Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - x^18 + 4*x^17 + 12*x^16 + 4*x^15 - 20*x^14 - 28*x^13 - 68*x^12 - 24*x^11 + 70*x^10 + 170*x^9 + 249*x^8 + 258*x^7 + 271*x^6 + 184*x^5 + 126*x^4 + 70*x^3 + 34*x^2 + 8*x + 1)
gp: K = bnfinit(y^20 - 2*y^19 - y^18 + 4*y^17 + 12*y^16 + 4*y^15 - 20*y^14 - 28*y^13 - 68*y^12 - 24*y^11 + 70*y^10 + 170*y^9 + 249*y^8 + 258*y^7 + 271*y^6 + 184*y^5 + 126*y^4 + 70*y^3 + 34*y^2 + 8*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 - x^18 + 4*x^17 + 12*x^16 + 4*x^15 - 20*x^14 - 28*x^13 - 68*x^12 - 24*x^11 + 70*x^10 + 170*x^9 + 249*x^8 + 258*x^7 + 271*x^6 + 184*x^5 + 126*x^4 + 70*x^3 + 34*x^2 + 8*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 - x^18 + 4*x^17 + 12*x^16 + 4*x^15 - 20*x^14 - 28*x^13 - 68*x^12 - 24*x^11 + 70*x^10 + 170*x^9 + 249*x^8 + 258*x^7 + 271*x^6 + 184*x^5 + 126*x^4 + 70*x^3 + 34*x^2 + 8*x + 1)
\( x^{20} - 2 x^{19} - x^{18} + 4 x^{17} + 12 x^{16} + 4 x^{15} - 20 x^{14} - 28 x^{13} - 68 x^{12} + \cdots + 1 \)
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: | $[2, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-1201657195483347978027008\)
\(\medspace = -\,2^{30}\cdot 47^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.00\) | 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: | $2^{3/2}47^{1/2}\approx 19.390719429665317$ | ||
| Ramified primes: |
\(2\), \(47\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-47}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{4831}a^{18}+\frac{1313}{4831}a^{17}-\frac{1603}{4831}a^{16}+\frac{1245}{4831}a^{15}+\frac{967}{4831}a^{14}-\frac{2415}{4831}a^{13}+\frac{229}{4831}a^{12}-\frac{180}{4831}a^{11}-\frac{1642}{4831}a^{10}-\frac{2089}{4831}a^{9}+\frac{934}{4831}a^{8}-\frac{1461}{4831}a^{7}-\frac{506}{4831}a^{6}-\frac{632}{4831}a^{5}-\frac{1167}{4831}a^{4}+\frac{875}{4831}a^{3}-\frac{360}{4831}a^{2}-\frac{1633}{4831}a-\frac{2150}{4831}$, $\frac{1}{20\!\cdots\!77}a^{19}-\frac{1557362169739}{20\!\cdots\!77}a^{18}+\frac{88\!\cdots\!24}{20\!\cdots\!77}a^{17}+\frac{10\!\cdots\!55}{20\!\cdots\!77}a^{16}+\frac{937834589060012}{20\!\cdots\!77}a^{15}+\frac{38\!\cdots\!56}{20\!\cdots\!77}a^{14}-\frac{91\!\cdots\!25}{20\!\cdots\!77}a^{13}+\frac{88\!\cdots\!16}{20\!\cdots\!77}a^{12}-\frac{88\!\cdots\!90}{20\!\cdots\!77}a^{11}+\frac{41688562597430}{20\!\cdots\!77}a^{10}-\frac{10\!\cdots\!61}{20\!\cdots\!77}a^{9}-\frac{42\!\cdots\!17}{20\!\cdots\!77}a^{8}+\frac{44\!\cdots\!98}{20\!\cdots\!77}a^{7}+\frac{68\!\cdots\!66}{20\!\cdots\!77}a^{6}-\frac{37\!\cdots\!37}{20\!\cdots\!77}a^{5}-\frac{138043565413421}{664652411251567}a^{4}+\frac{87\!\cdots\!12}{20\!\cdots\!77}a^{3}-\frac{31\!\cdots\!56}{20\!\cdots\!77}a^{2}-\frac{29\!\cdots\!05}{20\!\cdots\!77}a+\frac{22144199823811}{20\!\cdots\!77}$
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 $1$
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: | $10$ | 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: |
$\frac{98\!\cdots\!44}{20\!\cdots\!77}a^{19}-\frac{20\!\cdots\!06}{20\!\cdots\!77}a^{18}-\frac{79\!\cdots\!68}{20\!\cdots\!77}a^{17}+\frac{42\!\cdots\!12}{20\!\cdots\!77}a^{16}+\frac{11\!\cdots\!34}{20\!\cdots\!77}a^{15}+\frac{21\!\cdots\!84}{20\!\cdots\!77}a^{14}-\frac{20\!\cdots\!38}{20\!\cdots\!77}a^{13}-\frac{24\!\cdots\!27}{20\!\cdots\!77}a^{12}-\frac{62\!\cdots\!42}{20\!\cdots\!77}a^{11}-\frac{15\!\cdots\!41}{20\!\cdots\!77}a^{10}+\frac{74\!\cdots\!82}{20\!\cdots\!77}a^{9}+\frac{15\!\cdots\!69}{20\!\cdots\!77}a^{8}+\frac{22\!\cdots\!54}{20\!\cdots\!77}a^{7}+\frac{21\!\cdots\!54}{20\!\cdots\!77}a^{6}+\frac{22\!\cdots\!56}{20\!\cdots\!77}a^{5}+\frac{45\!\cdots\!41}{664652411251567}a^{4}+\frac{91\!\cdots\!80}{20\!\cdots\!77}a^{3}+\frac{50\!\cdots\!90}{20\!\cdots\!77}a^{2}+\frac{21\!\cdots\!82}{20\!\cdots\!77}a+\frac{36\!\cdots\!17}{20\!\cdots\!77}$, $\frac{52\!\cdots\!68}{20\!\cdots\!77}a^{19}-\frac{11\!\cdots\!02}{20\!\cdots\!77}a^{18}-\frac{48\!\cdots\!52}{20\!\cdots\!77}a^{17}+\frac{23\!\cdots\!49}{20\!\cdots\!77}a^{16}+\frac{61\!\cdots\!90}{20\!\cdots\!77}a^{15}+\frac{10\!\cdots\!44}{20\!\cdots\!77}a^{14}-\frac{11\!\cdots\!16}{20\!\cdots\!77}a^{13}-\frac{13\!\cdots\!36}{20\!\cdots\!77}a^{12}-\frac{31\!\cdots\!52}{20\!\cdots\!77}a^{11}-\frac{57\!\cdots\!97}{20\!\cdots\!77}a^{10}+\frac{43\!\cdots\!22}{20\!\cdots\!77}a^{9}+\frac{84\!\cdots\!99}{20\!\cdots\!77}a^{8}+\frac{11\!\cdots\!24}{20\!\cdots\!77}a^{7}+\frac{10\!\cdots\!96}{20\!\cdots\!77}a^{6}+\frac{10\!\cdots\!18}{20\!\cdots\!77}a^{5}+\frac{21\!\cdots\!68}{664652411251567}a^{4}+\frac{45\!\cdots\!30}{20\!\cdots\!77}a^{3}+\frac{25\!\cdots\!29}{20\!\cdots\!77}a^{2}+\frac{11\!\cdots\!18}{20\!\cdots\!77}a+\frac{30\!\cdots\!82}{20\!\cdots\!77}$, $\frac{61\!\cdots\!13}{20\!\cdots\!77}a^{19}-\frac{11\!\cdots\!82}{20\!\cdots\!77}a^{18}-\frac{10\!\cdots\!67}{20\!\cdots\!77}a^{17}+\frac{27\!\cdots\!55}{20\!\cdots\!77}a^{16}+\frac{79\!\cdots\!99}{20\!\cdots\!77}a^{15}+\frac{31\!\cdots\!67}{20\!\cdots\!77}a^{14}-\frac{13\!\cdots\!55}{20\!\cdots\!77}a^{13}-\frac{20\!\cdots\!64}{20\!\cdots\!77}a^{12}-\frac{41\!\cdots\!77}{20\!\cdots\!77}a^{11}-\frac{16\!\cdots\!78}{20\!\cdots\!77}a^{10}+\frac{53\!\cdots\!40}{20\!\cdots\!77}a^{9}+\frac{11\!\cdots\!98}{20\!\cdots\!77}a^{8}+\frac{15\!\cdots\!67}{20\!\cdots\!77}a^{7}+\frac{15\!\cdots\!62}{20\!\cdots\!77}a^{6}+\frac{14\!\cdots\!34}{20\!\cdots\!77}a^{5}+\frac{31\!\cdots\!81}{664652411251567}a^{4}+\frac{56\!\cdots\!91}{20\!\cdots\!77}a^{3}+\frac{33\!\cdots\!99}{20\!\cdots\!77}a^{2}+\frac{14\!\cdots\!01}{20\!\cdots\!77}a+\frac{37\!\cdots\!64}{20\!\cdots\!77}$, $\frac{14\!\cdots\!41}{20\!\cdots\!77}a^{19}-\frac{16\!\cdots\!04}{20\!\cdots\!77}a^{18}-\frac{42\!\cdots\!73}{20\!\cdots\!77}a^{17}+\frac{45\!\cdots\!86}{20\!\cdots\!77}a^{16}+\frac{23\!\cdots\!71}{20\!\cdots\!77}a^{15}+\frac{21\!\cdots\!82}{20\!\cdots\!77}a^{14}-\frac{27\!\cdots\!47}{20\!\cdots\!77}a^{13}-\frac{74\!\cdots\!24}{20\!\cdots\!77}a^{12}-\frac{13\!\cdots\!51}{20\!\cdots\!77}a^{11}-\frac{11\!\cdots\!78}{20\!\cdots\!77}a^{10}+\frac{10\!\cdots\!41}{20\!\cdots\!77}a^{9}+\frac{38\!\cdots\!60}{20\!\cdots\!77}a^{8}+\frac{59\!\cdots\!72}{20\!\cdots\!77}a^{7}+\frac{65\!\cdots\!72}{20\!\cdots\!77}a^{6}+\frac{60\!\cdots\!31}{20\!\cdots\!77}a^{5}+\frac{14\!\cdots\!48}{664652411251567}a^{4}+\frac{25\!\cdots\!19}{20\!\cdots\!77}a^{3}+\frac{14\!\cdots\!74}{20\!\cdots\!77}a^{2}+\frac{71\!\cdots\!02}{20\!\cdots\!77}a+\frac{91\!\cdots\!40}{20\!\cdots\!77}$, $\frac{75\!\cdots\!29}{20\!\cdots\!77}a^{19}-\frac{12\!\cdots\!21}{20\!\cdots\!77}a^{18}-\frac{14\!\cdots\!33}{20\!\cdots\!77}a^{17}+\frac{31\!\cdots\!38}{20\!\cdots\!77}a^{16}+\frac{10\!\cdots\!46}{20\!\cdots\!77}a^{15}+\frac{50\!\cdots\!11}{20\!\cdots\!77}a^{14}-\frac{17\!\cdots\!40}{20\!\cdots\!77}a^{13}-\frac{27\!\cdots\!40}{20\!\cdots\!77}a^{12}-\frac{53\!\cdots\!89}{20\!\cdots\!77}a^{11}-\frac{25\!\cdots\!46}{20\!\cdots\!77}a^{10}+\frac{64\!\cdots\!81}{20\!\cdots\!77}a^{9}+\frac{15\!\cdots\!89}{20\!\cdots\!77}a^{8}+\frac{20\!\cdots\!73}{20\!\cdots\!77}a^{7}+\frac{20\!\cdots\!15}{20\!\cdots\!77}a^{6}+\frac{19\!\cdots\!08}{20\!\cdots\!77}a^{5}+\frac{43\!\cdots\!14}{664652411251567}a^{4}+\frac{77\!\cdots\!90}{20\!\cdots\!77}a^{3}+\frac{42\!\cdots\!68}{20\!\cdots\!77}a^{2}+\frac{21\!\cdots\!97}{20\!\cdots\!77}a+\frac{43\!\cdots\!03}{20\!\cdots\!77}$, $\frac{44\!\cdots\!56}{20\!\cdots\!77}a^{19}-\frac{96\!\cdots\!11}{20\!\cdots\!77}a^{18}-\frac{18\!\cdots\!58}{20\!\cdots\!77}a^{17}+\frac{15\!\cdots\!80}{20\!\cdots\!77}a^{16}+\frac{50\!\cdots\!18}{20\!\cdots\!77}a^{15}+\frac{16\!\cdots\!27}{20\!\cdots\!77}a^{14}-\frac{77\!\cdots\!04}{20\!\cdots\!77}a^{13}-\frac{11\!\cdots\!42}{20\!\cdots\!77}a^{12}-\frac{32\!\cdots\!18}{20\!\cdots\!77}a^{11}-\frac{84\!\cdots\!86}{20\!\cdots\!77}a^{10}+\frac{28\!\cdots\!63}{20\!\cdots\!77}a^{9}+\frac{74\!\cdots\!52}{20\!\cdots\!77}a^{8}+\frac{11\!\cdots\!33}{20\!\cdots\!77}a^{7}+\frac{11\!\cdots\!23}{20\!\cdots\!77}a^{6}+\frac{11\!\cdots\!51}{20\!\cdots\!77}a^{5}+\frac{22\!\cdots\!44}{664652411251567}a^{4}+\frac{53\!\cdots\!01}{20\!\cdots\!77}a^{3}+\frac{27\!\cdots\!59}{20\!\cdots\!77}a^{2}+\frac{92\!\cdots\!10}{20\!\cdots\!77}a+\frac{93\!\cdots\!19}{20\!\cdots\!77}$, $\frac{1667635568524}{4851477454391}a^{19}-\frac{4110440217292}{4851477454391}a^{18}-\frac{13341249839}{4851477454391}a^{17}+\frac{7143956359748}{4851477454391}a^{16}+\frac{16988261652766}{4851477454391}a^{15}-\frac{1846126582789}{4851477454391}a^{14}-\frac{36295480613946}{4851477454391}a^{13}-\frac{31768441452483}{4851477454391}a^{12}-\frac{92996394060020}{4851477454391}a^{11}+\frac{13711806383930}{4851477454391}a^{10}+\frac{130382692368086}{4851477454391}a^{9}+\frac{227434397629884}{4851477454391}a^{8}+\frac{289810394333042}{4851477454391}a^{7}+\frac{235055411067829}{4851477454391}a^{6}+\frac{267782312019647}{4851477454391}a^{5}+\frac{121543816360277}{4851477454391}a^{4}+\frac{96269195850465}{4851477454391}a^{3}+\frac{53112284285833}{4851477454391}a^{2}+\frac{18592904696792}{4851477454391}a+\frac{6164613149572}{4851477454391}$, $\frac{446431219534975}{20\!\cdots\!77}a^{19}-\frac{41\!\cdots\!47}{20\!\cdots\!77}a^{18}+\frac{85\!\cdots\!96}{20\!\cdots\!77}a^{17}-\frac{244049798121112}{20\!\cdots\!77}a^{16}-\frac{99\!\cdots\!34}{20\!\cdots\!77}a^{15}-\frac{25\!\cdots\!14}{20\!\cdots\!77}a^{14}+\frac{49\!\cdots\!06}{20\!\cdots\!77}a^{13}+\frac{55\!\cdots\!62}{20\!\cdots\!77}a^{12}+\frac{37\!\cdots\!37}{20\!\cdots\!77}a^{11}+\frac{15\!\cdots\!93}{20\!\cdots\!77}a^{10}-\frac{33\!\cdots\!88}{20\!\cdots\!77}a^{9}-\frac{17\!\cdots\!98}{20\!\cdots\!77}a^{8}-\frac{23\!\cdots\!63}{20\!\cdots\!77}a^{7}-\frac{32\!\cdots\!27}{20\!\cdots\!77}a^{6}-\frac{21\!\cdots\!69}{20\!\cdots\!77}a^{5}-\frac{11\!\cdots\!72}{664652411251567}a^{4}-\frac{47\!\cdots\!59}{20\!\cdots\!77}a^{3}-\frac{98\!\cdots\!83}{20\!\cdots\!77}a^{2}-\frac{46\!\cdots\!67}{20\!\cdots\!77}a+\frac{772146545193268}{20\!\cdots\!77}$, $\frac{16\!\cdots\!72}{20\!\cdots\!77}a^{19}-\frac{66\!\cdots\!28}{20\!\cdots\!77}a^{18}+\frac{61\!\cdots\!67}{20\!\cdots\!77}a^{17}+\frac{74\!\cdots\!94}{20\!\cdots\!77}a^{16}+\frac{55\!\cdots\!47}{20\!\cdots\!77}a^{15}-\frac{26\!\cdots\!61}{20\!\cdots\!77}a^{14}-\frac{30\!\cdots\!88}{20\!\cdots\!77}a^{13}+\frac{21\!\cdots\!30}{20\!\cdots\!77}a^{12}-\frac{54\!\cdots\!13}{20\!\cdots\!77}a^{11}+\frac{14\!\cdots\!61}{20\!\cdots\!77}a^{10}+\frac{12\!\cdots\!20}{20\!\cdots\!77}a^{9}+\frac{46\!\cdots\!27}{20\!\cdots\!77}a^{8}-\frac{24\!\cdots\!14}{20\!\cdots\!77}a^{7}-\frac{15\!\cdots\!43}{20\!\cdots\!77}a^{6}-\frac{12\!\cdots\!63}{20\!\cdots\!77}a^{5}-\frac{11\!\cdots\!38}{664652411251567}a^{4}-\frac{13\!\cdots\!63}{20\!\cdots\!77}a^{3}-\frac{16\!\cdots\!24}{20\!\cdots\!77}a^{2}-\frac{69\!\cdots\!92}{20\!\cdots\!77}a-\frac{45\!\cdots\!14}{20\!\cdots\!77}$, $\frac{30\!\cdots\!11}{20\!\cdots\!77}a^{19}-\frac{79\!\cdots\!42}{20\!\cdots\!77}a^{18}+\frac{15\!\cdots\!12}{20\!\cdots\!77}a^{17}+\frac{12\!\cdots\!84}{20\!\cdots\!77}a^{16}+\frac{25\!\cdots\!24}{20\!\cdots\!77}a^{15}-\frac{56\!\cdots\!19}{20\!\cdots\!77}a^{14}-\frac{55\!\cdots\!96}{20\!\cdots\!77}a^{13}-\frac{37\!\cdots\!09}{20\!\cdots\!77}a^{12}-\frac{17\!\cdots\!02}{20\!\cdots\!77}a^{11}+\frac{15\!\cdots\!74}{20\!\cdots\!77}a^{10}+\frac{19\!\cdots\!29}{20\!\cdots\!77}a^{9}+\frac{33\!\cdots\!00}{20\!\cdots\!77}a^{8}+\frac{52\!\cdots\!78}{20\!\cdots\!77}a^{7}+\frac{49\!\cdots\!61}{20\!\cdots\!77}a^{6}+\frac{62\!\cdots\!42}{20\!\cdots\!77}a^{5}+\frac{11\!\cdots\!02}{664652411251567}a^{4}+\frac{27\!\cdots\!11}{20\!\cdots\!77}a^{3}+\frac{18\!\cdots\!39}{20\!\cdots\!77}a^{2}+\frac{54\!\cdots\!13}{20\!\cdots\!77}a+\frac{23\!\cdots\!35}{20\!\cdots\!77}$
