Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 190*x^16 + 15219*x^14 - 670985*x^12 + 17820195*x^10 - 292417885*x^8 + 2904860828*x^6 - 16125600181*x^4 + 40252586306*x^2 - 16862569939)
gp: K = bnfinit(y^18 - 190*y^16 + 15219*y^14 - 670985*y^12 + 17820195*y^10 - 292417885*y^8 + 2904860828*y^6 - 16125600181*y^4 + 40252586306*y^2 - 16862569939, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 190*x^16 + 15219*x^14 - 670985*x^12 + 17820195*x^10 - 292417885*x^8 + 2904860828*x^6 - 16125600181*x^4 + 40252586306*x^2 - 16862569939);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 190*x^16 + 15219*x^14 - 670985*x^12 + 17820195*x^10 - 292417885*x^8 + 2904860828*x^6 - 16125600181*x^4 + 40252586306*x^2 - 16862569939)
\( x^{18} - 190 x^{16} + 15219 x^{14} - 670985 x^{12} + 17820195 x^{10} - 292417885 x^{8} + \cdots - 16862569939 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[18, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1275032635857445963924166132749828096\)
\(\medspace = 2^{18}\cdot 19^{17}\cdot 31^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(101.36\) | 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: | $2\cdot 19^{17/18}31^{1/2}\approx 179.64807867551454$ | ||
| Ramified primes: |
\(2\), \(19\), \(31\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{19}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{31}a^{8}-\frac{4}{31}a^{6}-\frac{2}{31}a^{4}+\frac{10}{31}a^{2}$, $\frac{1}{31}a^{9}-\frac{4}{31}a^{7}-\frac{2}{31}a^{5}+\frac{10}{31}a^{3}$, $\frac{1}{961}a^{10}-\frac{4}{961}a^{8}+\frac{60}{961}a^{6}+\frac{382}{961}a^{4}+\frac{10}{31}a^{2}$, $\frac{1}{961}a^{11}-\frac{4}{961}a^{9}+\frac{60}{961}a^{7}+\frac{382}{961}a^{5}+\frac{10}{31}a^{3}$, $\frac{1}{29791}a^{12}-\frac{4}{29791}a^{10}+\frac{60}{29791}a^{8}-\frac{6345}{29791}a^{6}-\frac{455}{961}a^{4}+\frac{14}{31}a^{2}$, $\frac{1}{29791}a^{13}-\frac{4}{29791}a^{11}+\frac{60}{29791}a^{9}-\frac{6345}{29791}a^{7}-\frac{455}{961}a^{5}+\frac{14}{31}a^{3}$, $\frac{1}{923521}a^{14}-\frac{4}{923521}a^{12}+\frac{60}{923521}a^{10}-\frac{6345}{923521}a^{8}-\frac{14870}{29791}a^{6}+\frac{262}{961}a^{4}-\frac{8}{31}a^{2}$, $\frac{1}{923521}a^{15}-\frac{4}{923521}a^{13}+\frac{60}{923521}a^{11}-\frac{6345}{923521}a^{9}-\frac{14870}{29791}a^{7}+\frac{262}{961}a^{5}-\frac{8}{31}a^{3}$, $\frac{1}{76\!\cdots\!97}a^{16}-\frac{22215006197}{76\!\cdots\!97}a^{14}-\frac{535691093073}{76\!\cdots\!97}a^{12}-\frac{27922767191947}{76\!\cdots\!97}a^{10}+\frac{15071427809196}{24\!\cdots\!87}a^{8}+\frac{33650395962851}{79148703388277}a^{6}+\frac{314351971226}{2553183980267}a^{4}+\frac{20218176838}{82360773557}a^{2}+\frac{849938195}{2656799147}$, $\frac{1}{76\!