}a^{5}-\frac{6}{23}a^{4}-\frac{10}{23}a^{3}+\frac{1}{23}a^{2}+\frac{5}{23}a+\frac{11}{23}$, $\frac{1}{23}a^{18}-\frac{9}{23}a^{16}+\frac{8}{23}a^{15}+\frac{7}{23}a^{14}+\frac{8}{23}a^{13}-\frac{7}{23}a^{12}-\frac{10}{23}a^{11}-\frac{10}{23}a^{10}+\frac{11}{23}a^{9}-\frac{2}{23}a^{8}-\frac{8}{23}a^{7}-\frac{6}{23}a^{6}+\frac{5}{23}a^{5}+\frac{7}{23}a^{4}-\frac{9}{23}a^{3}+\frac{6}{23}a^{2}-\frac{7}{23}a+\frac{11}{23}$, $\frac{1}{10\!\cdots\!53}a^{19}-\frac{75\!\cdots\!38}{10\!\cdots\!53}a^{18}+\frac{13\!\cdots\!22}{10\!\cdots\!53}a^{17}-\frac{16\!\cdots\!02}{10\!\cdots\!53}a^{16}+\frac{78\!\cdots\!46}{10\!\cdots\!53}a^{15}+\frac{20\!\cdots\!39}{10\!\cdots\!53}a^{14}+\frac{33\!\cdots\!24}{10\!\cdots\!53}a^{13}-\frac{48\!\cdots\!40}{10\!\cdots\!53}a^{12}+\frac{45\!\cdots\!09}{10\!\cdots\!53}a^{11}-\frac{75\!\cdots\!49}{10\!\cdots\!53}a^{10}+\frac{15\!\cdots\!10}{10\!\cdots\!53}a^{9}-\frac{41\!\cdots\!87}{10\!\cdots\!53}a^{8}-\frac{22\!\cdots\!14}{10\!\cdots\!53}a^{7}+\frac{36\!\cdots\!18}{10\!\cdots\!53}a^{6}+\frac{21\!\cdots\!27}{10\!\cdots\!53}a^{5}-\frac{46\!\cdots\!18}{10\!\cdots\!53}a^{4}-\frac{11\!\cdots\!36}{10\!\cdots\!53}a^{3}-\frac{38\!\cdots\!47}{10\!\cdots\!53}a^{2}+\frac{15\!\cdots\!74}{10\!\cdots\!53}a-\frac{40\!\cdots\!63}{10\!\cdots\!53}$
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: |
$\frac{38\!\cdots\!84}{39\!\cdots\!31}a^{19}-\frac{15\!\cdots\!10}{39\!\cdots\!31}a^{18}+\frac{89\!\cdots\!44}{39\!\cdots\!31}a^{17}-\frac{28\!\cdots\!38}{39\!\cdots\!31}a^{16}+\frac{10\!\cdots\!98}{39\!\cdots\!31}a^{15}-\frac{10\!\cdots\!10}{16\!\cdots\!97}a^{14}+\frac{45\!\cdots\!80}{39\!\cdots\!31}a^{13}-\frac{11\!\cdots\!97}{39\!\cdots\!31}a^{12}+\frac{10\!\cdots\!46}{39\!\cdots\!31}a^{11}-\frac{48\!\cdots\!36}{39\!\cdots\!31}a^{10}+\frac{12\!\cdots\!44}{39\!\cdots\!31}a^{9}-\frac{53\!\cdots\!96}{39\!\cdots\!31}a^{8}+\frac{16\!\cdots\!92}{39\!\cdots\!31}a^{7}-\frac{32\!\cdots\!24}{39\!\cdots\!31}a^{6}+\frac{60\!\cdots\!74}{39\!\cdots\!31}a^{5}-\frac{50\!\cdots\!09}{39\!\cdots\!31}a^{4}+\frac{52\!\cdots\!96}{39\!\cdots\!31}a^{3}+\frac{32\!\cdots\!80}{39\!\cdots\!31}a^{2}-\frac{19\!\cdots\!32}{39\!\cdots\!31}a+\frac{51\!\cdots\!61}{39\!\cdots\!31}$, $\frac{29\!\cdots\!86}{39\!\cdots\!31}a^{19}-\frac{31\!\cdots\!02}{39\!\cdots\!31}a^{18}+\frac{30\!\cdots\!66}{39\!\cdots\!31}a^{17}-\frac{17\!\cdots\!09}{39\!\cdots\!31}a^{16}+\frac{15\!\cdots\!26}{39\!\cdots\!31}a^{15}+\frac{25\!\cdots\!54}{16\!\cdots\!97}a^{14}-\frac{25\!\cdots\!00}{39\!\cdots\!31}a^{13}+\frac{10\!\cdots\!58}{39\!\cdots\!31}a^{12}-\frac{20\!\cdots\!76}{39\!\cdots\!31}a^{11}-\frac{43\!\cdots\!26}{39\!\cdots\!31}a^{10}+\frac{59\!\cdots\!60}{39\!\cdots\!31}a^{9}-\frac{54\!\cdots\!54}{39\!\cdots\!31}a^{8}+\frac{29\!\cdots\!30}{39\!\cdots\!31}a^{7}+\frac{13\!\cdots\!66}{39\!\cdots\!31}a^{6}-\frac{30\!\cdots\!84}{39\!\cdots\!31}a^{5}+\frac{10\!\cdots\!65}{39\!\cdots\!31}a^{4}-\frac{11\!\cdots\!38}{39\!\cdots\!31}a^{3}+\frac{15\!\cdots\!90}{39\!\cdots\!31}a^{2}-\frac{69\!\cdots\!88}{39\!\cdots\!31}a+\frac{27\!\cdots\!63}{39\!\cdots\!31}$, $\frac{31\!\cdots\!40}{39\!\cdots\!31}a^{19}-\frac{23\!\cdots\!28}{39\!\cdots\!31}a^{18}+\frac{12\!\cdots\!72}{39\!\cdots\!31}a^{17}-\frac{48\!\cdots\!89}{39\!\cdots\!31}a^{16}+\frac{17\!\cdots\!88}{39\!\cdots\!31}a^{15}-\frac{22\!\cdots\!28}{16\!\cdots\!97}a^{14}+\frac{11\!\cdots\!04}{39\!\cdots\!31}a^{13}-\frac{22\!\cdots\!50}{39\!\cdots\!31}a^{12}+\frac{43\!\cdots\!68}{39\!\cdots\!31}a^{11}-\frac{82\!\cdots\!48}{39\!\cdots\!31}a^{10}+\frac{22\!\cdots\!80}{39\!\cdots\!31}a^{9}-\frac{89\!\cdots\!23}{39\!\cdots\!31}a^{8}+\frac{28\!\cdots\!96}{39\!\cdots\!31}a^{7}-\frac{74\!\cdots\!08}{39\!\cdots\!31}a^{6}+\frac{14\!\cdots\!68}{39\!\cdots\!31}a^{5}-\frac{21\!\cdots\!58}{39\!\cdots\!31}a^{4}+\frac{24\!\cdots\!56}{39\!\cdots\!31}a^{3}-\frac{11\!\cdots\!20}{39\!\cdots\!31}a^{2}+\frac{45\!\cdots\!44}{39\!\cdots\!31}a+\frac{10\!\cdots\!56}{39\!\cdots\!31}$, $\frac{86\!\cdots\!98}{39\!\cdots\!31}a^{19}-\frac{12\!\cdots\!08}{39\!\cdots\!31}a^{18}+\frac{58\!\cdots\!78}{39\!\cdots\!31}a^{17}-\frac{26\!\cdots\!29}{39\!\cdots\!31}a^{16}+\frac{93\!\cdots\!72}{39\!\cdots\!31}a^{15}-\frac{13\!\cdots\!64}{16\!\cdots\!97}a^{14}+\frac{70\!\cdots\!80}{39\!\cdots\!31}a^{13}-\frac{12\!\cdots\!55}{39\!\cdots\!31}a^{12}+\frac{30\!\cdots\!22}{39\!\cdots\!31}a^{11}-\frac{44\!\cdots\!10}{39\!\cdots\!31}a^{10}+\frac{12\!\cdots\!84}{39\!\cdots\!31}a^{9}-\frac{47\!\cdots\!42}{39\!\cdots\!31}a^{8}+\frac{15\!\cdots\!62}{39\!\cdots\!31}a^{7}-\frac{45\!\cdots\!90}{39\!\cdots\!31}a^{6}+\frac{90\!\cdots\!58}{39\!\cdots\!31}a^{5}-\frac{15\!\cdots\!74}{39\!\cdots\!31}a^{4}+\frac{16\!\cdots\!34}{39\!\cdots\!31}a^{3}-\frac{11\!\cdots\!10}{39\!\cdots\!31}a^{2}+\frac{50\!\cdots\!56}{39\!\cdots\!31}a+\frac{10\!\cdots\!67}{39\!\cdots\!31}$, $\frac{45\!\cdots\!77}{10\!\cdots\!53}a^{19}-\frac{21\!\cdots\!87}{10\!\cdots\!53}a^{18}+\frac{11\!\cdots\!87}{10\!\cdots\!53}a^{17}-\frac{41\!\cdots\!38}{10\!\cdots\!53}a^{16}+\frac{15\!\cdots\!93}{10\!\cdots\!53}a^{15}-\frac{39\!\cdots\!86}{10\!\cdots\!53}a^{14}+\frac{78\!\cdots\!88}{10\!\cdots\!53}a^{13}-\frac{19\!\cdots\!48}{10\!\cdots\!53}a^{12}+\frac{25\!\cdots\!71}{10\!\cdots\!53}a^{11}-\frac{72\!\cdots\!45}{10\!\cdots\!53}a^{10}+\frac{20\!\cdots\!50}{10\!\cdots\!53}a^{9}-\frac{76\!\cdots\!86}{10\!\cdots\!53}a^{8}+\frac{24\!\cdots\!38}{10\!\cdots\!53}a^{7}-\frac{23\!\cdots\!64}{44\!\cdots\!11}a^{6}+\frac{10\!\cdots\!28}{10\!\cdots\!53}a^{5}-\frac{13\!\cdots\!20}{10\!\cdots\!53}a^{4}+\frac{14\!\cdots\!69}{10\!\cdots\!53}a^{3}-\frac{69\!\cdots\!79}{10\!\cdots\!53}a^{2}+\frac{12\!\cdots\!43}{44\!\cdots\!11}a+\frac{47\!\cdots\!64}{10\!\cdots\!53}$, $\frac{42\!\cdots\!11}{10\!\cdots\!53}a^{19}-\frac{23\!\cdots\!56}{10\!\cdots\!53}a^{18}+\frac{12\!\cdots\!57}{10\!\cdots\!53}a^{17}-\frac{46\!\cdots\!02}{10\!\cdots\!53}a^{16}+\frac{17\!\cdots\!73}{10\!\cdots\!53}a^{15}-\frac{45\!\cdots\!79}{10\!\cdots\!53}a^{14}+\frac{93\!\cdots\!08}{10\!\cdots\!53}a^{13}-\frac{20\!\cdots\!14}{10\!\cdots\!53}a^{12}+\frac{32\!\cdots\!76}{10\!\cdots\!53}a^{11}-\frac{78\!\cdots\!13}{10\!\cdots\!53}a^{10}+\frac{21\!\cdots\!18}{10\!\cdots\!53}a^{9}-\frac{85\!\cdots\!35}{10\!\cdots\!53}a^{8}+\frac{27\!\cdots\!35}{10\!\cdots\!53}a^{7}-\frac{63\!\cdots\!11}{10\!\cdots\!53}a^{6}+\frac{12\!\cdots\!08}{10\!\cdots\!53}a^{5}-\frac{16\!\cdots\!89}{10\!\cdots\!53}a^{4}+\frac{17\!\cdots\!81}{10\!\cdots\!53}a^{3}-\frac{81\!\cdots\!74}{10\!\cdots\!53}a^{2}+\frac{31\!\cdots\!90}{10\!\cdots\!53}a+\frac{30\!\cdots\!30}{10\!\cdots\!53}$, $\frac{39\!\cdots\!06}{10\!\cdots\!53}a^{19}-\frac{13\!\cdots\!26}{10\!\cdots\!53}a^{18}+\frac{78\!\cdots\!48}{10\!\cdots\!53}a^{17}-\frac{23\!\cdots\!31}{10\!\cdots\!53}a^{16}+\frac{40\!\cdots\!24}{44\!\cdots\!11}a^{15}-\frac{19\!\cdots\!07}{10\!\cdots\!53}a^{14}+\frac{32\!\cdots\!57}{10\!\cdots\!53}a^{13}-\frac{11\!\cdots\!96}{10\!\cdots\!53}a^{12}+\frac{66\!\cdots\!09}{10\!\cdots\!53}a^{11}-\frac{43\!\cdots\!92}{10\!\cdots\!53}a^{10}+\frac{12\!\cdots\!57}{10\!\cdots\!53}a^{9}-\frac{44\!\cdots\!88}{10\!\cdots\!53}a^{8}+\frac{13\!\cdots\!63}{10\!\cdots\!53}a^{7}-\frac{23\!\cdots\!07}{10\!\cdots\!53}a^{6}+\frac{48\!\cdots\!92}{10\!\cdots\!53}a^{5}-\frac{43\!\cdots\!00}{10\!\cdots\!53}a^{4}+\frac{53\!\cdots\!89}{10\!\cdots\!53}a^{3}-\frac{14\!\cdots\!99}{10\!\cdots\!53}a^{2}+\frac{61\!\cdots\!99}{10\!\cdots\!53}a+\frac{26\!\cdots\!31}{10\!\cdots\!53}$, $\frac{94\!\cdots\!53}{23\!\cdots\!71}a^{19}-\frac{44\!\cdots\!00}{23\!\cdots\!71}a^{18}+\frac{24\!\cdots\!95}{23\!\cdots\!71}a^{17}-\frac{85\!\cdots\!03}{23\!\cdots\!71}a^{16}+\frac{32\!\cdots\!22}{23\!\cdots\!71}a^{15}-\frac{81\!\cdots\!92}{23\!\cdots\!71}a^{14}+\frac{69\!\cdots\!01}{10\!\cdots\!77}a^{13}-\frac{38\!\cdots\!32}{23\!\cdots\!71}a^{12}+\frac{51\!\cdots\!19}{23\!\cdots\!71}a^{11}-\frac{14\!\cdots\!16}{23\!\cdots\!71}a^{10}+\frac{41\!\cdots\!05}{23\!\cdots\!71}a^{9}-\frac{15\!\cdots\!46}{23\!\cdots\!71}a^{8}+\frac{50\!\cdots\!91}{23\!\cdots\!71}a^{7}-\frac{11\!\cdots\!66}{23\!\cdots\!71}a^{6}+\frac{21\!\cdots\!33}{23\!\cdots\!71}a^{5}-\frac{26\!\cdots\!72}{23\!\cdots\!71}a^{4}+\frac{28\!\cdots\!84}{23\!\cdots\!71}a^{3}-\frac{12\!\cdots\!31}{23\!\cdots\!71}a^{2}+\frac{48\!\cdots\!24}{23\!\cdots\!71}a-\frac{75\!\cdots\!32}{10\!\cdots\!77}$, $\frac{97\!\cdots\!02}{10\!\cdots\!53}a^{19}+\frac{54\!\cdots\!87}{10\!\cdots\!53}a^{18}-\frac{58\!\cdots\!04}{10\!\cdots\!53}a^{17}+\frac{25\!\cdots\!07}{10\!\cdots\!53}a^{16}-\frac{15\!\cdots\!01}{10\!\cdots\!53}a^{15}+\frac{52\!\cdots\!78}{10\!\cdots\!53}a^{14}-\frac{22\!\cdots\!17}{10\!\cdots\!53}a^{13}-\frac{81\!\cdots\!74}{10\!\cdots\!53}a^{12}-\frac{27\!\cdots\!96}{10\!\cdots\!53}a^{11}+\frac{64\!\cdots\!07}{10\!\cdots\!53}a^{10}+\frac{79\!\cdots\!61}{10\!\cdots\!53}a^{9}+\frac{61\!\cdots\!23}{10\!\cdots\!53}a^{8}+\frac{53\!\cdots\!45}{10\!\cdots\!53}a^{7}+\frac{93\!\cdots\!20}{10\!\cdots\!53}a^{6}-\frac{70\!\cdots\!57}{44\!\cdots\!11}a^{5}-\frac{53\!\cdots\!29}{10\!\cdots\!53}a^{4}+\frac{24\!\cdots\!43}{10\!\cdots\!53}a^{3}-\frac{23\!\cdots\!40}{10\!\cdots\!53}a^{2}+\frac{12\!\cdots\!99}{10\!\cdots\!53}a-\frac{36\!\cdots\!86}{10\!\cdots\!53}$
| 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 - 6*x^19 + 34*x^18 - 132*x^17 + 502*x^16 - 1450*x^15 + 3398*x^14 - 7814*x^13 + 13793*x^12 - 30364*x^11 + 73744*x^10 - 254376*x^9 + 822868*x^8 - 2139784*x^7 + 4734462*x^6 - 7831492*x^5 + 11001528*x^4 - 10686038*x^3 + 8803206*x^2 - 3761584*x + 1294987)
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 - 6*x^19 + 34*x^18 - 132*x^17 + 502*x^16 - 1450*x^15 + 3398*x^14 - 7814*x^13 + 13793*x^12 - 30364*x^11 + 73744*x^10 - 254376*x^9 + 822868*x^8 - 2139784*x^7 + 4734462*x^6 - 7831492*x^5 + 11001528*x^4 - 10686038*x^3 + 8803206*x^2 - 3761584*x + 1294987, 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 - 6*x^19 + 34*x^18 - 132*x^17 + 502*x^16 - 1450*x^15 + 3398*x^14 - 7814*x^13 + 13793*x^12 - 30364*x^11 + 73744*x^10 - 254376*x^9 + 822868*x^8 - 2139784*x^7 + 4734462*x^6 - 7831492*x^5 + 11001528*x^4 - 10686038*x^3 + 8803206*x^2 - 3761584*x + 1294987);
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 - 6*x^19 + 34*x^18 - 132*x^17 + 502*x^16 - 1450*x^15 + 3398*x^14 - 7814*x^13 + 13793*x^12 - 30364*x^11 + 73744*x^10 - 254376*x^9 + 822868*x^8 - 2139784*x^7 + 4734462*x^6 - 7831492*x^5 + 11001528*x^4 - 10686038*x^3 + 8803206*x^2 - 3761584*x + 1294987);
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 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |
|
\(23\)
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |