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Properties

Label 18.0.396...664.1
Degree $18$
Signature $[0, 9]$
Discriminant $-3.966\times 10^{46}$
Root discriminant \(387.97\)
Ramified primes $2,3,7,17$
Class number $9720$ (GRH)
Class group [3, 6, 6, 90] (GRH)
Galois group $C_3^2:D_6$ (as 18T52)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 186*x^16 - 796*x^15 + 20886*x^14 - 43086*x^13 + 1499866*x^12 - 4087764*x^11 + 87804711*x^10 - 169457551*x^9 + 3935286084*x^8 - 1528031184*x^7 + 87470587978*x^6 + 226709260608*x^5 + 1850764572768*x^4 + 12656412331040*x^3 + 48056367294720*x^2 + 195986267080704*x + 353454875222016)
 
gp: K = bnfinit(y^18 - 3*y^17 + 186*y^16 - 796*y^15 + 20886*y^14 - 43086*y^13 + 1499866*y^12 - 4087764*y^11 + 87804711*y^10 - 169457551*y^9 + 3935286084*y^8 - 1528031184*y^7 + 87470587978*y^6 + 226709260608*y^5 + 1850764572768*y^4 + 12656412331040*y^3 + 48056367294720*y^2 + 195986267080704*y + 353454875222016, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 + 186*x^16 - 796*x^15 + 20886*x^14 - 43086*x^13 + 1499866*x^12 - 4087764*x^11 + 87804711*x^10 - 169457551*x^9 + 3935286084*x^8 - 1528031184*x^7 + 87470587978*x^6 + 226709260608*x^5 + 1850764572768*x^4 + 12656412331040*x^3 + 48056367294720*x^2 + 195986267080704*x + 353454875222016);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 3*x^17 + 186*x^16 - 796*x^15 + 20886*x^14 - 43086*x^13 + 1499866*x^12 - 4087764*x^11 + 87804711*x^10 - 169457551*x^9 + 3935286084*x^8 - 1528031184*x^7 + 87470587978*x^6 + 226709260608*x^5 + 1850764572768*x^4 + 12656412331040*x^3 + 48056367294720*x^2 + 195986267080704*x + 353454875222016)
 

\( x^{18} - 3 x^{17} + 186 x^{16} - 796 x^{15} + 20886 x^{14} - 43086 x^{13} + 1499866 x^{12} + \cdots + 353454875222016 \) 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:   \(-39655567350356468482252381576612622224820465664\) \(\medspace = -\,2^{12}\cdot 3^{30}\cdot 7^{15}\cdot 17^{13}\) 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:  \(387.97\)
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:  $2^{2/3}3^{11/6}7^{5/6}17^{5/6}\approx 638.3096819753366$
Ramified primes:   \(2\), \(3\), \(7\), \(17\) 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:  \(\Q(\sqrt{-119}) \)
$\card{ \Aut(K/\Q) }$:  $6$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{14}a^{6}-\frac{1}{2}a^{5}+\frac{1}{14}a^{4}-\frac{1}{2}a^{3}-\frac{1}{7}a^{2}+\frac{3}{7}$, $\frac{1}{14}a^{7}-\frac{3}{7}a^{5}+\frac{5}{14}a^{3}+\frac{3}{7}a$, $\frac{1}{28}a^{8}-\frac{1}{2}a^{5}+\frac{11}{28}a^{4}-\frac{1}{2}a^{3}-\frac{3}{14}a^{2}-\frac{1}{2}a+\frac{2}{7}$, $\frac{1}{84}a^{9}-\frac{1}{28}a^{5}+\frac{2}{21}a^{3}-\frac{1}{2}a^{2}+\frac{3}{7}a$, $\frac{1}{168}a^{10}-\frac{1}{168}a^{9}-\frac{1}{28}a^{7}-\frac{1}{56}a^{6}-\frac{15}{56}a^{5}-\frac{19}{42}a^{4}+\frac{1}{42}a^{3}+\frac{13}{28}a^{2}-\frac{3}{7}a$, $\frac{1}{168}a^{11}-\frac{1}{168}a^{9}+\frac{1}{56}a^{7}+\frac{59}{168}a^{5}+\frac{1}{4}a^{4}+\frac{29}{84}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{79968}a^{12}-\frac{1}{952}a^{11}+\frac{73}{26656}a^{10}+\frac{19}{5712}a^{9}+\frac{349}{26656}a^{8}+\frac{5}{952}a^{7}-\frac{2701}{79968}a^{6}-\frac{12}{119}a^{5}+\frac{3491}{13328}a^{4}+\frac{1025}{5712}a^{3}-\frac{205}{833}a^{2}-\frac{396}{833}$, $\frac{1}{239904}a^{13}-\frac{173}{239904}a^{11}-\frac{3}{1904}a^{10}-\frac{1361}{239904}a^{9}-\frac{3}{238}a^{8}+\frac{6875}{239904}a^{7}-\frac{3}{952}a^{6}+\frac{12601}{119952}a^{5}-\frac{65}{336}a^{4}+\frac{4939}{29988}a^{3}-\frac{97}{357}a^{2}-\frac{13}{833}a-\frac{94}{357}$, $\frac{1}{718752384}a^{14}+\frac{1315}{718752384}a^{13}+\frac{319}{718752384}a^{12}-\frac{1560113}{718752384}a^{11}+\frac{1060165}{718752384}a^{10}-\frac{1113515}{718752384}a^{9}+\frac{5147711}{718752384}a^{8}+\frac{12605165}{718752384}a^{7}-\frac{11687555}{359376192}a^{6}+\frac{25511531}{179688096}a^{5}+\frac{111654505}{359376192}a^{4}-\frac{12526075}{44922024}a^{3}-\frac{561457}{7487004}a^{2}-\frac{1957609}{7487004}a-\frac{630772}{1871751}$, $\frac{1}{2156257152}a^{15}-\frac{5}{3358656}a^{13}+\frac{125}{39930688}a^{12}+\frac{176815}{89844048}a^{11}+\frac{86287}{39930688}a^{10}-\frac{8017}{269532144}a^{9}-\frac{354237}{19965344}a^{8}-\frac{25317151}{718752384}a^{7}+\frac{2558267}{359376192}a^{6}+\frac{55764605}{359376192}a^{5}+\frac{44803249}{119792064}a^{4}-\frac{16612885}{269532144}a^{3}+\frac{242174}{1871751}a^{2}-\frac{531701}{2495668}a+\frac{1863622}{5615253}$, $\frac{1}{12920292854784}a^{16}+\frac{359}{2153382142464}a^{15}+\frac{31}{51271003392}a^{14}+\frac{1028771}{538345535616}a^{13}-\frac{8552107}{2153382142464}a^{12}+\frac{1069373}{538345535616}a^{11}-\frac{34472959}{6460146427392}a^{10}-\frac{8901582287}{2153382142464}a^{9}+\frac{916228703}{478529364992}a^{8}+\frac{8447110537}{717794047488}a^{7}+\frac{68500044049}{2153382142464}a^{6}+\frac{3772764983}{20125066752}a^{5}-\frac{402230567063}{807518303424}a^{4}+\frac{2206781629}{5607765996}a^{3}+\frac{3040984181}{6408875424}a^{2}-\frac{3316078205}{8411648994}a+\frac{195336524}{467313833}$, $\frac{1}{12\!\cdots\!48}a^{17}+\frac{16\!\cdots\!15}{13\!\cdots\!72}a^{16}+\frac{33\!\cdots\!33}{20\!\cdots\!08}a^{15}+\frac{68\!\cdots\!69}{11\!\cdots\!56}a^{14}-\frac{13\!\cdots\!75}{20\!\cdots\!08}a^{13}+\frac{98\!\cdots\!83}{69\!\cdots\!36}a^{12}+\frac{13\!\cdots\!25}{62\!\cdots\!24}a^{11}-\frac{99\!\cdots\!95}{85\!\cdots\!96}a^{10}-\frac{22\!\cdots\!07}{41\!\cdots\!16}a^{9}+\frac{20\!\cdots\!29}{41\!\cdots\!16}a^{8}+\frac{17\!\cdots\!33}{65\!\cdots\!44}a^{7}-\frac{78\!\cdots\!57}{34\!\cdots\!68}a^{6}+\frac{12\!\cdots\!95}{62\!\cdots\!24}a^{5}-\frac{24\!\cdots\!05}{51\!\cdots\!76}a^{4}+\frac{43\!\cdots\!91}{13\!\cdots\!88}a^{3}+\frac{25\!\cdots\!41}{13\!\cdots\!88}a^{2}-\frac{11\!\cdots\!33}{90\!\cdots\!02}a-\frac{21\!\cdots\!62}{13\!\cdots\!03}$ 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:  $2$, $3$

Class group and class number

$C_{3}\times C_{6}\times C_{6}\times C_{90}$, which has order $9720$ (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:   $\frac{32\!\cdots\!69}{89\!\cdots\!96}a^{17}-\frac{52\!\cdots\!27}{30\!\cdots\!56}a^{16}+\frac{10\!\cdots\!17}{15\!\cdots\!28}a^{15}-\frac{25\!\cdots\!79}{62\!\cdots\!72}a^{14}+\frac{60\!\cdots\!99}{75\!\cdots\!64}a^{13}-\frac{14\!\cdots\!97}{50\!\cdots\!76}a^{12}+\frac{42\!\cdots\!59}{75\!\cdots\!64}a^{11}-\frac{36\!\cdots\!89}{15\!\cdots\!28}a^{10}+\frac{17\!\cdots\!51}{50\!\cdots\!76}a^{9}-\frac{11\!\cdots\!57}{10\!\cdots\!52}a^{8}+\frac{23\!\cdots\!03}{15\!\cdots\!28}a^{7}-\frac{15\!\cdots\!19}{50\!\cdots\!76}a^{6}+\frac{51\!\cdots\!07}{15\!\cdots\!28}a^{5}+\frac{36\!\cdots\!91}{37\!\cdots\!32}a^{4}+\frac{15\!\cdots\!03}{23\!\cdots\!52}a^{3}+\frac{78\!\cdots\!17}{31\!\cdots\!36}a^{2}+\frac{94\!\cdots\!75}{78\!\cdots\!84}a+\frac{67\!\cdots\!95}{19\!\cdots\!21}$, $\frac{17\!\cdots\!75}{13\!\cdots\!88}a^{17}-\frac{62\!\cdots\!75}{62\!\cdots\!24}a^{16}+\frac{97\!\cdots\!89}{31\!\cdots\!12}a^{15}-\frac{26\!\cdots\!45}{13\!\cdots\!88}a^{14}+\frac{18\!\cdots\!21}{52\!\cdots\!52}a^{13}-\frac{12\!\cdots\!69}{10\!\cdots\!04}a^{12}+\frac{86\!\cdots\!65}{52\!\cdots\!52}a^{11}-\frac{82\!\cdots\!37}{31\!\cdots\!12}a^{10}+\frac{12\!\cdots\!73}{31\!\cdots\!12}a^{9}+\frac{52\!\cdots\!23}{20\!\cdots\!08}a^{8}+\frac{83\!\cdots\!89}{11\!\cdots\!56}a^{7}+\frac{11\!\cdots\!45}{11\!\cdots\!56}a^{6}+\frac{41\!\cdots\!25}{10\!\cdots\!04}a^{5}+\frac{23\!\cdots\!15}{78\!\cdots\!28}a^{4}+\frac{20\!\cdots\!83}{12\!\cdots\!27}a^{3}+\frac{50\!\cdots\!17}{72\!\cdots\!16}a^{2}+\frac{38\!\cdots\!95}{16\!\cdots\!36}a+\frac{19\!\cdots\!87}{40\!\cdots\!09}$, $\frac{55\!\cdots\!15}{39\!\cdots\!64}a^{17}+\frac{11\!\cdots\!95}{12\!\cdots\!24}a^{16}-\frac{12\!\cdots\!59}{31\!\cdots\!12}a^{15}+\frac{29\!\cdots\!43}{21\!\cdots\!48}a^{14}-\frac{21\!\cdots\!27}{52\!\cdots\!52}a^{13}+\frac{51\!\cdots\!57}{34\!\cdots\!68}a^{12}-\frac{59\!\cdots\!57}{15\!\cdots\!56}a^{11}+\frac{95\!\cdots\!97}{10\!\cdots\!04}a^{10}-\frac{71\!\cdots\!91}{31\!\cdots\!12}a^{9}+\frac{11\!\cdots\!63}{20\!\cdots\!08}a^{8}-\frac{25\!\cdots\!47}{10\!\cdots\!04}a^{7}+\frac{22\!\cdots\!29}{10\!\cdots\!04}a^{6}+\frac{30\!\cdots\!99}{31\!\cdots\!12}a^{5}+\frac{60\!\cdots\!57}{26\!\cdots\!76}a^{4}+\frac{21\!\cdots\!61}{98\!\cdots\!16}a^{3}+\frac{84\!\cdots\!61}{65\!\cdots\!44}a^{2}+\frac{50\!\cdots\!47}{77\!\cdots\!16}a+\frac{26\!\cdots\!27}{24\!\cdots\!77}$, $\frac{92\!\cdots\!15}{39\!\cdots\!64}a^{17}+\frac{68\!\cdots\!61}{20\!\cdots\!08}a^{16}+\frac{18\!\cdots\!95}{31\!\cdots\!12}a^{15}+\frac{30\!\cdots\!87}{58\!\cdots\!28}a^{14}+\frac{74\!\cdots\!25}{13\!\cdots\!88}a^{13}+\frac{59\!\cdots\!79}{11\!\cdots\!56}a^{12}+\frac{60\!\cdots\!20}{12\!\cdots\!27}a^{11}+\frac{37\!\cdots\!45}{10\!\cdots\!04}a^{10}+\frac{71\!\cdots\!93}{31\!\cdots\!12}a^{9}+\frac{28\!\cdots\!53}{20\!\cdots\!08}a^{8}+\frac{80\!\cdots\!09}{10\!\cdots\!04}a^{7}+\frac{64\!\cdots\!87}{10\!\cdots\!04}a^{6}+\frac{12\!\cdots\!13}{31\!\cdots\!12}a^{5}+\frac{15\!\cdots\!53}{65\!\cdots\!44}a^{4}+\frac{13\!\cdots\!83}{12\!\cdots\!27}a^{3}+\frac{14\!\cdots\!71}{38\!\cdots\!32}a^{2}+\frac{24\!\cdots\!83}{27\!\cdots\!06}a+\frac{39\!\cdots\!75}{40\!\cdots\!09}$, $\frac{14\!\cdots\!89}{39\!\cdots\!64}a^{17}+\frac{24\!\cdots\!73}{62\!\cdots\!24}a^{16}+\frac{11\!\cdots\!85}{31\!\cdots\!12}a^{15}+\frac{17\!\cdots\!21}{26\!\cdots\!76}a^{14}+\frac{28\!\cdots\!27}{52\!\cdots\!52}a^{13}+\frac{10\!\cdots\!75}{10\!\cdots\!04}a^{12}+\frac{14\!\cdots\!77}{15\!\cdots\!56}a^{11}+\frac{19\!\cdots\!59}{31\!\cdots\!12}a^{10}-\frac{29\!\cdots\!71}{31\!\cdots\!12}a^{9}+\frac{84\!\cdots\!47}{20\!\cdots\!08}a^{8}-\frac{56\!\cdots\!43}{10\!\cdots\!04}a^{7}+\frac{20\!\cdots\!73}{10\!\cdots\!04}a^{6}-\frac{75\!\cdots\!09}{31\!\cdots\!12}a^{5}+\frac{29\!\cdots\!13}{78\!\cdots\!28}a^{4}+\frac{18\!\cdots\!27}{12\!\cdots\!27}a^{3}+\frac{35\!\cdots\!53}{65\!\cdots\!44}a^{2}+\frac{95\!\cdots\!61}{16\!\cdots\!36}a+\frac{52\!\cdots\!43}{40\!\cdots\!09}$, $\frac{85\!\cdots\!05}{69\!\cdots\!36}a^{17}+\frac{25\!\cdots\!25}{26\!\cdots\!76}a^{16}-\frac{19\!\cdots\!84}{40\!\cdots\!09}a^{15}+\frac{10\!\cdots\!89}{58\!\cdots\!28}a^{14}-\frac{49\!\cdots\!31}{34\!\cdots\!68}a^{13}+\frac{13\!\cdots\!11}{43\!\cdots\!96}a^{12}-\frac{42\!\cdots\!65}{34\!\cdots\!68}a^{11}+\frac{12\!\cdots\!77}{10\!\cdots\!04}a^{10}-\frac{13\!\cdots\!49}{20\!\cdots\!08}a^{9}+\frac{90\!\cdots\!61}{58\!\cdots\!28}a^{8}-\frac{10\!\cdots\!41}{34\!\cdots\!68}a^{7}+\frac{17\!\cdots\!89}{34\!\cdots\!68}a^{6}+\frac{41\!\cdots\!07}{58\!\cdots\!28}a^{5}+\frac{35\!\cdots\!65}{26\!\cdots\!76}a^{4}+\frac{37\!\cdots\!37}{38\!\cdots\!32}a^{3}+\frac{15\!\cdots\!47}{36\!\cdots\!08}a^{2}+\frac{10\!\cdots\!67}{54\!\cdots\!12}a+\frac{42\!\cdots\!77}{13\!\cdots\!03}$, $\frac{27\!\cdots\!61}{25\!\cdots\!44}a^{17}+\frac{70\!\cdots\!17}{30\!\cdots\!28}a^{16}+\frac{67\!\cdots\!89}{17\!\cdots\!96}a^{15}+\frac{58\!\cdots\!73}{25\!\cdots\!44}a^{14}+\frac{43\!\cdots\!03}{18\!\cdots\!96}a^{13}+\frac{63\!\cdots\!51}{47\!\cdots\!28}a^{12}+\frac{11\!\cdots\!83}{60\!\cdots\!32}a^{11}+\frac{21\!\cdots\!83}{21\!\cdots\!52}a^{10}+\frac{70\!\cdots\!33}{72\!\cdots\!84}a^{9}+\frac{45\!\cdots\!73}{14\!\cdots\!68}a^{8}+\frac{30\!\cdots\!51}{10\!\cdots\!12}a^{7}+\frac{22\!\cdots\!57}{14\!\cdots\!84}a^{6}+\frac{82\!\cdots\!47}{72\!\cdots\!84}a^{5}+\frac{18\!\cdots\!77}{27\!\cdots\!44}a^{4}+\frac{27\!\cdots\!67}{99\!\cdots\!49}a^{3}+\frac{97\!\cdots\!31}{10\!\cdots\!56}a^{2}+\frac{17\!\cdots\!20}{99\!\cdots\!49}a+\frac{35\!\cdots\!21}{33\!\cdots\!83}$, $\frac{12\!\cdots\!41}{78\!\cdots\!28}a^{17}-\frac{12\!\cdots\!33}{20\!\cdots\!08}a^{16}+\frac{33\!\cdots\!07}{10\!\cdots\!04}a^{15}-\frac{66\!\cdots\!91}{65\!\cdots\!44}a^{14}+\frac{26\!\cdots\!89}{52\!\cdots\!52}a^{13}-\frac{10\!\cdots\!61}{10\!\cdots\!04}a^{12}+\frac{37\!\cdots\!31}{15\!\cdots\!56}a^{11}-\frac{57\!\cdots\!91}{10\!\cdots\!04}a^{10}+\frac{27\!\cdots\!07}{11\!\cdots\!56}a^{9}-\frac{89\!\cdots\!05}{23\!\cdots\!12}a^{8}+\frac{74\!\cdots\!81}{10\!\cdots\!04}a^{7}-\frac{99\!\cdots\!59}{10\!\cdots\!04}a^{6}-\frac{13\!\cdots\!81}{18\!\cdots\!36}a^{5}-\frac{11\!\cdots\!21}{87\!\cdots\!92}a^{4}-\frac{35\!\cdots\!21}{32\!\cdots\!72}a^{3}-\frac{30\!\cdots\!39}{65\!\cdots\!44}a^{2}-\frac{44\!\cdots\!83}{18\!\cdots\!04}a-\frac{65\!\cdots\!83}{13\!\cdots\!03}$ 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:  \( 396536534938351.06 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 396536534938351.06 \cdot 9720}{2\cdot\sqrt{39655567350356468482252381576612622224820465664}}\cr\approx \mathstrut & 147.701942820269 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 186*x^16 - 796*x^15 + 20886*x^14 - 43086*x^13 + 1499866*x^12 - 4087764*x^11 + 87804711*x^10 - 169457551*x^9 + 3935286084*x^8 - 1528031184*x^7 + 87470587978*x^6 + 226709260608*x^5 + 1850764572768*x^4 + 12656412331040*x^3 + 48056367294720*x^2 + 195986267080704*x + 353454875222016)
 
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 - 3*x^17 + 186*x^16 - 796*x^15 + 20886*x^14 - 43086*x^13 + 1499866*x^12 - 4087764*x^11 + 87804711*x^10 - 169457551*x^9 + 3935286084*x^8 - 1528031184*x^7 + 87470587978*x^6 + 226709260608*x^5 + 1850764572768*x^4 + 12656412331040*x^3 + 48056367294720*x^2 + 195986267080704*x + 353454875222016, 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 - 3*x^17 + 186*x^16 - 796*x^15 + 20886*x^14 - 43086*x^13 + 1499866*x^12 - 4087764*x^11 + 87804711*x^10 - 169457551*x^9 + 3935286084*x^8 - 1528031184*x^7 + 87470587978*x^6 + 226709260608*x^5 + 1850764572768*x^4 + 12656412331040*x^3 + 48056367294720*x^2 + 195986267080704*x + 353454875222016);
 
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 - 3*x^17 + 186*x^16 - 796*x^15 + 20886*x^14 - 43086*x^13 + 1499866*x^12 - 4087764*x^11 + 87804711*x^10 - 169457551*x^9 + 3935286084*x^8 - 1528031184*x^7 + 87470587978*x^6 + 226709260608*x^5 + 1850764572768*x^4 + 12656412331040*x^3 + 48056367294720*x^2 + 195986267080704*x + 353454875222016);
 
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

$C_3^2:D_6$ (as 18T52):

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 108
The 20 conjugacy class representatives for $C_3^2:D_6$
Character table for $C_3^2:D_6$

Intermediate fields

\(\Q(\sqrt{-119}) \), 3.1.972.2, 6.0.1592111260656.10, 9.3.9023659939580352192.1

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 18 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 18.0.109172181872552006090639177294248664675479296.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ R ${\href{/padicField/11.6.0.1}{6} }^{3}$ ${\href{/padicField/13.6.0.1}{6} }^{3}$ R ${\href{/padicField/19.6.0.1}{6} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }^{3}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ ${\href{/padicField/43.3.0.1}{3} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{3}$

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 2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
\(3\) Copy content Toggle raw display 3.3.5.2$x^{3} + 18 x + 3$$3$$1$$5$$S_3$$[5/2]_{2}$
3.3.5.2$x^{3} + 18 x + 3$$3$$1$$5$$S_3$$[5/2]_{2}$
3.3.5.2$x^{3} + 18 x + 3$$3$$1$$5$$S_3$$[5/2]_{2}$
3.3.5.2$x^{3} + 18 x + 3$$3$$1$$5$$S_3$$[5/2]_{2}$
3.3.5.2$x^{3} + 18 x + 3$$3$$1$$5$$S_3$$[5/2]_{2}$
3.3.5.2$x^{3} + 18 x + 3$$3$$1$$5$$S_3$$[5/2]_{2}$
\(7\) Copy content Toggle raw display 7.6.5.2$x^{6} + 42$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.3$x^{6} + 35$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.1$x^{6} + 21$$6$$1$$5$$C_6$$[\ ]_{6}$
\(17\) Copy content Toggle raw display 17.6.3.2$x^{6} + 289 x^{2} - 68782$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.12.10.2$x^{12} - 3060 x^{6} - 197676$$6$$2$$10$$C_6\times S_3$$[\ ]_{6}^{6}$