\cdots\!97}a^{17}-\frac{22215006197}{76\!\cdots\!97}a^{15}-\frac{535691093073}{76\!\cdots\!97}a^{13}-\frac{27922767191947}{76\!\cdots\!97}a^{11}+\frac{15071427809196}{24\!\cdots\!87}a^{9}+\frac{33650395962851}{79148703388277}a^{7}+\frac{314351971226}{2553183980267}a^{5}+\frac{20218176838}{82360773557}a^{3}+\frac{849938195}{2656799147}a$
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 $4$ (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: | $17$ | 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: |
$\frac{259604366124}{76\!\cdots\!97}a^{16}-\frac{45044151842305}{76\!\cdots\!97}a^{14}+\frac{32\!\cdots\!24}{76\!\cdots\!97}a^{12}-\frac{12\!\cdots\!94}{76\!\cdots\!97}a^{10}+\frac{84\!\cdots\!23}{24\!\cdots\!87}a^{8}-\frac{33\!\cdots\!47}{79148703388277}a^{6}+\frac{72\!\cdots\!42}{2553183980267}a^{4}-\frac{668631199398959}{82360773557}a^{2}+\frac{9272724761336}{2656799147}$, $\frac{253983753851}{76\!\cdots\!97}a^{16}-\frac{43990016927630}{76\!\cdots\!97}a^{14}+\frac{31\!\cdots\!61}{76\!\cdots\!97}a^{12}-\frac{11\!\cdots\!90}{76\!\cdots\!97}a^{10}+\frac{82\!\cdots\!35}{24\!\cdots\!87}a^{8}-\frac{32\!\cdots\!22}{79148703388277}a^{6}+\frac{69\!\cdots\!05}{2553183980267}a^{4}-\frac{642671580029695}{82360773557}a^{2}+\frac{8912489927114}{2656799147}$, $\frac{497613895431}{76\!\cdots\!97}a^{16}-\frac{86280864937824}{76\!\cdots\!97}a^{14}+\frac{61\!\cdots\!08}{76\!\cdots\!97}a^{12}-\frac{23\!\cdots\!59}{76\!\cdots\!97}a^{10}+\frac{16\!\cdots\!76}{24\!\cdots\!87}a^{8}-\frac{64\!\cdots\!53}{79148703388277}a^{6}+\frac{13\!\cdots\!52}{2553183980267}a^{4}-\frac{12\!\cdots\!76}{82360773557}a^{2}+\frac{17637782313019}{2656799147}$, $\frac{117706970870}{76\!\cdots\!97}a^{16}-\frac{20516438728194}{76\!\cdots\!97}a^{14}+\frac{14\!\cdots\!01}{76\!\cdots\!97}a^{12}-\frac{55\!\cdots\!16}{76\!\cdots\!97}a^{10}+\frac{39\!\cdots\!78}{24\!\cdots\!87}a^{8}-\frac{15\!\cdots\!05}{79148703388277}a^{6}+\frac{34\!\cdots\!41}{2553183980267}a^{4}-\frac{317865048011070}{82360773557}a^{2}+\frac{4408300474761}{2656799147}$, $\frac{334231664161}{76\!\cdots\!97}a^{16}-\frac{58056099424966}{76\!\cdots\!97}a^{14}+\frac{41\!\cdots\!15}{76\!\cdots\!97}a^{12}-\frac{15\!\cdots\!37}{76\!\cdots\!97}a^{10}+\frac{10\!\cdots\!34}{24\!\cdots\!87}a^{8}-\frac{44\!\cdots\!92}{79148703388277}a^{6}+\frac{94\!\cdots\!20}{2553183980267}a^{4}-\frac{869881721464908}{82360773557}a^{2}+\frac{12066705876124}{2656799147}$, $\frac{7238638734}{24\!\cdots\!87}a^{16}-\frac{1255672604279}{24\!\cdots\!87}a^{14}+\frac{89406091843936}{24\!\cdots\!87}a^{12}-\frac{33\!\cdots\!76}{24\!\cdots\!87}a^{10}+\frac{73\!\cdots\!93}{24\!\cdots\!87}a^{8}-\frac{29\!\cdots\!84}{79148703388277}a^{6}+\frac{62\!\cdots\!74}{2553183980267}a^{4}-\frac{576768447569346}{82360773557}a^{2}+\frac{7996573130287}{2656799147}$, $\frac{160411387060}{76\!\cdots\!97}a^{16}-\frac{27825347735039}{76\!\cdots\!97}a^{14}+\frac{19\!\cdots\!14}{76\!\cdots\!97}a^{12}-\frac{74\!\cdots\!74}{76\!\cdots\!97}a^{10}+\frac{52\!\cdots\!36}{24\!\cdots\!87}a^{8}-\frac{20\!\cdots\!28}{79148703388277}a^{6}+\frac{44\!\cdots\!89}{2553183980267}a^{4}-\frac{411620190043144}{82360773557}a^{2}+\frac{5714933318969}{2656799147}$, $\frac{107184095344}{76\!\cdots\!97}a^{16}-\frac{18475531555461}{76\!\cdots\!97}a^{14}+\frac{13\!\cdots\!15}{76\!\cdots\!97}a^{12}-\frac{48\!\cdots\!25}{76\!\cdots\!97}a^{10}+\frac{10\!\cdots\!89}{79148703388277}a^{8}-\frac{13\!\cdots\!16}{79148703388277}a^{6}+\frac{28\!\cdots\!52}{2553183980267}a^{4}-\frac{258144238149922}{82360773557}a^{2}+\frac{3568697828688}{2656799147}$, $\frac{3239284788453}{76\!\cdots\!97}a^{17}+\frac{10590383221700}{76\!\cdots\!97}a^{16}-\frac{563546927402686}{76\!\cdots\!97}a^{15}-\frac{18\!\cdots\!16}{76\!\cdots\!97}a^{14}+\frac{40\!\cdots\!50}{76\!\cdots\!97}a^{13}+\frac{13\!\cdots\!26}{76\!\cdots\!97}a^{12}-\frac{15\!\cdots\!25}{76\!\cdots\!97}a^{11}-\frac{50\!\cdots\!57}{76\!\cdots\!97}a^{10}+\frac{10\!\cdots\!38}{24\!\cdots\!87}a^{9}+\frac{35\!\cdots\!53}{24\!\cdots\!87}a^{8}-\frac{43\!\cdots\!10}{79148703388277}a^{7}-\frac{14\!\cdots\!78}{79148703388277}a^{6}+\frac{92\!\cdots\!24}{2553183980267}a^{5}+\frac{30\!\cdots\!51}{2553183980267}a^{4}-\frac{85\!\cdots\!71}{82360773557}a^{3}-\frac{28\!\cdots\!37}{82360773557}a^{2}+\frac{119054300502960}{2656799147}a+\frac{390519332213771}{2656799147}$, $\frac{475932414416}{76\!\cdots\!97}a^{17}-\frac{3427651174394}{76\!\cdots\!97}a^{16}-\frac{82911796175679}{76\!\cdots\!97}a^{15}+\frac{596171336021939}{76\!\cdots\!97}a^{14}+\frac{59\!\cdots\!50}{76\!\cdots\!97}a^{13}-\frac{42\!\cdots\!04}{76\!\cdots\!97}a^{12}-\frac{22\!\cdots\!89}{76\!\cdots\!97}a^{11}+\frac{16\!\cdots\!20}{76\!\cdots\!97}a^{10}+\frac{15\!\cdots\!41}{24\!\cdots\!87}a^{9}-\frac{11\!\cdots\!84}{24\!\cdots\!87}a^{8}-\frac{63\!\cdots\!10}{79148703388277}a^{7}+\frac{45\!\cdots\!89}{79148703388277}a^{6}+\frac{13\!\cdots\!43}{2553183980267}a^{5}-\frac{97\!\cdots\!51}{2553183980267}a^{4}-\frac{12\!\cdots\!41}{82360773557}a^{3}+\frac{90\!\cdots\!98}{82360773557}a^{2}+\frac{17753233239710}{2656799147}a-\frac{125629096674900}{2656799147}$, $\frac{186575087883}{76\!\cdots\!97}a^{17}+\frac{708996202948}{76\!\cdots\!97}a^{16}-\frac{32377641715823}{76\!\cdots\!97}a^{15}-\frac{123015739424677}{76\!\cdots\!97}a^{14}+\frac{23\!\cdots\!06}{76\!\cdots\!97}a^{13}+\frac{87\!\cdots\!24}{76\!\cdots\!97}a^{12}-\frac{87\!\cdots\!05}{76\!\cdots\!97}a^{11}-\frac{33\!\cdots\!45}{76\!\cdots\!97}a^{10}+\frac{61\!\cdots\!39}{24\!\cdots\!87}a^{9}+\frac{23\!\cdots\!49}{24\!\cdots\!87}a^{8}-\frac{24\!\cdots\!49}{79148703388277}a^{7}-\frac{92\!\cdots\!17}{79148703388277}a^{6}+\frac{52\!\cdots\!36}{2553183980267}a^{5}+\frac{19\!\cdots\!54}{2553183980267}a^{4}-\frac{490411818991408}{82360773557}a^{3}-\frac{18\!\cdots\!96}{82360773557}a^{2}+\frac{7507486638596}{2656799147}a+\frac{25891538836957}{2656799147}$, $\frac{78592855877}{76\!\cdots\!97}a^{17}-\frac{289773093522}{76\!\cdots\!97}a^{16}-\frac{13621841193877}{76\!\cdots\!97}a^{15}+\frac{49934438962018}{76\!\cdots\!97}a^{14}+\frac{968894979134377}{76\!\cdots\!97}a^{13}-\frac{35\!\cdots\!49}{76\!\cdots\!97}a^{12}-\frac{36\!\cdots\!98}{76\!\cdots\!97}a^{11}+\frac{13\!\cdots\!44}{76\!\cdots\!97}a^{10}+\frac{25\!\cdots\!66}{24\!\cdots\!87}a^{9}-\frac{91\!\cdots\!32}{24\!\cdots\!87}a^{8}-\frac{10\!\cdots\!16}{79148703388277}a^{7}+\frac{36\!\cdots\!23}{79148703388277}a^{6}+\frac{21\!\cdots\!80}{2553183980267}a^{5}-\frac{76\!\cdots\!15}{2553183980267}a^{4}-\frac{200199798058162}{82360773557}a^{3}+\frac{693861481674662}{82360773557}a^{2}+\frac{2764064666822}{2656799147}a-\frac{9609946899881}{2656799147}$, $\frac{227571658170}{76\!\cdots\!97}a^{17}-\frac{305117886592}{76\!\cdots\!97}a^{16}-\frac{39548812585556}{76\!\cdots\!97}a^{15}+\frac{52148623767825}{76\!\cdots\!97}a^{14}+\frac{28\!\cdots\!50}{76\!\cdots\!97}a^{13}-\frac{36\!\cdots\!91}{76\!\cdots\!97}a^{12}-\frac{10\!\cdots\!76}{76\!\cdots\!97}a^{11}+\frac{13\!\cdots\!57}{76\!\cdots\!97}a^{10}+\frac{24\!\cdots\!16}{79148703388277}a^{9}-\frac{91\!\cdots\!29}{24\!\cdots\!87}a^{8}-\frac{30\!\cdots\!89}{79148703388277}a^{7}+\frac{35\!\cdots\!29}{79148703388277}a^{6}+\frac{64\!\cdots\!29}{2553183980267}a^{5}-\frac{74\!\cdots\!20}{2553183980267}a^{4}-\frac{594149019098225}{82360773557}a^{3}+\frac{672223708319496}{82360773557}a^{2}+\frac{8217905536497}{2656799147}a-\frac{9264314999649}{2656799147}$, $\frac{2433405312705}{76\!\cdots\!97}a^{17}-\frac{7906022717021}{76\!\cdots\!97}a^{16}-\frac{421601697529202}{76\!\cdots\!97}a^{15}+\frac{13\!\cdots\!72}{76\!\cdots\!97}a^{14}+\frac{29\!\cdots\!87}{76\!\cdots\!97}a^{13}-\frac{97\!\cdots\!15}{76\!\cdots\!97}a^{12}-\frac{11\!\cdots\!60}{76\!\cdots\!97}a^{11}+\frac{36\!\cdots\!53}{76\!\cdots\!97}a^{10}+\frac{78\!\cdots\!90}{24\!\cdots\!87}a^{9}-\frac{25\!\cdots\!49}{24\!\cdots\!87}a^{8}-\frac{31\!\cdots\!45}{79148703388277}a^{7}+\frac{10\!\cdots\!12}{79148703388277}a^{6}+\frac{67\!\cdots\!35}{2553183980267}a^{5}-\frac{21\!\cdots\!80}{2553183980267}a^{4}-\frac{61\!\cdots\!94}{82360773557}a^{3}+\frac{19\!\cdots\!05}{82360773557}a^{2}+\frac{85652711523292}{2656799147}a-\frac{274242835809539}{2656799147}$, $\frac{212512262000}{76\!\cdots\!97}a^{17}+\frac{1182483755005}{76\!\cdots\!97}a^{16}-\frac{37021352665940}{76\!\cdots\!97}a^{15}-\frac{205717772763145}{76\!\cdots\!97}a^{14}+\frac{26\!\cdots\!36}{76\!\cdots\!97}a^{13}+\frac{14\!\cdots\!45}{76\!\cdots\!97}a^{12}-\frac{10\!\cdots\!77}{76\!\cdots\!97}a^{11}-\frac{55\!\cdots\!40}{76\!\cdots\!97}a^{10}+\frac{70\!\cdots\!34}{24\!\cdots\!87}a^{9}+\frac{39\!\cdots\!60}{24\!\cdots\!87}a^{8}-\frac{28\!\cdots\!49}{79148703388277}a^{7}-\frac{15\!\cdots\!89}{79148703388277}a^{6}+\frac{61\!\cdots\!31}{2553183980267}a^{5}+\frac{33\!\cdots\!12}{2553183980267}a^{4}-\frac{575610700817171}{82360773557}a^{3}-\frac{31\!\cdots\!30}{82360773557}a^{2}+\frac{8008245583182}{2656799147}a+\frac{43671017788669}{2656799147}$, $\frac{3355256822748}{76\!\cdots\!97}a^{17}-\frac{14349560941850}{76\!\cdots\!97}a^{16}-\frac{580993427662273}{76\!\cdots\!97}a^{15}+\frac{24\!\cdots\!49}{76\!\cdots\!97}a^{14}+\frac{41\!\cdots\!93}{76\!\cdots\!97}a^{13}-\frac{17\!\cdots\!87}{76\!\cdots\!97}a^{12}-\frac{15\!\cdots\!83}{76\!\cdots\!97}a^{11}+\frac{66\!\cdots\!10}{76\!\cdots\!97}a^{10}+\frac{10\!\cdots\!71}{24\!\cdots\!87}a^{9}-\frac{46\!\cdots\!55}{24\!\cdots\!87}a^{8}-\frac{43\!\cdots\!18}{79148703388277}a^{7}+\frac{18\!\cdots\!01}{79148703388277}a^{6}+\frac{92\!\cdots\!77}{2553183980267}a^{5}-\frac{39\!\cdots\!20}{2553183980267}a^{4}-\frac{84\!\cdots\!83}{82360773557}a^{3}+\frac{36\!\cdots\!08}{82360773557}a^{2}+\frac{117342930383932}{2656799147}a-\frac{501956500596458}{2656799147}$, $\frac{2137553583286}{76\!\cdots\!97}a^{17}-\frac{8608757826915}{76\!\cdots\!97}a^{16}-\frac{371099854663624}{76\!\cdots\!97}a^{15}+\frac{14\!\cdots\!53}{76\!\cdots\!97}a^{14}+\frac{26\!\cdots\!53}{76\!\cdots\!97}a^{13}-\frac{10\!\cdots\!15}{76\!\cdots\!97}a^{12}-\frac{10\!\cdots\!48}{76\!\cdots\!97}a^{11}+\frac{40\!\cdots\!10}{76\!\cdots\!97}a^{10}+\frac{69\!\cdots\!70}{24\!\cdots\!87}a^{9}-\frac{28\!\cdots\!05}{24\!\cdots\!87}a^{8}-\frac{28\!\cdots\!98}{79148703388277}a^{7}+\frac{11\!\cdots\!57}{79148703388277}a^{6}+\frac{60\!\cdots\!34}{2553183980267}a^{5}-\frac{24\!\cdots\!09}{2553183980267}a^{4}-\frac{55\!\cdots\!74}{82360773557}a^{3}+\frac{22\!\cdots\!53}{82360773557}a^{2}+\frac{77383092126259}{2656799147}a-\frac{308197479388405}{2656799147}$
| 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: | \( 402218304851 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 402218304851 \cdot 4}{2\cdot\sqrt{1275032635857445963924166132749828096}}\cr\approx \mathstrut & 0.186754509928825 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 190*x^16 + 15219*x^14 - 670985*x^12 + 17820195*x^10 - 292417885*x^8 + 2904860828*x^6 - 16125600181*x^4 + 40252586306*x^2 - 16862569939)
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^18 - 190*x^16 + 15219*x^14 - 670985*x^12 + 17820195*x^10 - 292417885*x^8 + 2904860828*x^6 - 16125600181*x^4 + 40252586306*x^2 - 16862569939, 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^18 - 190*x^16 + 15219*x^14 - 670985*x^12 + 17820195*x^10 - 292417885*x^8 + 2904860828*x^6 - 16125600181*x^4 + 40252586306*x^2 - 16862569939);
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^18 - 190*x^16 + 15219*x^14 - 670985*x^12 + 17820195*x^10 - 292417885*x^8 + 2904860828*x^6 - 16125600181*x^4 + 40252586306*x^2 - 16862569939);
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
$C_2^2:C_{18}$ (as 18T26):
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 72 |
| The 24 conjugacy class representatives for $C_2^2:C_{18}$ |
| Character table for $C_2^2:C_{18}$ |
Intermediate fields
| 3.3.361.1, 6.6.152289992896.1, \(\Q(\zeta_{19})^+\) |
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 36 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 | R | ${\href{/padicField/3.9.0.1}{9} }^{2}$ | ${\href{/padicField/5.9.0.1}{9} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{4}$ | $18$ | ${\href{/padicField/17.9.0.1}{9} }^{2}$ | R | $18$ | $18$ | R | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{12}$ | $18$ | $18$ | $18$ | $18$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.18.118 | $x^{18} + 18 x^{17} + 198 x^{16} + 1536 x^{15} + 9312 x^{14} + 45696 x^{13} + 187776 x^{12} + 655872 x^{11} + 2010400 x^{10} + 5500224 x^{9} + 14116288 x^{8} + 34058240 x^{7} + 79898624 x^{6} + 169960448 x^{5} + 335809536 x^{4} + 542121984 x^{3} + 798549248 x^{2} + 783239680 x + 807955968$ | $2$ | $9$ | $18$ | $C_{18}$ | $[2]^{9}$ |
|
\(19\)
| 19.18.17.1 | $x^{18} + 342$ | $18$ | $1$ | $17$ | $C_{18}$ | $[\ ]_{18}$ |
|
\(31\)
| 31.6.3.2 | $x^{6} + 95 x^{4} + 56 x^{3} + 2884 x^{2} - 5152 x + 28684$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 31.6.3.1 | $x^{6} + 961 x^{2} - 834148$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 31.6.0.1 | $x^{6} + 19 x^{3} + 16 x^{2} + 8 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |