Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 12*x^15 + 36*x^14 - 33*x^12 + 117*x^11 - 171*x^10 - 2*x^9 + 135*x^8 - 81*x^7 - 30*x^6 + 36*x^5 + 81*x^4 - 393*x^3 + 531*x^2 - 270*x + 44)
gp: K = bnfinit(y^18 - 12*y^15 + 36*y^14 - 33*y^12 + 117*y^11 - 171*y^10 - 2*y^9 + 135*y^8 - 81*y^7 - 30*y^6 + 36*y^5 + 81*y^4 - 393*y^3 + 531*y^2 - 270*y + 44, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 12*x^15 + 36*x^14 - 33*x^12 + 117*x^11 - 171*x^10 - 2*x^9 + 135*x^8 - 81*x^7 - 30*x^6 + 36*x^5 + 81*x^4 - 393*x^3 + 531*x^2 - 270*x + 44);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 12*x^15 + 36*x^14 - 33*x^12 + 117*x^11 - 171*x^10 - 2*x^9 + 135*x^8 - 81*x^7 - 30*x^6 + 36*x^5 + 81*x^4 - 393*x^3 + 531*x^2 - 270*x + 44)
\( x^{18} - 12 x^{15} + 36 x^{14} - 33 x^{12} + 117 x^{11} - 171 x^{10} - 2 x^{9} + 135 x^{8} - 81 x^{7} + \cdots + 44 \)
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: | $[6, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(36644198070556426025390625\)
\(\medspace = 3^{36}\cdot 5^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(26.32\) | 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: | $3^{58/27}5^{3/4}\approx 35.412341444574025$ | ||
| Ramified primes: |
\(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{12}+\frac{1}{4}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{13}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{952}a^{16}+\frac{9}{952}a^{15}+\frac{19}{238}a^{14}+\frac{151}{952}a^{13}+\frac{9}{136}a^{12}-\frac{47}{238}a^{11}-\frac{1}{7}a^{10}+\frac{309}{952}a^{9}+\frac{31}{68}a^{8}+\frac{65}{136}a^{7}-\frac{40}{119}a^{6}-\frac{167}{476}a^{4}+\frac{181}{476}a^{3}+\frac{249}{952}a^{2}-\frac{219}{476}a-\frac{93}{238}$, $\frac{1}{31\!\cdots\!12}a^{17}-\frac{105841414556345}{31\!\cdots\!12}a^{16}-\frac{46\!\cdots\!19}{78\!\cdots\!78}a^{15}+\frac{49\!\cdots\!29}{31\!\cdots\!12}a^{14}-\frac{31\!\cdots\!77}{31\!\cdots\!12}a^{13}-\frac{17\!\cdots\!95}{78\!\cdots\!78}a^{12}+\frac{59\!\cdots\!36}{39\!\cdots\!39}a^{11}+\frac{84\!\cdots\!69}{31\!\cdots\!12}a^{10}-\frac{38\!\cdots\!25}{78\!\cdots\!78}a^{9}+\frac{15\!\cdots\!87}{45\!\cdots\!16}a^{8}+\frac{42\!\cdots\!25}{15\!\cdots\!56}a^{7}-\frac{24\!\cdots\!31}{78\!\cdots\!78}a^{6}+\frac{29\!\cdots\!03}{78\!\cdots\!78}a^{5}-\frac{16\!\cdots\!23}{39\!\cdots\!39}a^{4}+\frac{11\!\cdots\!09}{31\!\cdots\!12}a^{3}+\frac{68\!\cdots\!09}{15\!\cdots\!56}a^{2}+\frac{68\!\cdots\!94}{39\!\cdots\!39}a+\frac{14\!\cdots\!92}{39\!\cdots\!39}$
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
Trivial group, which has order $1$ (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: | $11$ | 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: |
$\frac{641937521115}{10860908935042}a^{17}+\frac{79116760827}{5430454467521}a^{16}+\frac{26764657449}{5430454467521}a^{15}-\frac{3931760030940}{5430454467521}a^{14}+\frac{10570081998636}{5430454467521}a^{13}+\frac{2349750710280}{5430454467521}a^{12}-\frac{17287431279039}{10860908935042}a^{11}+\frac{64571716853325}{10860908935042}a^{10}-\frac{91019866630781}{10860908935042}a^{9}-\frac{15425530973811}{5430454467521}a^{8}+\frac{67883497525053}{10860908935042}a^{7}-\frac{18927509618457}{10860908935042}a^{6}-\frac{17173191890457}{5430454467521}a^{5}+\frac{16438189858146}{5430454467521}a^{4}+\frac{41382068045949}{10860908935042}a^{3}-\frac{216890839459953}{10860908935042}a^{2}+\frac{250464765196773}{10860908935042}a-\frac{33149262805525}{5430454467521}$, $\frac{190182269157885}{92\!\cdots\!68}a^{17}+\frac{39\!\cdots\!75}{15\!\cdots\!56}a^{16}+\frac{52\!\cdots\!07}{15\!\cdots\!56}a^{15}-\frac{92\!\cdots\!03}{39\!\cdots\!39}a^{14}+\frac{18\!\cdots\!29}{39\!\cdots\!39}a^{13}+\frac{76\!\cdots\!67}{15\!\cdots\!56}a^{12}+\frac{32\!\cdots\!87}{78\!\cdots\!78}a^{11}+\frac{13\!\cdots\!15}{92\!\cdots\!68}a^{10}-\frac{66\!\cdots\!01}{39\!\cdots\!39}a^{9}-\frac{19\!\cdots\!13}{11\!\cdots\!54}a^{8}-\frac{85\!\cdots\!35}{78\!\cdots\!78}a^{7}+\frac{11\!\cdots\!09}{78\!\cdots\!78}a^{6}-\frac{28\!\cdots\!43}{92\!\cdots\!68}a^{5}+\frac{78\!\cdots\!17}{22\!\cdots\!08}a^{4}-\frac{63\!\cdots\!07}{15\!\cdots\!56}a^{3}-\frac{54\!\cdots\!51}{22\!\cdots\!08}a^{2}-\frac{22\!\cdots\!19}{78\!\cdots\!78}a+\frac{99\!\cdots\!42}{39\!\cdots\!39}$, $\frac{21\!\cdots\!45}{31\!\cdots\!12}a^{17}+\frac{15\!\cdots\!27}{31\!\cdots\!12}a^{16}+\frac{32\!\cdots\!53}{78\!\cdots\!78}a^{15}-\frac{24\!\cdots\!83}{31\!\cdots\!12}a^{14}+\frac{57\!\cdots\!23}{31\!\cdots\!12}a^{13}+\frac{51\!\cdots\!82}{39\!\cdots\!39}a^{12}-\frac{82\!\cdots\!29}{78\!\cdots\!78}a^{11}+\frac{21\!\cdots\!29}{31\!\cdots\!12}a^{10}-\frac{29\!\cdots\!95}{46\!\cdots\!34}a^{9}-\frac{20\!\cdots\!49}{45\!\cdots\!16}a^{8}+\frac{77\!\cdots\!75}{15\!\cdots\!56}a^{7}-\frac{13\!\cdots\!71}{78\!\cdots\!78}a^{6}-\frac{27\!\cdots\!33}{78\!\cdots\!78}a^{5}+\frac{24\!\cdots\!35}{78\!\cdots\!78}a^{4}+\frac{22\!\cdots\!75}{45\!\cdots\!16}a^{3}-\frac{34\!\cdots\!77}{15\!\cdots\!56}a^{2}+\frac{74\!\cdots\!19}{39\!\cdots\!39}a-\frac{18\!\cdots\!74}{39\!\cdots\!39}$, $\frac{21\!\cdots\!45}{31\!\cdots\!12}a^{17}+\frac{15\!\cdots\!27}{31\!\cdots\!12}a^{16}+\frac{32\!\cdots\!53}{78\!\cdots\!78}a^{15}-\frac{24\!\cdots\!83}{31\!\cdots\!12}a^{14}+\frac{57\!\cdots\!23}{31\!\cdots\!12}a^{13}+\frac{51\!\cdots\!82}{39\!\cdots\!39}a^{12}-\frac{82\!\cdots\!29}{78\!\cdots\!78}a^{11}+\frac{21\!\cdots\!29}{31\!\cdots\!12}a^{10}-\frac{29\!\cdots\!95}{46\!\cdots\!34}a^{9}-\frac{20\!\cdots\!49}{45\!\cdots\!16}a^{8}+\frac{77\!\cdots\!75}{15\!\cdots\!56}a^{7}-\frac{13\!\cdots\!71}{78\!\cdots\!78}a^{6}-\frac{27\!\cdots\!33}{78\!\cdots\!78}a^{5}+\frac{24\!\cdots\!35}{78\!\cdots\!78}a^{4}+\frac{22\!\cdots\!75}{45\!\cdots\!16}a^{3}-\frac{34\!\cdots\!77}{15\!\cdots\!56}a^{2}+\frac{74\!\cdots\!19}{39\!\cdots\!39}a-\frac{14\!\cdots\!35}{39\!\cdots\!39}$, $\frac{312641150013499}{92\!\cdots\!68}a^{17}+\frac{14\!\cdots\!97}{31\!\cdots\!12}a^{16}+\frac{14\!\cdots\!75}{31\!\cdots\!12}a^{15}-\frac{14\!\cdots\!50}{39\!\cdots\!39}a^{14}+\frac{21\!\cdots\!51}{31\!\cdots\!12}a^{13}+\frac{35\!\cdots\!47}{31\!\cdots\!12}a^{12}+\frac{21\!\cdots\!53}{78\!\cdots\!78}a^{11}+\frac{28\!\cdots\!83}{92\!\cdots\!68}a^{10}-\frac{20\!\cdots\!17}{31\!\cdots\!12}a^{9}-\frac{19\!\cdots\!74}{56\!\cdots\!77}a^{8}+\frac{82\!\cdots\!39}{31\!\cdots\!12}a^{7}+\frac{27\!\cdots\!77}{15\!\cdots\!56}a^{6}-\frac{16\!\cdots\!25}{92\!\cdots\!68}a^{5}+\frac{63\!\cdots\!77}{22\!\cdots\!08}a^{4}+\frac{23\!\cdots\!03}{15\!\cdots\!56}a^{3}-\frac{35\!\cdots\!09}{45\!\cdots\!16}a^{2}+\frac{10\!\cdots\!29}{15\!\cdots\!56}a+\frac{36\!\cdots\!15}{78\!\cdots\!78}$, $\frac{12\!\cdots\!21}{31\!\cdots\!12}a^{17}-\frac{11\!\cdots\!77}{15\!\cdots\!56}a^{16}-\frac{52\!\cdots\!83}{31\!\cdots\!12}a^{15}-\frac{22\!\cdots\!49}{31\!\cdots\!12}a^{14}+\frac{33\!\cdots\!29}{15\!\cdots\!56}a^{13}-\frac{23\!\cdots\!97}{31\!\cdots\!12}a^{12}-\frac{18\!\cdots\!31}{39\!\cdots\!39}a^{11}+\frac{44\!\cdots\!69}{31\!\cdots\!12}a^{10}-\frac{50\!\cdots\!49}{31\!\cdots\!12}a^{9}-\frac{14\!\cdots\!93}{45\!\cdots\!16}a^{8}+\frac{50\!\cdots\!23}{31\!\cdots\!12}a^{7}-\frac{98\!\cdots\!49}{39\!\cdots\!39}a^{6}+\frac{45\!\cdots\!69}{15\!\cdots\!56}a^{5}+\frac{950116934760115}{56\!\cdots\!77}a^{4}-\frac{45\!\cdots\!37}{31\!\cdots\!12}a^{3}+\frac{54\!\cdots\!65}{45\!\cdots\!16}a^{2}+\frac{62\!\cdots\!43}{15\!\cdots\!56}a-\frac{81\!\cdots\!13}{78\!\cdots\!78}$, $\frac{41\!\cdots\!63}{31\!\cdots\!12}a^{17}+\frac{25\!\cdots\!33}{31\!\cdots\!12}a^{16}+\frac{770774712031579}{11\!\cdots\!54}a^{15}-\frac{67\!\cdots\!89}{45\!\cdots\!16}a^{14}+\frac{11\!\cdots\!69}{31\!\cdots\!12}a^{13}+\frac{34\!\cdots\!67}{15\!\cdots\!56}a^{12}-\frac{19\!\cdots\!03}{78\!\cdots\!78}a^{11}+\frac{43\!\cdots\!63}{31\!\cdots\!12}a^{10}-\frac{10\!\cdots\!65}{78\!\cdots\!78}a^{9}-\frac{33\!\cdots\!87}{45\!\cdots\!16}a^{8}+\frac{53\!\cdots\!35}{46\!\cdots\!34}a^{7}-\frac{76\!\cdots\!45}{15\!\cdots\!56}a^{6}-\frac{11\!\cdots\!53}{15\!\cdots\!56}a^{5}+\frac{19\!\cdots\!85}{39\!\cdots\!39}a^{4}+\frac{33\!\cdots\!61}{31\!\cdots\!12}a^{3}-\frac{16\!\cdots\!01}{39\!\cdots\!39}a^{2}+\frac{33\!\cdots\!09}{78\!\cdots\!78}a-\frac{44\!\cdots\!13}{39\!\cdots\!39}$, $\frac{27\!\cdots\!51}{15\!\cdots\!56}a^{17}-\frac{51\!\cdots\!15}{15\!\cdots\!56}a^{16}+\frac{598508537524152}{39\!\cdots\!39}a^{15}-\frac{41\!\cdots\!55}{15\!\cdots\!56}a^{14}+\frac{16\!\cdots\!09}{15\!\cdots\!56}a^{13}-\frac{11\!\cdots\!31}{78\!\cdots\!78}a^{12}+\frac{59\!\cdots\!47}{78\!\cdots\!78}a^{11}+\frac{46\!\cdots\!45}{15\!\cdots\!56}a^{10}-\frac{31\!\cdots\!35}{78\!\cdots\!78}a^{9}+\frac{86\!\cdots\!93}{22\!\cdots\!08}a^{8}-\frac{10\!\cdots\!95}{78\!\cdots\!78}a^{7}+\frac{84\!\cdots\!85}{39\!\cdots\!39}a^{6}-\frac{42\!\cdots\!25}{78\!\cdots\!78}a^{5}+\frac{30\!\cdots\!09}{39\!\cdots\!39}a^{4}-\frac{13\!\cdots\!53}{15\!\cdots\!56}a^{3}+\frac{22\!\cdots\!86}{39\!\cdots\!39}a^{2}+\frac{18\!\cdots\!87}{78\!\cdots\!78}a-\frac{56\!\cdots\!98}{39\!\cdots\!39}$, $\frac{10\!\cdots\!87}{78\!\cdots\!78}a^{17}+\frac{25\!\cdots\!31}{31\!\cdots\!12}a^{16}+\frac{25\!\cdots\!87}{31\!\cdots\!12}a^{15}-\frac{12\!\cdots\!73}{78\!\cdots\!78}a^{14}+\frac{12\!\cdots\!25}{31\!\cdots\!12}a^{13}+\frac{60\!\cdots\!25}{31\!\cdots\!12}a^{12}-\frac{10\!\cdots\!07}{56\!\cdots\!77}a^{11}+\frac{51\!\cdots\!94}{39\!\cdots\!39}a^{10}-\frac{41\!\cdots\!13}{31\!\cdots\!12}a^{9}-\frac{15\!\cdots\!29}{22\!\cdots\!08}a^{8}+\frac{31\!\cdots\!17}{31\!\cdots\!12}a^{7}-\frac{39\!\cdots\!37}{39\!\cdots\!39}a^{6}-\frac{63\!\cdots\!49}{78\!\cdots\!78}a^{5}+\frac{56\!\cdots\!11}{15\!\cdots\!56}a^{4}+\frac{93\!\cdots\!29}{15\!\cdots\!56}a^{3}-\frac{12\!\cdots\!85}{31\!\cdots\!12}a^{2}+\frac{57\!\cdots\!55}{15\!\cdots\!56}a-\frac{61\!\cdots\!93}{78\!\cdots\!78}$, $\frac{24\!\cdots\!49}{31\!\cdots\!12}a^{17}+\frac{20\!\cdots\!65}{31\!\cdots\!12}a^{16}+\frac{66\!\cdots\!37}{15\!\cdots\!56}a^{15}-\frac{28\!\cdots\!85}{31\!\cdots\!12}a^{14}+\frac{64\!\cdots\!29}{31\!\cdots\!12}a^{13}+\frac{14\!\cdots\!71}{78\!\cdots\!78}a^{12}-\frac{60\!\cdots\!36}{39\!\cdots\!39}a^{11}+\frac{26\!\cdots\!05}{31\!\cdots\!12}a^{10}-\frac{11\!\cdots\!71}{15\!\cdots\!56}a^{9}-\frac{30\!\cdots\!33}{45\!\cdots\!16}a^{8}+\frac{10\!\cdots\!55}{15\!\cdots\!56}a^{7}-\frac{31\!\cdots\!61}{15\!\cdots\!56}a^{6}-\frac{39\!\cdots\!93}{15\!\cdots\!56}a^{5}-\frac{10\!\cdots\!65}{56\!\cdots\!77}a^{4}+\frac{24\!\cdots\!63}{31\!\cdots\!12}a^{3}-\frac{63\!\cdots\!85}{22\!\cdots\!08}a^{2}+\frac{92\!\cdots\!69}{39\!\cdots\!39}a-\frac{13\!\cdots\!19}{39\!\cdots\!39}$, $\frac{35\!\cdots\!74}{39\!\cdots\!39}a^{17}+\frac{353064137604767}{92\!\cdots\!68}a^{16}+\frac{17\!\cdots\!83}{78\!\cdots\!78}a^{15}-\frac{42\!\cdots\!78}{39\!\cdots\!39}a^{14}+\frac{22\!\cdots\!29}{78\!\cdots\!78}a^{13}+\frac{17\!\cdots\!99}{15\!\cdots\!56}a^{12}-\frac{93\!\cdots\!94}{39\!\cdots\!39}a^{11}+\frac{38\!\cdots\!67}{39\!\cdots\!39}a^{10}-\frac{17\!\cdots\!91}{15\!\cdots\!56}a^{9}-\frac{63\!\cdots\!07}{13\!\cdots\!24}a^{8}+\frac{40\!\cdots\!60}{39\!\cdots\!39}a^{7}-\frac{66\!\cdots\!03}{15\!\cdots\!56}a^{6}-\frac{33\!\cdots\!75}{78\!\cdots\!78}a^{5}+\frac{41\!\cdots\!47}{15\!\cdots\!56}a^{4}+\frac{85\!\cdots\!27}{13\!\cdots\!24}a^{3}-\frac{48\!\cdots\!21}{15\!\cdots\!56}a^{2}+\frac{76\!\cdots\!85}{23\!\cdots\!67}a-\frac{37\!\cdots\!27}{39\!\cdots\!39}$
| 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: | \( 1783677.91642 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 1783677.91642 \cdot 1}{2\cdot\sqrt{36644198070556426025390625}}\cr\approx \mathstrut & 0.580153628567 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 12*x^15 + 36*x^14 - 33*x^12 + 117*x^11 - 171*x^10 - 2*x^9 + 135*x^8 - 81*x^7 - 30*x^6 + 36*x^5 + 81*x^4 - 393*x^3 + 531*x^2 - 270*x + 44)
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 - 12*x^15 + 36*x^14 - 33*x^12 + 117*x^11 - 171*x^10 - 2*x^9 + 135*x^8 - 81*x^7 - 30*x^6 + 36*x^5 + 81*x^4 - 393*x^3 + 531*x^2 - 270*x + 44, 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 - 12*x^15 + 36*x^14 - 33*x^12 + 117*x^11 - 171*x^10 - 2*x^9 + 135*x^8 - 81*x^7 - 30*x^6 + 36*x^5 + 81*x^4 - 393*x^3 + 531*x^2 - 270*x + 44);
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 - 12*x^15 + 36*x^14 - 33*x^12 + 117*x^11 - 171*x^10 - 2*x^9 + 135*x^8 - 81*x^7 - 30*x^6 + 36*x^5 + 81*x^4 - 393*x^3 + 531*x^2 - 270*x + 44);
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
$\He_3:C_4$ (as 18T49):
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 14 conjugacy class representatives for $\He_3:C_4$ |
| Character table for $\He_3:C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 6.2.4100625.1 |
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 sibling: | data not computed |
| Degree 27 sibling: | data not computed |
| 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 | ${\href{/padicField/2.4.0.1}{4} }^{3}{,}\,{\href{/padicField/2.2.0.1}{2} }^{3}$ | R | R | ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{5}{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.3.0.1}{3} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $18$ | $9$ | $2$ | $36$ | |||
|
\(5\)
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 5.12.9.2 | $x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |