Properties

Label 18.0.563...448.1
Degree $18$
Signature $[0, 9]$
Discriminant $-5.632\times 10^{30}$
Root discriminant \(51.09\)
Ramified primes $2,3,127$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

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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 - 2*x^16 + 27*x^15 - 63*x^14 - 182*x^13 + 326*x^12 + 24*x^11 + 1072*x^10 + 3160*x^9 + 1576*x^8 + 3344*x^7 + 7792*x^6 + 7808*x^5 + 30080*x^4 + 60160*x^3 + 60672*x^2 + 26624*x + 4096)
 
gp: K = bnfinit(y^18 - 3*y^17 - 2*y^16 + 27*y^15 - 63*y^14 - 182*y^13 + 326*y^12 + 24*y^11 + 1072*y^10 + 3160*y^9 + 1576*y^8 + 3344*y^7 + 7792*y^6 + 7808*y^5 + 30080*y^4 + 60160*y^3 + 60672*y^2 + 26624*y + 4096, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 - 2*x^16 + 27*x^15 - 63*x^14 - 182*x^13 + 326*x^12 + 24*x^11 + 1072*x^10 + 3160*x^9 + 1576*x^8 + 3344*x^7 + 7792*x^6 + 7808*x^5 + 30080*x^4 + 60160*x^3 + 60672*x^2 + 26624*x + 4096);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 - 2*x^16 + 27*x^15 - 63*x^14 - 182*x^13 + 326*x^12 + 24*x^11 + 1072*x^10 + 3160*x^9 + 1576*x^8 + 3344*x^7 + 7792*x^6 + 7808*x^5 + 30080*x^4 + 60160*x^3 + 60672*x^2 + 26624*x + 4096)
 

\( x^{18} - 3 x^{17} - 2 x^{16} + 27 x^{15} - 63 x^{14} - 182 x^{13} + 326 x^{12} + 24 x^{11} + 1072 x^{10} + 3160 x^{9} + 1576 x^{8} + 3344 x^{7} + 7792 x^{6} + 7808 x^{5} + 30080 x^{4} + \cdots + 4096 \) 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:   \(-5632074928630950758566400360448\) \(\medspace = -\,2^{18}\cdot 3^{9}\cdot 127^{10}\) 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:  \(51.09\)
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:  $2^{3/2}3^{1/2}127^{5/6}\approx 277.50792385820444$
Ramified primes:   \(2\), \(3\), \(127\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Aut(K/\Q) }$:  $3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{1}{8}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{10}-\frac{1}{8}a^{9}+\frac{1}{16}a^{8}-\frac{1}{16}a^{7}+\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a$, $\frac{1}{32}a^{12}-\frac{1}{32}a^{11}-\frac{1}{16}a^{10}+\frac{1}{32}a^{9}+\frac{3}{32}a^{8}+\frac{1}{16}a^{7}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{13}-\frac{1}{32}a^{11}+\frac{1}{32}a^{10}-\frac{1}{8}a^{9}-\frac{1}{32}a^{8}+\frac{1}{8}a^{6}+\frac{1}{4}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{64}a^{14}-\frac{1}{64}a^{13}-\frac{1}{64}a^{11}-\frac{1}{64}a^{10}-\frac{1}{16}a^{9}+\frac{1}{32}a^{8}-\frac{1}{16}a^{7}+\frac{1}{8}a^{6}-\frac{1}{8}a^{5}+\frac{1}{8}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{256}a^{15}+\frac{1}{256}a^{14}-\frac{1}{128}a^{13}-\frac{1}{256}a^{12}+\frac{5}{256}a^{11}-\frac{7}{128}a^{10}-\frac{11}{128}a^{9}-\frac{1}{32}a^{8}-\frac{1}{32}a^{7}-\frac{5}{32}a^{6}-\frac{3}{32}a^{5}-\frac{3}{16}a^{4}-\frac{7}{16}a^{3}-\frac{1}{4}a$, $\frac{1}{512}a^{16}-\frac{1}{512}a^{15}-\frac{1}{128}a^{14}+\frac{3}{512}a^{13}+\frac{7}{512}a^{12}+\frac{1}{64}a^{11}-\frac{13}{256}a^{10}+\frac{9}{128}a^{9}+\frac{5}{64}a^{8}-\frac{7}{64}a^{7}-\frac{9}{64}a^{6}-\frac{1}{32}a^{4}+\frac{3}{16}a^{3}-\frac{1}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{11\!\cdots\!32}a^{17}+\frac{98\!\cdots\!17}{11\!\cdots\!32}a^{16}-\frac{53\!\cdots\!39}{55\!\cdots\!16}a^{15}-\frac{56\!\cdots\!45}{11\!\cdots\!32}a^{14}+\frac{55\!\cdots\!29}{11\!\cdots\!32}a^{13}+\frac{34\!\cdots\!99}{55\!\cdots\!16}a^{12}+\frac{40\!\cdots\!99}{55\!\cdots\!16}a^{11}+\frac{25\!\cdots\!43}{69\!\cdots\!52}a^{10}+\frac{60\!\cdots\!55}{69\!\cdots\!52}a^{9}+\frac{53\!\cdots\!87}{13\!\cdots\!04}a^{8}+\frac{10\!\cdots\!89}{13\!\cdots\!04}a^{7}-\frac{11\!\cdots\!37}{69\!\cdots\!52}a^{6}-\frac{95\!\cdots\!51}{54\!\cdots\!76}a^{5}+\frac{34\!\cdots\!61}{17\!\cdots\!88}a^{4}-\frac{19\!\cdots\!39}{87\!\cdots\!44}a^{3}-\frac{14\!\cdots\!85}{43\!\cdots\!72}a^{2}+\frac{18\!\cdots\!85}{43\!\cdots\!72}a-\frac{22\!\cdots\!91}{10\!\cdots\!68}$ 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

$C_{2}$, which has order $2$ (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:   \( -\frac{693570908493955}{432739011821809664} a^{17} + \frac{2340308272355869}{432739011821809664} a^{16} + \frac{262576486421757}{216369505910904832} a^{15} - \frac{18976322657208169}{432739011821809664} a^{14} + \frac{50802472029220505}{432739011821809664} a^{13} + \frac{53823693559736835}{216369505910904832} a^{12} - \frac{133874644475789629}{216369505910904832} a^{11} + \frac{5121641378728171}{27046188238863104} a^{10} - \frac{47915558313508693}{27046188238863104} a^{9} - \frac{238981647733541057}{54092376477726208} a^{8} - \frac{45313337058619219}{54092376477726208} a^{7} - \frac{133093303032384041}{27046188238863104} a^{6} - \frac{290897290481288057}{27046188238863104} a^{5} - \frac{56579125697596667}{6761547059715776} a^{4} - \frac{152010996061114863}{3380773529857888} a^{3} - \frac{134790296096409437}{1690386764928944} a^{2} - \frac{112464181087348891}{1690386764928944} a - \frac{6773562299056895}{422596691232236} \)  (order $6$) 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:   $\frac{37\!\cdots\!83}{11\!\cdots\!32}a^{17}-\frac{12\!\cdots\!21}{11\!\cdots\!32}a^{16}-\frac{16\!\cdots\!37}{55\!\cdots\!16}a^{15}+\frac{10\!\cdots\!77}{11\!\cdots\!32}a^{14}-\frac{27\!\cdots\!73}{11\!\cdots\!32}a^{13}-\frac{29\!\cdots\!31}{55\!\cdots\!16}a^{12}+\frac{71\!\cdots\!61}{55\!\cdots\!16}a^{11}-\frac{11\!\cdots\!19}{34\!\cdots\!76}a^{10}+\frac{25\!\cdots\!61}{69\!\cdots\!52}a^{9}+\frac{13\!\cdots\!37}{13\!\cdots\!04}a^{8}+\frac{29\!\cdots\!07}{13\!\cdots\!04}a^{7}+\frac{72\!\cdots\!49}{69\!\cdots\!52}a^{6}+\frac{12\!\cdots\!99}{54\!\cdots\!76}a^{5}+\frac{32\!\cdots\!65}{17\!\cdots\!88}a^{4}+\frac{82\!\cdots\!83}{87\!\cdots\!44}a^{3}+\frac{74\!\cdots\!47}{43\!\cdots\!72}a^{2}+\frac{63\!\cdots\!23}{43\!\cdots\!72}a+\frac{43\!\cdots\!59}{10\!\cdots\!68}$, $\frac{18\!\cdots\!57}{11\!\cdots\!32}a^{17}+\frac{55\!\cdots\!77}{11\!\cdots\!32}a^{16}-\frac{24\!\cdots\!79}{55\!\cdots\!16}a^{15}+\frac{71\!\cdots\!55}{11\!\cdots\!32}a^{14}+\frac{19\!\cdots\!53}{11\!\cdots\!32}a^{13}-\frac{70\!\cdots\!25}{55\!\cdots\!16}a^{12}-\frac{24\!\cdots\!17}{55\!\cdots\!16}a^{11}+\frac{36\!\cdots\!95}{69\!\cdots\!52}a^{10}-\frac{21\!\cdots\!61}{69\!\cdots\!52}a^{9}+\frac{21\!\cdots\!03}{13\!\cdots\!04}a^{8}+\frac{35\!\cdots\!73}{13\!\cdots\!04}a^{7}-\frac{17\!\cdots\!45}{69\!\cdots\!52}a^{6}+\frac{21\!\cdots\!45}{54\!\cdots\!76}a^{5}+\frac{10\!\cdots\!37}{17\!\cdots\!88}a^{4}+\frac{51\!\cdots\!37}{87\!\cdots\!44}a^{3}+\frac{14\!\cdots\!99}{43\!\cdots\!72}a^{2}+\frac{19\!\cdots\!57}{43\!\cdots\!72}a+\frac{16\!\cdots\!05}{10\!\cdots\!68}$, $\frac{22\!\cdots\!69}{87\!\cdots\!16}a^{17}-\frac{76\!\cdots\!75}{87\!\cdots\!16}a^{16}-\frac{78\!\cdots\!23}{43\!\cdots\!08}a^{15}+\frac{61\!\cdots\!75}{87\!\cdots\!16}a^{14}-\frac{16\!\cdots\!71}{87\!\cdots\!16}a^{13}-\frac{17\!\cdots\!05}{43\!\cdots\!08}a^{12}+\frac{43\!\cdots\!43}{43\!\cdots\!08}a^{11}-\frac{83\!\cdots\!41}{27\!\cdots\!88}a^{10}+\frac{16\!\cdots\!93}{54\!\cdots\!76}a^{9}+\frac{76\!\cdots\!27}{10\!\cdots\!52}a^{8}+\frac{17\!\cdots\!81}{10\!\cdots\!52}a^{7}+\frac{42\!\cdots\!91}{54\!\cdots\!76}a^{6}+\frac{94\!\cdots\!11}{54\!\cdots\!76}a^{5}+\frac{19\!\cdots\!99}{13\!\cdots\!44}a^{4}+\frac{50\!\cdots\!51}{68\!\cdots\!72}a^{3}+\frac{44\!\cdots\!73}{34\!\cdots\!36}a^{2}+\frac{37\!\cdots\!49}{34\!\cdots\!36}a+\frac{24\!\cdots\!33}{85\!\cdots\!84}$, $\frac{15\!\cdots\!55}{21\!\cdots\!04}a^{17}-\frac{66\!\cdots\!67}{21\!\cdots\!04}a^{16}+\frac{10\!\cdots\!09}{54\!\cdots\!76}a^{15}+\frac{41\!\cdots\!13}{21\!\cdots\!04}a^{14}-\frac{15\!\cdots\!15}{21\!\cdots\!04}a^{13}-\frac{15\!\cdots\!27}{27\!\cdots\!88}a^{12}+\frac{37\!\cdots\!93}{10\!\cdots\!52}a^{11}-\frac{19\!\cdots\!33}{54\!\cdots\!76}a^{10}+\frac{27\!\cdots\!51}{27\!\cdots\!88}a^{9}+\frac{31\!\cdots\!07}{27\!\cdots\!88}a^{8}-\frac{25\!\cdots\!27}{27\!\cdots\!88}a^{7}+\frac{41\!\cdots\!69}{17\!\cdots\!68}a^{6}+\frac{36\!\cdots\!57}{13\!\cdots\!44}a^{5}+\frac{50\!\cdots\!41}{68\!\cdots\!72}a^{4}+\frac{61\!\cdots\!81}{34\!\cdots\!36}a^{3}+\frac{32\!\cdots\!03}{17\!\cdots\!68}a^{2}+\frac{14\!\cdots\!38}{21\!\cdots\!71}a+\frac{16\!\cdots\!61}{21\!\cdots\!71}$, $\frac{62\!\cdots\!69}{55\!\cdots\!16}a^{17}-\frac{17\!\cdots\!15}{55\!\cdots\!16}a^{16}-\frac{28\!\cdots\!75}{27\!\cdots\!08}a^{15}+\frac{12\!\cdots\!03}{55\!\cdots\!16}a^{14}-\frac{34\!\cdots\!83}{55\!\cdots\!16}a^{13}-\frac{43\!\cdots\!69}{27\!\cdots\!08}a^{12}+\frac{42\!\cdots\!03}{27\!\cdots\!08}a^{11}-\frac{26\!\cdots\!23}{34\!\cdots\!76}a^{10}+\frac{41\!\cdots\!43}{17\!\cdots\!88}a^{9}+\frac{21\!\cdots\!55}{69\!\cdots\!52}a^{8}+\frac{21\!\cdots\!29}{69\!\cdots\!52}a^{7}+\frac{12\!\cdots\!11}{34\!\cdots\!76}a^{6}+\frac{82\!\cdots\!53}{27\!\cdots\!88}a^{5}+\frac{15\!\cdots\!65}{87\!\cdots\!44}a^{4}+\frac{51\!\cdots\!69}{10\!\cdots\!68}a^{3}+\frac{13\!\cdots\!85}{21\!\cdots\!36}a^{2}+\frac{16\!\cdots\!95}{21\!\cdots\!36}a+\frac{11\!\cdots\!25}{54\!\cdots\!34}$, $\frac{51\!\cdots\!99}{13\!\cdots\!04}a^{17}-\frac{44\!\cdots\!15}{34\!\cdots\!76}a^{16}-\frac{40\!\cdots\!33}{13\!\cdots\!04}a^{15}+\frac{14\!\cdots\!95}{13\!\cdots\!04}a^{14}-\frac{18\!\cdots\!13}{69\!\cdots\!52}a^{13}-\frac{81\!\cdots\!77}{13\!\cdots\!04}a^{12}+\frac{25\!\cdots\!13}{17\!\cdots\!88}a^{11}-\frac{17\!\cdots\!43}{69\!\cdots\!52}a^{10}+\frac{36\!\cdots\!37}{87\!\cdots\!44}a^{9}+\frac{40\!\cdots\!09}{43\!\cdots\!72}a^{8}+\frac{10\!\cdots\!27}{87\!\cdots\!44}a^{7}+\frac{16\!\cdots\!71}{17\!\cdots\!88}a^{6}+\frac{28\!\cdots\!93}{17\!\cdots\!68}a^{5}+\frac{98\!\cdots\!49}{87\!\cdots\!44}a^{4}+\frac{21\!\cdots\!81}{21\!\cdots\!36}a^{3}+\frac{36\!\cdots\!07}{21\!\cdots\!36}a^{2}+\frac{71\!\cdots\!71}{54\!\cdots\!34}a+\frac{68\!\cdots\!71}{27\!\cdots\!17}$, $\frac{14\!\cdots\!53}{55\!\cdots\!16}a^{17}-\frac{49\!\cdots\!75}{55\!\cdots\!16}a^{16}-\frac{61\!\cdots\!45}{27\!\cdots\!08}a^{15}+\frac{40\!\cdots\!47}{55\!\cdots\!16}a^{14}-\frac{10\!\cdots\!99}{55\!\cdots\!16}a^{13}-\frac{11\!\cdots\!27}{27\!\cdots\!08}a^{12}+\frac{28\!\cdots\!67}{27\!\cdots\!08}a^{11}-\frac{20\!\cdots\!23}{69\!\cdots\!52}a^{10}+\frac{10\!\cdots\!19}{34\!\cdots\!76}a^{9}+\frac{51\!\cdots\!87}{69\!\cdots\!52}a^{8}+\frac{10\!\cdots\!05}{69\!\cdots\!52}a^{7}+\frac{29\!\cdots\!65}{34\!\cdots\!76}a^{6}+\frac{48\!\cdots\!41}{27\!\cdots\!88}a^{5}+\frac{15\!\cdots\!25}{10\!\cdots\!68}a^{4}+\frac{32\!\cdots\!75}{43\!\cdots\!72}a^{3}+\frac{36\!\cdots\!99}{27\!\cdots\!17}a^{2}+\frac{24\!\cdots\!85}{21\!\cdots\!36}a+\frac{16\!\cdots\!87}{54\!\cdots\!34}$, $\frac{11\!\cdots\!07}{55\!\cdots\!16}a^{17}-\frac{38\!\cdots\!73}{55\!\cdots\!16}a^{16}-\frac{20\!\cdots\!43}{27\!\cdots\!08}a^{15}+\frac{30\!\cdots\!25}{55\!\cdots\!16}a^{14}-\frac{85\!\cdots\!17}{55\!\cdots\!16}a^{13}-\frac{81\!\cdots\!93}{27\!\cdots\!08}a^{12}+\frac{22\!\cdots\!21}{27\!\cdots\!08}a^{11}-\frac{22\!\cdots\!27}{69\!\cdots\!52}a^{10}+\frac{80\!\cdots\!29}{34\!\cdots\!76}a^{9}+\frac{36\!\cdots\!69}{69\!\cdots\!52}a^{8}+\frac{45\!\cdots\!43}{69\!\cdots\!52}a^{7}+\frac{22\!\cdots\!71}{34\!\cdots\!76}a^{6}+\frac{34\!\cdots\!75}{27\!\cdots\!88}a^{5}+\frac{42\!\cdots\!93}{43\!\cdots\!72}a^{4}+\frac{24\!\cdots\!85}{43\!\cdots\!72}a^{3}+\frac{10\!\cdots\!41}{10\!\cdots\!68}a^{2}+\frac{16\!\cdots\!27}{21\!\cdots\!36}a+\frac{86\!\cdots\!49}{54\!\cdots\!34}$ 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:  \( 621058194.3189892 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 621058194.3189892 \cdot 2}{6\cdot\sqrt{5632074928630950758566400360448}}\cr\approx \mathstrut & 1.33136042959378 \end{aligned}\] (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 - 2*x^16 + 27*x^15 - 63*x^14 - 182*x^13 + 326*x^12 + 24*x^11 + 1072*x^10 + 3160*x^9 + 1576*x^8 + 3344*x^7 + 7792*x^6 + 7808*x^5 + 30080*x^4 + 60160*x^3 + 60672*x^2 + 26624*x + 4096)
 
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 - 2*x^16 + 27*x^15 - 63*x^14 - 182*x^13 + 326*x^12 + 24*x^11 + 1072*x^10 + 3160*x^9 + 1576*x^8 + 3344*x^7 + 7792*x^6 + 7808*x^5 + 30080*x^4 + 60160*x^3 + 60672*x^2 + 26624*x + 4096, 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 - 2*x^16 + 27*x^15 - 63*x^14 - 182*x^13 + 326*x^12 + 24*x^11 + 1072*x^10 + 3160*x^9 + 1576*x^8 + 3344*x^7 + 7792*x^6 + 7808*x^5 + 30080*x^4 + 60160*x^3 + 60672*x^2 + 26624*x + 4096);
 
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 - 2*x^16 + 27*x^15 - 63*x^14 - 182*x^13 + 326*x^12 + 24*x^11 + 1072*x^10 + 3160*x^9 + 1576*x^8 + 3344*x^7 + 7792*x^6 + 7808*x^5 + 30080*x^4 + 60160*x^3 + 60672*x^2 + 26624*x + 4096);
 
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

$C_3\times S_3^2$ (as 18T46):

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 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.1016.1, 6.0.435483.1, 6.0.27870912.1

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 12 sibling: data not computed
Degree 18 siblings: data not computed
Degree 27 sibling: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 12.0.12932938664955809431289856.3

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 ${\href{/padicField/5.6.0.1}{6} }^{3}$ ${\href{/padicField/7.3.0.1}{3} }^{6}$ ${\href{/padicField/11.6.0.1}{6} }^{3}$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ ${\href{/padicField/17.6.0.1}{6} }^{3}$ ${\href{/padicField/19.3.0.1}{3} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }^{3}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }^{3}$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{9}$ ${\href{/padicField/53.6.0.1}{6} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }^{3}$

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 2.2.0.1$x^{2} + x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} + x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} + x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.6.2$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$$2$$2$$6$$C_2^2$$[3]^{2}$
\(3\) Copy content Toggle raw display 3.6.3.2$x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.6.2$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
\(127\) Copy content Toggle raw display $\Q_{127}$$x + 124$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 124$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 124$$1$$1$$0$Trivial$[\ ]$
127.2.1.1$x^{2} + 381$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} + 381$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} + 381$$2$$1$$1$$C_2$$[\ ]_{2}$
127.3.2.1$x^{3} + 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.6.5.1$x^{6} + 635$$6$$1$$5$$C_6$$[\ ]_{6}$