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Properties

Label 18.0.562...696.1
Degree $18$
Signature $[0, 9]$
Discriminant $-5.627\times 10^{27}$
Root discriminant \(34.81\)
Ramified primes $2,11,17$
Class number $3$ (GRH)
Class group [3] (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 53*x^16 - 224*x^15 + 1059*x^14 - 3090*x^13 + 9543*x^12 - 18332*x^11 + 38972*x^10 - 39464*x^9 + 60140*x^8 + 22376*x^7 + 29584*x^6 + 113584*x^5 + 243424*x^4 - 171840*x^3 + 703168*x^2 - 494560*x + 271568)
 
gp: K = bnfinit(y^18 - 6*y^17 + 53*y^16 - 224*y^15 + 1059*y^14 - 3090*y^13 + 9543*y^12 - 18332*y^11 + 38972*y^10 - 39464*y^9 + 60140*y^8 + 22376*y^7 + 29584*y^6 + 113584*y^5 + 243424*y^4 - 171840*y^3 + 703168*y^2 - 494560*y + 271568, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 53*x^16 - 224*x^15 + 1059*x^14 - 3090*x^13 + 9543*x^12 - 18332*x^11 + 38972*x^10 - 39464*x^9 + 60140*x^8 + 22376*x^7 + 29584*x^6 + 113584*x^5 + 243424*x^4 - 171840*x^3 + 703168*x^2 - 494560*x + 271568);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 6*x^17 + 53*x^16 - 224*x^15 + 1059*x^14 - 3090*x^13 + 9543*x^12 - 18332*x^11 + 38972*x^10 - 39464*x^9 + 60140*x^8 + 22376*x^7 + 29584*x^6 + 113584*x^5 + 243424*x^4 - 171840*x^3 + 703168*x^2 - 494560*x + 271568)
 

\( x^{18} - 6 x^{17} + 53 x^{16} - 224 x^{15} + 1059 x^{14} - 3090 x^{13} + 9543 x^{12} - 18332 x^{11} + \cdots + 271568 \) Copy content Toggle raw display

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:  $[0, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-5627055140868788987624861696\) \(\medspace = -\,2^{12}\cdot 11^{9}\cdot 17^{12}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(34.81\)
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:  $2^{2/3}11^{1/2}17^{2/3}\approx 34.80825783744108$
Ramified primes:   \(2\), \(11\), \(17\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-11}) \)
$\card{ \Gal(K/\Q) }$:  $18$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{6}a^{9}-\frac{1}{2}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{12}a^{10}-\frac{1}{12}a^{9}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{3}a^{4}+\frac{1}{6}a^{3}-\frac{1}{6}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{12}a^{11}-\frac{1}{12}a^{9}-\frac{1}{4}a^{7}-\frac{5}{12}a^{5}-\frac{1}{2}a^{4}+\frac{1}{6}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{120}a^{12}-\frac{1}{30}a^{11}-\frac{1}{40}a^{10}+\frac{1}{30}a^{9}+\frac{7}{40}a^{8}-\frac{1}{5}a^{7}-\frac{29}{120}a^{6}-\frac{1}{3}a^{5}+\frac{11}{60}a^{4}+\frac{1}{15}a^{3}-\frac{4}{15}a^{2}+\frac{13}{30}a-\frac{7}{30}$, $\frac{1}{120}a^{13}+\frac{1}{120}a^{11}+\frac{1}{60}a^{10}+\frac{7}{120}a^{9}-\frac{1}{24}a^{7}-\frac{1}{20}a^{6}-\frac{7}{30}a^{5}+\frac{2}{15}a^{4}+\frac{1}{6}a^{3}-\frac{7}{15}a^{2}+\frac{1}{6}a+\frac{1}{15}$, $\frac{1}{120}a^{14}-\frac{1}{30}a^{11}-\frac{1}{30}a^{9}-\frac{13}{60}a^{8}-\frac{1}{10}a^{7}-\frac{29}{120}a^{6}-\frac{11}{30}a^{5}-\frac{7}{20}a^{4}-\frac{11}{30}a^{3}+\frac{13}{30}a^{2}-\frac{1}{30}a+\frac{7}{30}$, $\frac{1}{720}a^{15}-\frac{1}{360}a^{14}-\frac{1}{720}a^{13}-\frac{1}{360}a^{12}-\frac{11}{720}a^{11}+\frac{7}{180}a^{10}-\frac{7}{720}a^{9}+\frac{11}{360}a^{8}+\frac{17}{120}a^{7}+\frac{71}{360}a^{6}+\frac{43}{90}a^{5}+\frac{2}{5}a^{4}-\frac{41}{180}a^{3}+\frac{1}{5}a^{2}-\frac{1}{2}a+\frac{5}{18}$, $\frac{1}{4601520}a^{16}-\frac{439}{1533840}a^{15}+\frac{10111}{4601520}a^{14}-\frac{25}{83664}a^{13}-\frac{4891}{1533840}a^{12}+\frac{547}{306768}a^{11}-\frac{4547}{4601520}a^{10}-\frac{10181}{920304}a^{9}-\frac{249313}{1150380}a^{8}+\frac{148279}{1150380}a^{7}-\frac{153287}{1150380}a^{6}+\frac{33821}{104580}a^{5}-\frac{22102}{57519}a^{4}-\frac{1369}{2772}a^{3}+\frac{27029}{191730}a^{2}-\frac{12349}{52290}a-\frac{2302}{26145}$, $\frac{1}{33\!\cdots\!80}a^{17}-\frac{82\!\cdots\!31}{16\!\cdots\!24}a^{16}+\frac{37\!\cdots\!97}{33\!\cdots\!80}a^{15}-\frac{80\!\cdots\!59}{18\!\cdots\!36}a^{14}+\frac{95\!\cdots\!47}{33\!\cdots\!80}a^{13}-\frac{49\!\cdots\!07}{50\!\cdots\!80}a^{12}+\frac{47\!\cdots\!31}{66\!\cdots\!96}a^{11}-\frac{38\!\cdots\!73}{13\!\cdots\!40}a^{10}-\frac{51\!\cdots\!77}{76\!\cdots\!20}a^{9}-\frac{88\!\cdots\!51}{79\!\cdots\!40}a^{8}+\frac{19\!\cdots\!13}{83\!\cdots\!20}a^{7}-\frac{39\!\cdots\!97}{16\!\cdots\!24}a^{6}+\frac{40\!\cdots\!89}{83\!\cdots\!20}a^{5}+\frac{89\!\cdots\!89}{19\!\cdots\!60}a^{4}-\frac{13\!\cdots\!87}{29\!\cdots\!90}a^{3}+\frac{16\!\cdots\!11}{41\!\cdots\!60}a^{2}+\frac{58\!\cdots\!17}{12\!\cdots\!82}a-\frac{14\!\cdots\!97}{19\!\cdots\!30}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{3}$, which has order $3$ (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:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $\frac{33\!\cdots\!07}{83\!\cdots\!20}a^{17}-\frac{11\!\cdots\!07}{23\!\cdots\!20}a^{16}+\frac{68\!\cdots\!27}{23\!\cdots\!67}a^{15}-\frac{43\!\cdots\!59}{23\!\cdots\!20}a^{14}+\frac{17\!\cdots\!09}{27\!\cdots\!40}a^{13}-\frac{58\!\cdots\!97}{23\!\cdots\!20}a^{12}+\frac{12\!\cdots\!31}{25\!\cdots\!64}a^{11}-\frac{19\!\cdots\!19}{16\!\cdots\!40}a^{10}+\frac{65\!\cdots\!93}{20\!\cdots\!30}a^{9}-\frac{17\!\cdots\!59}{41\!\cdots\!60}a^{8}-\frac{31\!\cdots\!39}{41\!\cdots\!60}a^{7}+\frac{10\!\cdots\!51}{41\!\cdots\!60}a^{6}-\frac{11\!\cdots\!43}{69\!\cdots\!10}a^{5}-\frac{36\!\cdots\!59}{13\!\cdots\!20}a^{4}-\frac{27\!\cdots\!11}{20\!\cdots\!30}a^{3}-\frac{65\!\cdots\!58}{10\!\cdots\!15}a^{2}-\frac{94\!\cdots\!12}{10\!\cdots\!85}a+\frac{98\!\cdots\!42}{95\!\cdots\!65}$, $\frac{34\!\cdots\!97}{46\!\cdots\!40}a^{17}-\frac{41\!\cdots\!17}{16\!\cdots\!40}a^{16}+\frac{35\!\cdots\!43}{83\!\cdots\!20}a^{15}-\frac{80\!\cdots\!63}{55\!\cdots\!80}a^{14}+\frac{23\!\cdots\!27}{23\!\cdots\!70}a^{13}-\frac{96\!\cdots\!63}{33\!\cdots\!48}a^{12}+\frac{10\!\cdots\!79}{83\!\cdots\!20}a^{11}-\frac{37\!\cdots\!55}{15\!\cdots\!88}a^{10}+\frac{47\!\cdots\!01}{69\!\cdots\!10}a^{9}-\frac{90\!\cdots\!27}{14\!\cdots\!45}a^{8}+\frac{57\!\cdots\!11}{41\!\cdots\!60}a^{7}+\frac{64\!\cdots\!43}{69\!\cdots\!10}a^{6}+\frac{22\!\cdots\!97}{20\!\cdots\!30}a^{5}+\frac{23\!\cdots\!17}{59\!\cdots\!80}a^{4}+\frac{20\!\cdots\!41}{29\!\cdots\!90}a^{3}-\frac{89\!\cdots\!09}{34\!\cdots\!05}a^{2}+\frac{14\!\cdots\!86}{95\!\cdots\!65}a-\frac{54\!\cdots\!46}{95\!\cdots\!65}$, $\frac{95\!\cdots\!31}{33\!\cdots\!80}a^{17}-\frac{25\!\cdots\!61}{92\!\cdots\!80}a^{16}+\frac{69\!\cdots\!93}{33\!\cdots\!80}a^{15}-\frac{47\!\cdots\!91}{41\!\cdots\!60}a^{14}+\frac{55\!\cdots\!21}{11\!\cdots\!60}a^{13}-\frac{32\!\cdots\!51}{18\!\cdots\!60}a^{12}+\frac{14\!\cdots\!61}{30\!\cdots\!80}a^{11}-\frac{91\!\cdots\!21}{83\!\cdots\!20}a^{10}+\frac{30\!\cdots\!43}{16\!\cdots\!40}a^{9}-\frac{37\!\cdots\!87}{16\!\cdots\!40}a^{8}+\frac{76\!\cdots\!63}{10\!\cdots\!15}a^{7}+\frac{64\!\cdots\!27}{41\!\cdots\!60}a^{6}-\frac{38\!\cdots\!19}{83\!\cdots\!20}a^{5}-\frac{25\!\cdots\!09}{83\!\cdots\!12}a^{4}+\frac{19\!\cdots\!17}{13\!\cdots\!20}a^{3}-\frac{14\!\cdots\!01}{41\!\cdots\!60}a^{2}+\frac{23\!\cdots\!33}{95\!\cdots\!65}a-\frac{48\!\cdots\!02}{45\!\cdots\!65}$, $\frac{43\!\cdots\!53}{11\!\cdots\!60}a^{17}-\frac{51\!\cdots\!87}{16\!\cdots\!40}a^{16}+\frac{74\!\cdots\!91}{33\!\cdots\!80}a^{15}-\frac{18\!\cdots\!13}{16\!\cdots\!40}a^{14}+\frac{14\!\cdots\!57}{33\!\cdots\!80}a^{13}-\frac{55\!\cdots\!33}{38\!\cdots\!60}a^{12}+\frac{17\!\cdots\!27}{47\!\cdots\!40}a^{11}-\frac{13\!\cdots\!97}{16\!\cdots\!40}a^{10}+\frac{48\!\cdots\!99}{38\!\cdots\!60}a^{9}-\frac{10\!\cdots\!11}{55\!\cdots\!80}a^{8}+\frac{22\!\cdots\!55}{20\!\cdots\!03}a^{7}-\frac{16\!\cdots\!11}{10\!\cdots\!15}a^{6}-\frac{47\!\cdots\!11}{92\!\cdots\!80}a^{5}-\frac{52\!\cdots\!03}{83\!\cdots\!12}a^{4}+\frac{33\!\cdots\!51}{69\!\cdots\!10}a^{3}-\frac{25\!\cdots\!41}{13\!\cdots\!20}a^{2}+\frac{51\!\cdots\!15}{38\!\cdots\!46}a-\frac{45\!\cdots\!71}{63\!\cdots\!91}$, $\frac{44\!\cdots\!01}{79\!\cdots\!40}a^{17}-\frac{32\!\cdots\!93}{16\!\cdots\!24}a^{16}+\frac{38\!\cdots\!03}{18\!\cdots\!60}a^{15}-\frac{23\!\cdots\!33}{41\!\cdots\!60}a^{14}+\frac{48\!\cdots\!27}{16\!\cdots\!40}a^{13}-\frac{16\!\cdots\!21}{34\!\cdots\!05}a^{12}+\frac{94\!\cdots\!69}{55\!\cdots\!80}a^{11}-\frac{57\!\cdots\!19}{83\!\cdots\!20}a^{10}+\frac{17\!\cdots\!19}{41\!\cdots\!60}a^{9}+\frac{47\!\cdots\!47}{83\!\cdots\!20}a^{8}+\frac{11\!\cdots\!39}{16\!\cdots\!24}a^{7}+\frac{17\!\cdots\!51}{83\!\cdots\!20}a^{6}+\frac{12\!\cdots\!61}{41\!\cdots\!60}a^{5}+\frac{91\!\cdots\!41}{41\!\cdots\!60}a^{4}+\frac{59\!\cdots\!91}{10\!\cdots\!15}a^{3}-\frac{47\!\cdots\!61}{11\!\cdots\!35}a^{2}-\frac{98\!\cdots\!54}{95\!\cdots\!65}a+\frac{47\!\cdots\!99}{38\!\cdots\!46}$, $\frac{24\!\cdots\!89}{53\!\cdots\!60}a^{17}-\frac{25\!\cdots\!61}{69\!\cdots\!10}a^{16}+\frac{17\!\cdots\!27}{66\!\cdots\!96}a^{15}-\frac{55\!\cdots\!03}{41\!\cdots\!60}a^{14}+\frac{17\!\cdots\!01}{33\!\cdots\!80}a^{13}-\frac{29\!\cdots\!91}{16\!\cdots\!40}a^{12}+\frac{15\!\cdots\!69}{33\!\cdots\!80}a^{11}-\frac{21\!\cdots\!41}{20\!\cdots\!30}a^{10}+\frac{13\!\cdots\!01}{83\!\cdots\!20}a^{9}-\frac{34\!\cdots\!37}{16\!\cdots\!40}a^{8}+\frac{15\!\cdots\!67}{11\!\cdots\!35}a^{7}+\frac{52\!\cdots\!41}{83\!\cdots\!20}a^{6}-\frac{16\!\cdots\!23}{83\!\cdots\!20}a^{5}+\frac{16\!\cdots\!61}{13\!\cdots\!20}a^{4}+\frac{41\!\cdots\!17}{41\!\cdots\!06}a^{3}-\frac{19\!\cdots\!03}{13\!\cdots\!20}a^{2}+\frac{10\!\cdots\!16}{31\!\cdots\!55}a-\frac{22\!\cdots\!49}{19\!\cdots\!30}$, $\frac{68\!\cdots\!37}{47\!\cdots\!40}a^{17}-\frac{15\!\cdots\!87}{16\!\cdots\!40}a^{16}+\frac{20\!\cdots\!43}{33\!\cdots\!80}a^{15}-\frac{42\!\cdots\!91}{16\!\cdots\!40}a^{14}+\frac{54\!\cdots\!89}{66\!\cdots\!96}a^{13}-\frac{14\!\cdots\!07}{83\!\cdots\!20}a^{12}+\frac{83\!\cdots\!27}{60\!\cdots\!36}a^{11}+\frac{89\!\cdots\!41}{16\!\cdots\!40}a^{10}-\frac{84\!\cdots\!07}{27\!\cdots\!40}a^{9}+\frac{23\!\cdots\!49}{33\!\cdots\!48}a^{8}-\frac{10\!\cdots\!93}{83\!\cdots\!20}a^{7}+\frac{84\!\cdots\!81}{92\!\cdots\!68}a^{6}+\frac{30\!\cdots\!77}{16\!\cdots\!24}a^{5}-\frac{16\!\cdots\!65}{92\!\cdots\!68}a^{4}+\frac{15\!\cdots\!91}{69\!\cdots\!10}a^{3}+\frac{26\!\cdots\!83}{83\!\cdots\!12}a^{2}-\frac{23\!\cdots\!54}{10\!\cdots\!85}a+\frac{15\!\cdots\!23}{10\!\cdots\!85}$, $\frac{12\!\cdots\!11}{13\!\cdots\!20}a^{17}-\frac{55\!\cdots\!49}{12\!\cdots\!80}a^{16}+\frac{28\!\cdots\!16}{78\!\cdots\!55}a^{15}-\frac{50\!\cdots\!39}{41\!\cdots\!60}a^{14}+\frac{10\!\cdots\!85}{20\!\cdots\!88}a^{13}-\frac{98\!\cdots\!33}{11\!\cdots\!80}a^{12}+\frac{29\!\cdots\!89}{15\!\cdots\!10}a^{11}+\frac{96\!\cdots\!01}{13\!\cdots\!20}a^{10}-\frac{75\!\cdots\!63}{38\!\cdots\!60}a^{9}+\frac{50\!\cdots\!39}{31\!\cdots\!20}a^{8}-\frac{12\!\cdots\!03}{31\!\cdots\!20}a^{7}+\frac{70\!\cdots\!53}{34\!\cdots\!98}a^{6}+\frac{74\!\cdots\!33}{15\!\cdots\!10}a^{5}+\frac{66\!\cdots\!82}{78\!\cdots\!55}a^{4}+\frac{50\!\cdots\!69}{62\!\cdots\!64}a^{3}+\frac{53\!\cdots\!51}{52\!\cdots\!70}a^{2}-\frac{12\!\cdots\!03}{14\!\cdots\!10}a+\frac{46\!\cdots\!38}{71\!\cdots\!05}$ Copy content Toggle raw display (assuming GRH)
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:  \( 3932278.1508 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 3932278.1508 \cdot 3}{2\cdot\sqrt{5627055140868788987624861696}}\cr\approx \mathstrut & 1.2000897294 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 53*x^16 - 224*x^15 + 1059*x^14 - 3090*x^13 + 9543*x^12 - 18332*x^11 + 38972*x^10 - 39464*x^9 + 60140*x^8 + 22376*x^7 + 29584*x^6 + 113584*x^5 + 243424*x^4 - 171840*x^3 + 703168*x^2 - 494560*x + 271568)
 
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 - 6*x^17 + 53*x^16 - 224*x^15 + 1059*x^14 - 3090*x^13 + 9543*x^12 - 18332*x^11 + 38972*x^10 - 39464*x^9 + 60140*x^8 + 22376*x^7 + 29584*x^6 + 113584*x^5 + 243424*x^4 - 171840*x^3 + 703168*x^2 - 494560*x + 271568, 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 - 6*x^17 + 53*x^16 - 224*x^15 + 1059*x^14 - 3090*x^13 + 9543*x^12 - 18332*x^11 + 38972*x^10 - 39464*x^9 + 60140*x^8 + 22376*x^7 + 29584*x^6 + 113584*x^5 + 243424*x^4 - 171840*x^3 + 703168*x^2 - 494560*x + 271568);
 
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 - 6*x^17 + 53*x^16 - 224*x^15 + 1059*x^14 - 3090*x^13 + 9543*x^12 - 18332*x^11 + 38972*x^10 - 39464*x^9 + 60140*x^8 + 22376*x^7 + 29584*x^6 + 113584*x^5 + 243424*x^4 - 171840*x^3 + 703168*x^2 - 494560*x + 271568);
 
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

$C_3:S_3$ (as 18T4):

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 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-11}) \), 3.1.12716.2 x3, 3.1.44.1 x3, 3.1.12716.1 x3, 3.1.3179.1 x3, 6.0.1778663216.2, 6.0.21296.1, 6.0.1778663216.1, 6.0.111166451.1, 9.1.22617481454656.1 x9

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 9 sibling: 9.1.22617481454656.1
Minimal sibling: 9.1.22617481454656.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.3.0.1}{3} }^{6}$ ${\href{/padicField/5.3.0.1}{3} }^{6}$ ${\href{/padicField/7.2.0.1}{2} }^{9}$ R ${\href{/padicField/13.2.0.1}{2} }^{9}$ R ${\href{/padicField/19.2.0.1}{2} }^{9}$ ${\href{/padicField/23.3.0.1}{3} }^{6}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.2.0.1}{2} }^{9}$ ${\href{/padicField/47.3.0.1}{3} }^{6}$ ${\href{/padicField/53.1.0.1}{1} }^{18}$ ${\href{/padicField/59.3.0.1}{3} }^{6}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
\(11\) Copy content Toggle raw display 11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
\(17\) Copy content Toggle raw display 17.6.4.1$x^{6} + 48 x^{5} + 879 x^{4} + 7682 x^{3} + 44916 x^{2} + 265428 x + 127425$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 48 x^{5} + 879 x^{4} + 7682 x^{3} + 44916 x^{2} + 265428 x + 127425$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 48 x^{5} + 879 x^{4} + 7682 x^{3} + 44916 x^{2} + 265428 x + 127425$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$