| 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: | \( 6974.62635242 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{9}\cdot 6974.62635242 \cdot 1}{2\cdot\sqrt{1201657195483347978027008}}\cr\approx \mathstrut & 0.194213520866 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - x^18 + 4*x^17 + 12*x^16 + 4*x^15 - 20*x^14 - 28*x^13 - 68*x^12 - 24*x^11 + 70*x^10 + 170*x^9 + 249*x^8 + 258*x^7 + 271*x^6 + 184*x^5 + 126*x^4 + 70*x^3 + 34*x^2 + 8*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 - 2*x^19 - x^18 + 4*x^17 + 12*x^16 + 4*x^15 - 20*x^14 - 28*x^13 - 68*x^12 - 24*x^11 + 70*x^10 + 170*x^9 + 249*x^8 + 258*x^7 + 271*x^6 + 184*x^5 + 126*x^4 + 70*x^3 + 34*x^2 + 8*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 - 2*x^19 - x^18 + 4*x^17 + 12*x^16 + 4*x^15 - 20*x^14 - 28*x^13 - 68*x^12 - 24*x^11 + 70*x^10 + 170*x^9 + 249*x^8 + 258*x^7 + 271*x^6 + 184*x^5 + 126*x^4 + 70*x^3 + 34*x^2 + 8*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 - 2*x^19 - x^18 + 4*x^17 + 12*x^16 + 4*x^15 - 20*x^14 - 28*x^13 - 68*x^12 - 24*x^11 + 70*x^10 + 170*x^9 + 249*x^8 + 258*x^7 + 271*x^6 + 184*x^5 + 126*x^4 + 70*x^3 + 34*x^2 + 8*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
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 40 |
| The 13 conjugacy class representatives for $C_5:D_4$ |
| Character table for $C_5:D_4$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.2.3008.1, 5.1.2209.1, 10.2.159897387008.2 |
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
| Galois closure: | deg 40 |
| Degree 20 sibling: | 20.0.1723568365103679045632.1 |
| 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 | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{5}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{5}$ | ${\href{/padicField/13.4.0.1}{4} }^{5}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{5}$ | ${\href{/padicField/23.2.0.1}{2} }^{9}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{5}$ | ${\href{/padicField/31.2.0.1}{2} }^{9}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | R | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.15.1 | $x^{10} + 70 x^{8} + 2 x^{7} + 1960 x^{6} - 26 x^{5} + 27441 x^{4} - 2240 x^{3} + 192110 x^{2} - 14504 x + 537993$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ |
| 2.10.15.1 | $x^{10} + 70 x^{8} + 2 x^{7} + 1960 x^{6} - 26 x^{5} + 27441 x^{4} - 2240 x^{3} + 192110 x^{2} - 14504 x + 537993$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ | |
|
\(47\)
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |