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Properties

Label 18.0.312...096.1
Degree $18$
Signature $[0, 9]$
Discriminant $-3.122\times 10^{45}$
Root discriminant \(336.87\)
Ramified primes $2,3,7,17$
Class number $54$ (GRH)
Class group [3, 3, 6] (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 + 33*x^16 - 388*x^15 + 3138*x^14 + 13218*x^13 + 53812*x^12 - 1464732*x^11 + 5349033*x^10 - 8306493*x^9 + 255408471*x^8 - 450926892*x^7 - 2241288864*x^6 - 4931594442*x^5 + 43187005062*x^4 + 351848838744*x^3 + 1835309384631*x^2 + 4396464030753*x + 7107138128187)
 
gp: K = bnfinit(y^18 - 3*y^17 + 33*y^16 - 388*y^15 + 3138*y^14 + 13218*y^13 + 53812*y^12 - 1464732*y^11 + 5349033*y^10 - 8306493*y^9 + 255408471*y^8 - 450926892*y^7 - 2241288864*y^6 - 4931594442*y^5 + 43187005062*y^4 + 351848838744*y^3 + 1835309384631*y^2 + 4396464030753*y + 7107138128187, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 + 33*x^16 - 388*x^15 + 3138*x^14 + 13218*x^13 + 53812*x^12 - 1464732*x^11 + 5349033*x^10 - 8306493*x^9 + 255408471*x^8 - 450926892*x^7 - 2241288864*x^6 - 4931594442*x^5 + 43187005062*x^4 + 351848838744*x^3 + 1835309384631*x^2 + 4396464030753*x + 7107138128187);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 3*x^17 + 33*x^16 - 388*x^15 + 3138*x^14 + 13218*x^13 + 53812*x^12 - 1464732*x^11 + 5349033*x^10 - 8306493*x^9 + 255408471*x^8 - 450926892*x^7 - 2241288864*x^6 - 4931594442*x^5 + 43187005062*x^4 + 351848838744*x^3 + 1835309384631*x^2 + 4396464030753*x + 7107138128187)
 

\( x^{18} - 3 x^{17} + 33 x^{16} - 388 x^{15} + 3138 x^{14} + 13218 x^{13} + 53812 x^{12} + \cdots + 7107138128187 \) 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:   \(-3121575272477039793063598549762509621195780096\) \(\medspace = -\,2^{12}\cdot 3^{33}\cdot 7^{12}\cdot 17^{13}\) 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:  \(336.87\)
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:  $2^{2/3}3^{11/6}7^{2/3}17^{5/6}\approx 461.510683112854$
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:  \(\Q(\sqrt{-51}) \)
$\card{ \Aut(K/\Q) }$:  $6$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6}a^{6}-\frac{1}{2}a^{5}+\frac{1}{6}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{6}a^{7}-\frac{1}{2}a^{5}+\frac{1}{6}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{12}a^{8}-\frac{1}{6}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}$, $\frac{1}{12}a^{9}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{6}a^{3}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{72}a^{10}-\frac{1}{72}a^{9}+\frac{1}{72}a^{8}-\frac{1}{36}a^{7}+\frac{5}{72}a^{6}+\frac{13}{72}a^{5}+\frac{1}{8}a^{4}+\frac{5}{12}a^{3}-\frac{1}{24}a^{2}+\frac{3}{8}a+\frac{1}{8}$, $\frac{1}{72}a^{11}-\frac{1}{72}a^{8}+\frac{1}{24}a^{7}-\frac{1}{12}a^{6}+\frac{11}{36}a^{5}-\frac{11}{24}a^{4}+\frac{1}{24}a^{3}+\frac{1}{3}a^{2}-\frac{1}{2}a+\frac{1}{8}$, $\frac{1}{34272}a^{12}-\frac{1}{408}a^{11}+\frac{13}{3808}a^{10}-\frac{83}{2448}a^{9}-\frac{3}{238}a^{8}-\frac{29}{408}a^{7}+\frac{2671}{34272}a^{6}-\frac{113}{408}a^{5}-\frac{109}{1428}a^{4}+\frac{95}{272}a^{3}-\frac{379}{3808}a^{2}+\frac{1}{8}a+\frac{1845}{3808}$, $\frac{1}{34272}a^{13}+\frac{67}{11424}a^{11}+\frac{7}{2448}a^{10}-\frac{13}{476}a^{9}-\frac{1}{204}a^{8}+\frac{2251}{34272}a^{7}-\frac{13}{204}a^{6}+\frac{9}{119}a^{5}-\frac{5}{48}a^{4}-\frac{3461}{11424}a^{3}+\frac{1}{68}a^{2}-\frac{59}{3808}a+\frac{5}{68}$, $\frac{1}{44005248}a^{14}+\frac{1}{95872}a^{13}-\frac{19}{3667104}a^{12}+\frac{206933}{44005248}a^{11}-\frac{65357}{14668416}a^{10}+\frac{264445}{7334208}a^{9}+\frac{232837}{6286464}a^{8}-\frac{825941}{14668416}a^{7}+\frac{15665}{287616}a^{6}+\frac{247341}{814912}a^{5}+\frac{2095531}{4889472}a^{4}-\frac{2156291}{4889472}a^{3}+\frac{34537}{611184}a^{2}+\frac{745685}{1629824}a-\frac{408539}{1629824}$, $\frac{1}{44005248}a^{15}-\frac{37}{4889472}a^{13}-\frac{61}{6286464}a^{12}+\frac{4603}{814912}a^{11}-\frac{1151}{698496}a^{10}-\frac{192617}{6286464}a^{9}+\frac{1171}{32742}a^{8}+\frac{2605}{47936}a^{7}-\frac{62069}{2095488}a^{6}+\frac{595075}{14668416}a^{5}-\frac{1611}{58208}a^{4}-\frac{126901}{1629824}a^{3}+\frac{314621}{698496}a^{2}+\frac{268479}{814912}a+\frac{88195}{232832}$, $\frac{1}{791038338048}a^{16}+\frac{359}{131839723008}a^{15}-\frac{359}{65919861504}a^{14}-\frac{4638407}{395519169024}a^{13}+\frac{17639}{10986643584}a^{12}-\frac{74869873}{21973287168}a^{11}-\frac{85132507}{14125684608}a^{10}+\frac{3945911}{308036736}a^{9}-\frac{333327979}{37668492288}a^{8}+\frac{4489841}{514998918}a^{7}+\frac{39488461}{1569520512}a^{6}-\frac{1602473231}{7324429056}a^{5}+\frac{9024731245}{21973287168}a^{4}-\frac{4536501575}{14648858112}a^{3}+\frac{1469301767}{3662214528}a^{2}-\frac{2229834549}{4882952704}a+\frac{3073324109}{9765905408}$, $\frac{1}{14\!\cdots\!64}a^{17}-\frac{31\!\cdots\!33}{71\!\cdots\!32}a^{16}-\frac{54\!\cdots\!25}{11\!\cdots\!72}a^{15}+\frac{74\!\cdots\!81}{71\!\cdots\!32}a^{14}-\frac{94\!\cdots\!23}{17\!\cdots\!08}a^{13}+\frac{12\!\cdots\!29}{11\!\cdots\!72}a^{12}+\frac{23\!\cdots\!93}{44\!\cdots\!52}a^{11}+\frac{12\!\cdots\!71}{44\!\cdots\!52}a^{10}-\frac{14\!\cdots\!13}{47\!\cdots\!88}a^{9}+\frac{33\!\cdots\!07}{11\!\cdots\!72}a^{8}+\frac{16\!\cdots\!09}{29\!\cdots\!68}a^{7}-\frac{49\!\cdots\!13}{13\!\cdots\!08}a^{6}+\frac{12\!\cdots\!01}{23\!\cdots\!72}a^{5}+\frac{36\!\cdots\!09}{11\!\cdots\!64}a^{4}+\frac{26\!\cdots\!21}{66\!\cdots\!04}a^{3}+\frac{47\!\cdots\!57}{26\!\cdots\!16}a^{2}+\frac{33\!\cdots\!05}{17\!\cdots\!44}a-\frac{35\!\cdots\!35}{44\!\cdots\!36}$ 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

$C_{3}\times C_{3}\times C_{6}$, which has order $54$ (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:   $\frac{13\!\cdots\!35}{10\!\cdots\!76}a^{17}+\frac{17\!\cdots\!95}{22\!\cdots\!28}a^{16}-\frac{36\!\cdots\!85}{11\!\cdots\!64}a^{15}+\frac{13\!\cdots\!19}{25\!\cdots\!44}a^{14}+\frac{23\!\cdots\!85}{33\!\cdots\!92}a^{13}+\frac{55\!\cdots\!13}{84\!\cdots\!48}a^{12}+\frac{75\!\cdots\!79}{50\!\cdots\!88}a^{11}-\frac{19\!\cdots\!15}{84\!\cdots\!48}a^{10}-\frac{15\!\cdots\!67}{33\!\cdots\!92}a^{9}+\frac{10\!\cdots\!81}{67\!\cdots\!84}a^{8}+\frac{97\!\cdots\!51}{55\!\cdots\!16}a^{7}-\frac{48\!\cdots\!53}{14\!\cdots\!08}a^{6}-\frac{70\!\cdots\!63}{56\!\cdots\!32}a^{5}+\frac{17\!\cdots\!75}{13\!\cdots\!96}a^{4}+\frac{56\!\cdots\!43}{37\!\cdots\!88}a^{3}+\frac{55\!\cdots\!93}{62\!\cdots\!48}a^{2}+\frac{73\!\cdots\!33}{31\!\cdots\!24}a+\frac{92\!\cdots\!03}{25\!\cdots\!92}$, $\frac{13\!\cdots\!87}{16\!\cdots\!92}a^{17}-\frac{15\!\cdots\!31}{66\!\cdots\!68}a^{16}+\frac{20\!\cdots\!91}{11\!\cdots\!28}a^{15}-\frac{10\!\cdots\!93}{41\!\cdots\!48}a^{14}+\frac{69\!\cdots\!51}{33\!\cdots\!84}a^{13}+\frac{21\!\cdots\!35}{13\!\cdots\!16}a^{12}+\frac{44\!\cdots\!07}{82\!\cdots\!96}a^{11}-\frac{19\!\cdots\!81}{16\!\cdots\!92}a^{10}+\frac{22\!\cdots\!83}{55\!\cdots\!64}a^{9}+\frac{13\!\cdots\!43}{24\!\cdots\!84}a^{8}+\frac{62\!\cdots\!37}{55\!\cdots\!64}a^{7}+\frac{39\!\cdots\!87}{18\!\cdots\!88}a^{6}-\frac{28\!\cdots\!95}{61\!\cdots\!96}a^{5}+\frac{22\!\cdots\!75}{32\!\cdots\!48}a^{4}+\frac{27\!\cdots\!51}{12\!\cdots\!92}a^{3}+\frac{87\!\cdots\!89}{30\!\cdots\!48}a^{2}+\frac{14\!\cdots\!81}{12\!\cdots\!92}a+\frac{84\!\cdots\!85}{35\!\cdots\!12}$, $\frac{29\!\cdots\!97}{74\!\cdots\!92}a^{17}-\frac{52\!\cdots\!35}{29\!\cdots\!68}a^{16}+\frac{14\!\cdots\!59}{99\!\cdots\!56}a^{15}-\frac{96\!\cdots\!47}{59\!\cdots\!36}a^{14}+\frac{82\!\cdots\!93}{59\!\cdots\!36}a^{13}+\frac{40\!\cdots\!81}{99\!\cdots\!56}a^{12}+\frac{52\!\cdots\!23}{84\!\cdots\!48}a^{11}-\frac{33\!\cdots\!69}{59\!\cdots\!36}a^{10}+\frac{20\!\cdots\!09}{70\!\cdots\!04}a^{9}-\frac{10\!\cdots\!33}{19\!\cdots\!12}a^{8}+\frac{83\!\cdots\!71}{94\!\cdots\!72}a^{7}-\frac{42\!\cdots\!91}{19\!\cdots\!12}a^{6}-\frac{20\!\cdots\!73}{16\!\cdots\!76}a^{5}+\frac{80\!\cdots\!65}{22\!\cdots\!68}a^{4}+\frac{45\!\cdots\!83}{66\!\cdots\!04}a^{3}+\frac{18\!\cdots\!65}{11\!\cdots\!84}a^{2}+\frac{73\!\cdots\!13}{22\!\cdots\!68}a+\frac{37\!\cdots\!79}{31\!\cdots\!24}$, $\frac{32\!\cdots\!81}{17\!\cdots\!08}a^{17}+\frac{77\!\cdots\!11}{71\!\cdots\!32}a^{16}-\frac{10\!\cdots\!59}{11\!\cdots\!72}a^{15}-\frac{70\!\cdots\!51}{22\!\cdots\!76}a^{14}-\frac{15\!\cdots\!47}{35\!\cdots\!16}a^{13}+\frac{11\!\cdots\!17}{14\!\cdots\!84}a^{12}+\frac{41\!\cdots\!25}{12\!\cdots\!72}a^{11}-\frac{48\!\cdots\!91}{17\!\cdots\!08}a^{10}-\frac{15\!\cdots\!05}{84\!\cdots\!48}a^{9}+\frac{79\!\cdots\!23}{79\!\cdots\!48}a^{8}+\frac{49\!\cdots\!61}{84\!\cdots\!48}a^{7}+\frac{26\!\cdots\!37}{12\!\cdots\!04}a^{6}-\frac{40\!\cdots\!87}{22\!\cdots\!68}a^{5}-\frac{21\!\cdots\!43}{29\!\cdots\!84}a^{4}+\frac{16\!\cdots\!17}{77\!\cdots\!24}a^{3}+\frac{76\!\cdots\!75}{33\!\cdots\!52}a^{2}+\frac{17\!\cdots\!13}{25\!\cdots\!08}a+\frac{11\!\cdots\!69}{12\!\cdots\!96}$, $\frac{19\!\cdots\!21}{17\!\cdots\!08}a^{17}+\frac{20\!\cdots\!19}{71\!\cdots\!32}a^{16}-\frac{18\!\cdots\!57}{44\!\cdots\!36}a^{15}+\frac{21\!\cdots\!47}{44\!\cdots\!52}a^{14}-\frac{24\!\cdots\!43}{35\!\cdots\!16}a^{13}+\frac{48\!\cdots\!63}{49\!\cdots\!28}a^{12}-\frac{58\!\cdots\!53}{12\!\cdots\!72}a^{11}+\frac{19\!\cdots\!73}{17\!\cdots\!08}a^{10}-\frac{22\!\cdots\!91}{28\!\cdots\!16}a^{9}+\frac{13\!\cdots\!39}{79\!\cdots\!48}a^{8}+\frac{15\!\cdots\!05}{84\!\cdots\!48}a^{7}-\frac{49\!\cdots\!23}{12\!\cdots\!04}a^{6}+\frac{48\!\cdots\!59}{66\!\cdots\!04}a^{5}-\frac{11\!\cdots\!07}{12\!\cdots\!32}a^{4}-\frac{49\!\cdots\!85}{44\!\cdots\!36}a^{3}+\frac{35\!\cdots\!67}{33\!\cdots\!52}a^{2}+\frac{13\!\cdots\!53}{44\!\cdots\!36}a+\frac{15\!\cdots\!81}{12\!\cdots\!96}$, $\frac{20\!\cdots\!03}{59\!\cdots\!36}a^{17}+\frac{27\!\cdots\!23}{88\!\cdots\!72}a^{16}+\frac{51\!\cdots\!51}{44\!\cdots\!36}a^{15}-\frac{17\!\cdots\!41}{29\!\cdots\!68}a^{14}+\frac{13\!\cdots\!21}{44\!\cdots\!36}a^{13}+\frac{18\!\cdots\!25}{15\!\cdots\!29}a^{12}+\frac{14\!\cdots\!05}{10\!\cdots\!56}a^{11}-\frac{96\!\cdots\!25}{19\!\cdots\!12}a^{10}-\frac{74\!\cdots\!27}{28\!\cdots\!16}a^{9}+\frac{35\!\cdots\!65}{26\!\cdots\!16}a^{8}+\frac{78\!\cdots\!33}{28\!\cdots\!16}a^{7}+\frac{21\!\cdots\!17}{66\!\cdots\!04}a^{6}-\frac{67\!\cdots\!15}{22\!\cdots\!68}a^{5}-\frac{23\!\cdots\!49}{33\!\cdots\!52}a^{4}+\frac{11\!\cdots\!95}{44\!\cdots\!36}a^{3}+\frac{16\!\cdots\!91}{11\!\cdots\!84}a^{2}+\frac{38\!\cdots\!45}{44\!\cdots\!36}a+\frac{31\!\cdots\!25}{12\!\cdots\!96}$, $\frac{94\!\cdots\!03}{71\!\cdots\!32}a^{17}+\frac{29\!\cdots\!81}{20\!\cdots\!52}a^{16}+\frac{50\!\cdots\!07}{33\!\cdots\!92}a^{15}+\frac{19\!\cdots\!89}{25\!\cdots\!44}a^{14}+\frac{43\!\cdots\!79}{71\!\cdots\!32}a^{13}+\frac{42\!\cdots\!23}{59\!\cdots\!36}a^{12}+\frac{25\!\cdots\!67}{35\!\cdots\!16}a^{11}+\frac{66\!\cdots\!95}{17\!\cdots\!08}a^{10}+\frac{43\!\cdots\!33}{23\!\cdots\!44}a^{9}+\frac{45\!\cdots\!37}{52\!\cdots\!32}a^{8}+\frac{38\!\cdots\!29}{59\!\cdots\!36}a^{7}+\frac{45\!\cdots\!15}{99\!\cdots\!56}a^{6}+\frac{13\!\cdots\!87}{56\!\cdots\!32}a^{5}+\frac{19\!\cdots\!11}{19\!\cdots\!12}a^{4}+\frac{27\!\cdots\!57}{88\!\cdots\!72}a^{3}+\frac{94\!\cdots\!03}{13\!\cdots\!08}a^{2}+\frac{11\!\cdots\!87}{11\!\cdots\!84}a+\frac{14\!\cdots\!23}{17\!\cdots\!44}$, $\frac{15\!\cdots\!93}{14\!\cdots\!84}a^{17}+\frac{49\!\cdots\!63}{12\!\cdots\!72}a^{16}-\frac{36\!\cdots\!03}{10\!\cdots\!56}a^{15}-\frac{10\!\cdots\!37}{14\!\cdots\!08}a^{14}+\frac{24\!\cdots\!57}{52\!\cdots\!12}a^{13}+\frac{34\!\cdots\!35}{74\!\cdots\!92}a^{12}+\frac{12\!\cdots\!77}{29\!\cdots\!68}a^{11}-\frac{20\!\cdots\!43}{89\!\cdots\!04}a^{10}-\frac{19\!\cdots\!35}{55\!\cdots\!92}a^{9}+\frac{96\!\cdots\!67}{82\!\cdots\!88}a^{8}-\frac{20\!\cdots\!39}{29\!\cdots\!68}a^{7}-\frac{22\!\cdots\!21}{33\!\cdots\!52}a^{6}-\frac{11\!\cdots\!53}{35\!\cdots\!52}a^{5}-\frac{81\!\cdots\!61}{99\!\cdots\!56}a^{4}+\frac{26\!\cdots\!13}{33\!\cdots\!52}a^{3}+\frac{24\!\cdots\!91}{41\!\cdots\!44}a^{2}+\frac{18\!\cdots\!73}{11\!\cdots\!84}a+\frac{19\!\cdots\!39}{68\!\cdots\!24}$ 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:  \( 142607524896722.38 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 142607524896722.38 \cdot 54}{2\cdot\sqrt{3121575272477039793063598549762509621195780096}}\cr\approx \mathstrut & 1.05181198660491 \end{aligned}\] (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 + 33*x^16 - 388*x^15 + 3138*x^14 + 13218*x^13 + 53812*x^12 - 1464732*x^11 + 5349033*x^10 - 8306493*x^9 + 255408471*x^8 - 450926892*x^7 - 2241288864*x^6 - 4931594442*x^5 + 43187005062*x^4 + 351848838744*x^3 + 1835309384631*x^2 + 4396464030753*x + 7107138128187)
 
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 + 33*x^16 - 388*x^15 + 3138*x^14 + 13218*x^13 + 53812*x^12 - 1464732*x^11 + 5349033*x^10 - 8306493*x^9 + 255408471*x^8 - 450926892*x^7 - 2241288864*x^6 - 4931594442*x^5 + 43187005062*x^4 + 351848838744*x^3 + 1835309384631*x^2 + 4396464030753*x + 7107138128187, 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 + 33*x^16 - 388*x^15 + 3138*x^14 + 13218*x^13 + 53812*x^12 - 1464732*x^11 + 5349033*x^10 - 8306493*x^9 + 255408471*x^8 - 450926892*x^7 - 2241288864*x^6 - 4931594442*x^5 + 43187005062*x^4 + 351848838744*x^3 + 1835309384631*x^2 + 4396464030753*x + 7107138128187);
 
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 + 33*x^16 - 388*x^15 + 3138*x^14 + 13218*x^13 + 53812*x^12 - 1464732*x^11 + 5349033*x^10 - 8306493*x^9 + 255408471*x^8 - 450926892*x^7 - 2241288864*x^6 - 4931594442*x^5 + 43187005062*x^4 + 351848838744*x^3 + 1835309384631*x^2 + 4396464030753*x + 7107138128187);
 
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

\(\Q(\sqrt{-51}) \), 3.1.972.2, 6.0.13925171376.8, 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: 18.6.318286244526390688310901391528421762902272.1

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 ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ R ${\href{/padicField/19.3.0.1}{3} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{3}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ ${\href{/padicField/43.3.0.1}{3} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }^{3}$

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 2.6.4.1$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}$
\(3\) Copy content Toggle raw display 3.6.11.3$x^{6} + 6$$6$$1$$11$$D_{6}$$[5/2]_{2}^{2}$
3.6.11.3$x^{6} + 6$$6$$1$$11$$D_{6}$$[5/2]_{2}^{2}$
3.6.11.3$x^{6} + 6$$6$$1$$11$$D_{6}$$[5/2]_{2}^{2}$
\(7\) Copy content Toggle raw display 7.6.4.1$x^{6} + 14 x^{3} - 245$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.2$x^{6} - 42 x^{3} + 147$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
\(17\) 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}$