Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 33*x^16 + 106*x^15 + 365*x^14 - 1333*x^13 - 1544*x^12 + 7717*x^11 + 893*x^10 - 21464*x^9 + 9947*x^8 + 25981*x^7 - 22611*x^6 - 8420*x^5 + 13270*x^4 - 1416*x^3 - 1968*x^2 + 512*x - 4)
gp: K = bnfinit(y^18 - 3*y^17 - 33*y^16 + 106*y^15 + 365*y^14 - 1333*y^13 - 1544*y^12 + 7717*y^11 + 893*y^10 - 21464*y^9 + 9947*y^8 + 25981*y^7 - 22611*y^6 - 8420*y^5 + 13270*y^4 - 1416*y^3 - 1968*y^2 + 512*y - 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 - 33*x^16 + 106*x^15 + 365*x^14 - 1333*x^13 - 1544*x^12 + 7717*x^11 + 893*x^10 - 21464*x^9 + 9947*x^8 + 25981*x^7 - 22611*x^6 - 8420*x^5 + 13270*x^4 - 1416*x^3 - 1968*x^2 + 512*x - 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 - 33*x^16 + 106*x^15 + 365*x^14 - 1333*x^13 - 1544*x^12 + 7717*x^11 + 893*x^10 - 21464*x^9 + 9947*x^8 + 25981*x^7 - 22611*x^6 - 8420*x^5 + 13270*x^4 - 1416*x^3 - 1968*x^2 + 512*x - 4)
\( x^{18} - 3 x^{17} - 33 x^{16} + 106 x^{15} + 365 x^{14} - 1333 x^{13} - 1544 x^{12} + 7717 x^{11} + \cdots - 4 \)
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: | $[18, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(4031470892249635921400320000\)
\(\medspace = 2^{12}\cdot 5^{4}\cdot 37^{9}\cdot 59^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(34.17\) | 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: | $2^{2/3}5^{2/3}37^{1/2}59^{2/3}\approx 427.8933494155845$ | ||
| Ramified primes: |
\(2\), \(5\), \(37\), \(59\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{37}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{3}$, $\frac{1}{55796}a^{16}-\frac{2895}{13949}a^{15}+\frac{10253}{55796}a^{14}+\frac{6667}{55796}a^{13}-\frac{385}{27898}a^{12}-\frac{4441}{55796}a^{11}-\frac{403}{4292}a^{10}+\frac{3971}{13949}a^{9}+\frac{6575}{55796}a^{8}-\frac{4643}{55796}a^{7}-\frac{317}{962}a^{6}+\frac{1413}{4292}a^{5}+\frac{3602}{13949}a^{4}-\frac{2736}{13949}a^{3}-\frac{6181}{27898}a^{2}-\frac{6730}{13949}a+\frac{4634}{13949}$, $\frac{1}{24\!\cdots\!56}a^{17}+\frac{4652782105583}{64\!\cdots\!62}a^{16}+\frac{15\!\cdots\!95}{24\!\cdots\!56}a^{15}+\frac{51\!\cdots\!29}{24\!\cdots\!56}a^{14}-\frac{15\!\cdots\!55}{61\!\cdots\!39}a^{13}+\frac{46\!\cdots\!63}{66\!\cdots\!88}a^{12}-\frac{86\!\cdots\!29}{24\!\cdots\!56}a^{11}-\frac{31\!\cdots\!63}{12\!\cdots\!78}a^{10}+\frac{88\!\cdots\!69}{10\!\cdots\!72}a^{9}-\frac{95\!\cdots\!93}{24\!\cdots\!56}a^{8}+\frac{26\!\cdots\!70}{61\!\cdots\!39}a^{7}-\frac{53\!\cdots\!91}{24\!\cdots\!56}a^{6}-\frac{40\!\cdots\!19}{12\!\cdots\!78}a^{5}+\frac{29\!\cdots\!00}{61\!\cdots\!39}a^{4}-\frac{25\!\cdots\!77}{61\!\cdots\!39}a^{3}+\frac{22\!\cdots\!05}{61\!\cdots\!39}a^{2}+\frac{13\!\cdots\!74}{61\!\cdots\!39}a-\frac{26\!\cdots\!15}{61\!\cdots\!39}$
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: | $17$ | 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: |
$\frac{70\!\cdots\!29}{61\!\cdots\!39}a^{17}-\frac{33\!\cdots\!13}{12\!\cdots\!24}a^{16}-\frac{24\!\cdots\!15}{61\!\cdots\!39}a^{15}+\frac{22\!\cdots\!55}{24\!\cdots\!56}a^{14}+\frac{11\!\cdots\!81}{24\!\cdots\!56}a^{13}-\frac{71\!\cdots\!26}{61\!\cdots\!39}a^{12}-\frac{63\!\cdots\!51}{24\!\cdots\!56}a^{11}+\frac{16\!\cdots\!95}{24\!\cdots\!56}a^{10}+\frac{30\!\cdots\!71}{53\!\cdots\!86}a^{9}-\frac{16\!\cdots\!23}{85\!\cdots\!64}a^{8}-\frac{54\!\cdots\!17}{24\!\cdots\!56}a^{7}+\frac{16\!\cdots\!84}{61\!\cdots\!39}a^{6}-\frac{18\!\cdots\!69}{24\!\cdots\!56}a^{5}-\frac{16\!\cdots\!91}{12\!\cdots\!78}a^{4}+\frac{76\!\cdots\!55}{12\!\cdots\!78}a^{3}+\frac{24\!\cdots\!95}{12\!\cdots\!78}a^{2}-\frac{61\!\cdots\!08}{61\!\cdots\!39}a+\frac{35\!\cdots\!70}{61\!\cdots\!39}$, $\frac{24\!\cdots\!54}{61\!\cdots\!39}a^{17}-\frac{63\!\cdots\!67}{12\!\cdots\!24}a^{16}-\frac{86\!\cdots\!06}{61\!\cdots\!39}a^{15}+\frac{42\!\cdots\!75}{24\!\cdots\!56}a^{14}+\frac{45\!\cdots\!45}{24\!\cdots\!56}a^{13}-\frac{13\!\cdots\!41}{61\!\cdots\!39}a^{12}-\frac{21\!\cdots\!39}{18\!\cdots\!12}a^{11}+\frac{31\!\cdots\!31}{24\!\cdots\!56}a^{10}+\frac{17\!\cdots\!73}{53\!\cdots\!86}a^{9}-\frac{92\!\cdots\!67}{24\!\cdots\!56}a^{8}-\frac{11\!\cdots\!81}{24\!\cdots\!56}a^{7}+\frac{24\!\cdots\!03}{47\!\cdots\!03}a^{6}+\frac{65\!\cdots\!63}{24\!\cdots\!56}a^{5}-\frac{18\!\cdots\!70}{61\!\cdots\!39}a^{4}-\frac{55\!\cdots\!49}{12\!\cdots\!78}a^{3}+\frac{70\!\cdots\!67}{12\!\cdots\!78}a^{2}-\frac{58\!\cdots\!18}{61\!\cdots\!39}a-\frac{11\!\cdots\!42}{47\!\cdots\!03}$, $\frac{71\!\cdots\!23}{12\!\cdots\!78}a^{17}-\frac{18\!\cdots\!75}{12\!\cdots\!24}a^{16}-\frac{12\!\cdots\!94}{61\!\cdots\!39}a^{15}+\frac{12\!\cdots\!23}{24\!\cdots\!56}a^{14}+\frac{58\!\cdots\!01}{24\!\cdots\!56}a^{13}-\frac{77\!\cdots\!75}{12\!\cdots\!78}a^{12}-\frac{29\!\cdots\!99}{24\!\cdots\!56}a^{11}+\frac{91\!\cdots\!43}{24\!\cdots\!56}a^{10}+\frac{63\!\cdots\!78}{26\!\cdots\!93}a^{9}-\frac{26\!\cdots\!71}{24\!\cdots\!56}a^{8}+\frac{94\!\cdots\!99}{24\!\cdots\!56}a^{7}+\frac{17\!\cdots\!37}{12\!\cdots\!78}a^{6}-\frac{14\!\cdots\!55}{24\!\cdots\!56}a^{5}-\frac{44\!\cdots\!44}{61\!\cdots\!39}a^{4}+\frac{25\!\cdots\!08}{61\!\cdots\!39}a^{3}+\frac{13\!\cdots\!41}{12\!\cdots\!78}a^{2}-\frac{38\!\cdots\!20}{61\!\cdots\!39}a-\frac{24\!\cdots\!72}{61\!\cdots\!39}$, $\frac{56\!\cdots\!13}{12\!\cdots\!78}a^{17}-\frac{10\!\cdots\!13}{12\!\cdots\!24}a^{16}-\frac{20\!\cdots\!45}{12\!\cdots\!78}a^{15}+\frac{69\!\cdots\!33}{24\!\cdots\!56}a^{14}+\frac{51\!\cdots\!93}{24\!\cdots\!56}a^{13}-\frac{43\!\cdots\!09}{12\!\cdots\!78}a^{12}-\frac{10\!\cdots\!29}{85\!\cdots\!64}a^{11}+\frac{51\!\cdots\!03}{24\!\cdots\!56}a^{10}+\frac{92\!\cdots\!10}{26\!\cdots\!93}a^{9}-\frac{11\!\cdots\!13}{18\!\cdots\!12}a^{8}-\frac{10\!\cdots\!97}{24\!\cdots\!56}a^{7}+\frac{10\!\cdots\!93}{12\!\cdots\!78}a^{6}+\frac{39\!\cdots\!03}{24\!\cdots\!56}a^{5}-\frac{53\!\cdots\!79}{12\!\cdots\!78}a^{4}-\frac{63\!\cdots\!51}{12\!\cdots\!78}a^{3}+\frac{83\!\cdots\!45}{12\!\cdots\!78}a^{2}-\frac{15\!\cdots\!31}{61\!\cdots\!39}a-\frac{77\!\cdots\!98}{61\!\cdots\!39}$, $\frac{17\!\cdots\!43}{61\!\cdots\!39}a^{17}-\frac{58\!\cdots\!21}{64\!\cdots\!62}a^{16}-\frac{12\!\cdots\!15}{12\!\cdots\!78}a^{15}+\frac{19\!\cdots\!94}{61\!\cdots\!39}a^{14}+\frac{61\!\cdots\!01}{42\!\cdots\!82}a^{13}-\frac{21\!\cdots\!57}{61\!\cdots\!39}a^{12}-\frac{59\!\cdots\!49}{61\!\cdots\!39}a^{11}+\frac{22\!\cdots\!41}{12\!\cdots\!78}a^{10}+\frac{91\!\cdots\!39}{26\!\cdots\!93}a^{9}-\frac{34\!\cdots\!85}{61\!\cdots\!39}a^{8}-\frac{60\!\cdots\!87}{94\!\cdots\!06}a^{7}+\frac{60\!\cdots\!20}{61\!\cdots\!39}a^{6}+\frac{37\!\cdots\!70}{61\!\cdots\!39}a^{5}-\frac{55\!\cdots\!58}{61\!\cdots\!39}a^{4}-\frac{31\!\cdots\!39}{12\!\cdots\!78}a^{3}+\frac{22\!\cdots\!29}{61\!\cdots\!39}a^{2}+\frac{20\!\cdots\!39}{61\!\cdots\!39}a-\frac{28\!\cdots\!86}{61\!\cdots\!39}$, $\frac{13\!\cdots\!47}{12\!\cdots\!24}a^{17}-\frac{29\!\cdots\!91}{12\!\cdots\!24}a^{16}-\frac{49\!\cdots\!11}{12\!\cdots\!24}a^{15}+\frac{26\!\cdots\!24}{32\!\cdots\!81}a^{14}+\frac{64\!\cdots\!35}{12\!\cdots\!24}a^{13}-\frac{13\!\cdots\!35}{12\!\cdots\!24}a^{12}-\frac{98\!\cdots\!39}{32\!\cdots\!81}a^{11}+\frac{21\!\cdots\!87}{35\!\cdots\!52}a^{10}+\frac{51\!\cdots\!21}{56\!\cdots\!88}a^{9}-\frac{44\!\cdots\!30}{24\!\cdots\!37}a^{8}-\frac{16\!\cdots\!03}{12\!\cdots\!24}a^{7}+\frac{88\!\cdots\!91}{35\!\cdots\!52}a^{6}+\frac{82\!\cdots\!03}{12\!\cdots\!24}a^{5}-\frac{97\!\cdots\!71}{64\!\cdots\!62}a^{4}-\frac{90\!\cdots\!71}{17\!\cdots\!26}a^{3}+\frac{20\!\cdots\!81}{64\!\cdots\!62}a^{2}-\frac{46\!\cdots\!26}{32\!\cdots\!81}a-\frac{65\!\cdots\!98}{32\!\cdots\!81}$, $\frac{83\!\cdots\!37}{12\!\cdots\!78}a^{17}-\frac{79\!\cdots\!17}{64\!\cdots\!62}a^{16}-\frac{29\!\cdots\!25}{12\!\cdots\!78}a^{15}+\frac{26\!\cdots\!67}{61\!\cdots\!39}a^{14}+\frac{36\!\cdots\!83}{12\!\cdots\!78}a^{13}-\frac{33\!\cdots\!13}{61\!\cdots\!39}a^{12}-\frac{10\!\cdots\!12}{61\!\cdots\!39}a^{11}+\frac{10\!\cdots\!61}{33\!\cdots\!94}a^{10}+\frac{11\!\cdots\!05}{26\!\cdots\!93}a^{9}-\frac{56\!\cdots\!47}{61\!\cdots\!39}a^{8}-\frac{50\!\cdots\!35}{12\!\cdots\!78}a^{7}+\frac{20\!\cdots\!12}{16\!\cdots\!47}a^{6}-\frac{90\!\cdots\!23}{12\!\cdots\!78}a^{5}-\frac{37\!\cdots\!20}{61\!\cdots\!39}a^{4}+\frac{56\!\cdots\!25}{33\!\cdots\!94}a^{3}+\frac{51\!\cdots\!39}{61\!\cdots\!39}a^{2}-\frac{18\!\cdots\!97}{61\!\cdots\!39}a-\frac{14\!\cdots\!58}{61\!\cdots\!39}$, $\frac{17\!\cdots\!83}{24\!\cdots\!56}a^{17}-\frac{77\!\cdots\!61}{64\!\cdots\!62}a^{16}-\frac{60\!\cdots\!93}{24\!\cdots\!56}a^{15}+\frac{10\!\cdots\!33}{24\!\cdots\!56}a^{14}+\frac{19\!\cdots\!66}{61\!\cdots\!39}a^{13}-\frac{13\!\cdots\!17}{24\!\cdots\!56}a^{12}-\frac{43\!\cdots\!33}{24\!\cdots\!56}a^{11}+\frac{38\!\cdots\!27}{12\!\cdots\!78}a^{10}+\frac{50\!\cdots\!45}{10\!\cdots\!72}a^{9}-\frac{21\!\cdots\!57}{24\!\cdots\!56}a^{8}-\frac{30\!\cdots\!92}{61\!\cdots\!39}a^{7}+\frac{28\!\cdots\!25}{24\!\cdots\!56}a^{6}+\frac{16\!\cdots\!22}{61\!\cdots\!39}a^{5}-\frac{33\!\cdots\!93}{61\!\cdots\!39}a^{4}+\frac{65\!\cdots\!30}{61\!\cdots\!39}a^{3}+\frac{37\!\cdots\!57}{61\!\cdots\!39}a^{2}-\frac{30\!\cdots\!17}{16\!\cdots\!47}a+\frac{49\!\cdots\!48}{61\!\cdots\!39}$, $\frac{67\!\cdots\!51}{24\!\cdots\!56}a^{17}-\frac{49\!\cdots\!47}{12\!\cdots\!24}a^{16}-\frac{24\!\cdots\!91}{24\!\cdots\!56}a^{15}+\frac{12\!\cdots\!39}{94\!\cdots\!06}a^{14}+\frac{31\!\cdots\!87}{24\!\cdots\!56}a^{13}-\frac{31\!\cdots\!55}{18\!\cdots\!12}a^{12}-\frac{96\!\cdots\!37}{12\!\cdots\!78}a^{11}+\frac{24\!\cdots\!67}{24\!\cdots\!56}a^{10}+\frac{25\!\cdots\!49}{10\!\cdots\!72}a^{9}-\frac{34\!\cdots\!81}{12\!\cdots\!78}a^{8}-\frac{89\!\cdots\!19}{24\!\cdots\!56}a^{7}+\frac{93\!\cdots\!99}{24\!\cdots\!56}a^{6}+\frac{63\!\cdots\!77}{24\!\cdots\!56}a^{5}-\frac{90\!\cdots\!99}{47\!\cdots\!03}a^{4}-\frac{10\!\cdots\!15}{94\!\cdots\!06}a^{3}+\frac{35\!\cdots\!85}{12\!\cdots\!78}a^{2}+\frac{15\!\cdots\!53}{61\!\cdots\!39}a-\frac{12\!\cdots\!63}{61\!\cdots\!39}$, $\frac{19\!\cdots\!63}{61\!\cdots\!39}a^{17}-\frac{39\!\cdots\!49}{44\!\cdots\!56}a^{16}-\frac{13\!\cdots\!93}{12\!\cdots\!78}a^{15}+\frac{77\!\cdots\!69}{24\!\cdots\!56}a^{14}+\frac{23\!\cdots\!21}{18\!\cdots\!12}a^{13}-\frac{49\!\cdots\!57}{12\!\cdots\!78}a^{12}-\frac{15\!\cdots\!37}{24\!\cdots\!56}a^{11}+\frac{57\!\cdots\!79}{24\!\cdots\!56}a^{10}+\frac{22\!\cdots\!00}{20\!\cdots\!61}a^{9}-\frac{16\!\cdots\!89}{24\!\cdots\!56}a^{8}+\frac{18\!\cdots\!03}{24\!\cdots\!56}a^{7}+\frac{10\!\cdots\!61}{12\!\cdots\!78}a^{6}-\frac{25\!\cdots\!67}{66\!\cdots\!88}a^{5}-\frac{49\!\cdots\!41}{12\!\cdots\!78}a^{4}+\frac{30\!\cdots\!23}{12\!\cdots\!78}a^{3}+\frac{57\!\cdots\!55}{12\!\cdots\!78}a^{2}-\frac{17\!\cdots\!61}{47\!\cdots\!03}a+\frac{72\!\cdots\!67}{61\!\cdots\!39}$, $\frac{36\!\cdots\!36}{16\!\cdots\!47}a^{17}-\frac{83\!\cdots\!87}{12\!\cdots\!24}a^{16}-\frac{90\!\cdots\!89}{12\!\cdots\!78}a^{15}+\frac{56\!\cdots\!03}{24\!\cdots\!56}a^{14}+\frac{19\!\cdots\!19}{24\!\cdots\!56}a^{13}-\frac{17\!\cdots\!34}{61\!\cdots\!39}a^{12}-\frac{63\!\cdots\!47}{18\!\cdots\!12}a^{11}+\frac{41\!\cdots\!45}{24\!\cdots\!56}a^{10}+\frac{80\!\cdots\!85}{53\!\cdots\!86}a^{9}-\frac{11\!\cdots\!43}{24\!\cdots\!56}a^{8}+\frac{58\!\cdots\!49}{24\!\cdots\!56}a^{7}+\frac{29\!\cdots\!04}{47\!\cdots\!03}a^{6}-\frac{13\!\cdots\!97}{24\!\cdots\!56}a^{5}-\frac{34\!\cdots\!63}{12\!\cdots\!78}a^{4}+\frac{19\!\cdots\!70}{61\!\cdots\!39}a^{3}+\frac{36\!\cdots\!27}{12\!\cdots\!78}a^{2}-\frac{97\!\cdots\!51}{21\!\cdots\!91}a+\frac{14\!\cdots\!10}{47\!\cdots\!03}$, $\frac{11\!\cdots\!64}{61\!\cdots\!39}a^{17}-\frac{920639486243681}{12\!\cdots\!24}a^{16}-\frac{42\!\cdots\!29}{61\!\cdots\!39}a^{15}+\frac{55\!\cdots\!69}{66\!\cdots\!88}a^{14}+\frac{23\!\cdots\!85}{24\!\cdots\!56}a^{13}+\frac{55\!\cdots\!09}{12\!\cdots\!78}a^{12}-\frac{15\!\cdots\!29}{24\!\cdots\!56}a^{11}-\frac{14\!\cdots\!13}{24\!\cdots\!56}a^{10}+\frac{56\!\cdots\!34}{26\!\cdots\!93}a^{9}+\frac{63\!\cdots\!59}{24\!\cdots\!56}a^{8}-\frac{93\!\cdots\!09}{24\!\cdots\!56}a^{7}-\frac{62\!\cdots\!77}{12\!\cdots\!78}a^{6}+\frac{87\!\cdots\!61}{24\!\cdots\!56}a^{5}+\frac{50\!\cdots\!71}{12\!\cdots\!78}a^{4}-\frac{99\!\cdots\!02}{61\!\cdots\!39}a^{3}+\frac{18\!\cdots\!21}{12\!\cdots\!78}a^{2}+\frac{44\!\cdots\!41}{21\!\cdots\!91}a-\frac{19\!\cdots\!80}{61\!\cdots\!39}$, $\frac{22\!\cdots\!11}{24\!\cdots\!56}a^{17}-\frac{22\!\cdots\!53}{12\!\cdots\!24}a^{16}-\frac{79\!\cdots\!55}{24\!\cdots\!56}a^{15}+\frac{75\!\cdots\!23}{12\!\cdots\!78}a^{14}+\frac{34\!\cdots\!27}{85\!\cdots\!64}a^{13}-\frac{19\!\cdots\!55}{24\!\cdots\!56}a^{12}-\frac{28\!\cdots\!67}{12\!\cdots\!78}a^{11}+\frac{11\!\cdots\!99}{24\!\cdots\!56}a^{10}+\frac{63\!\cdots\!37}{10\!\cdots\!72}a^{9}-\frac{16\!\cdots\!69}{12\!\cdots\!78}a^{8}-\frac{13\!\cdots\!27}{24\!\cdots\!56}a^{7}+\frac{45\!\cdots\!35}{24\!\cdots\!56}a^{6}-\frac{29\!\cdots\!19}{24\!\cdots\!56}a^{5}-\frac{11\!\cdots\!79}{12\!\cdots\!78}a^{4}+\frac{32\!\cdots\!11}{12\!\cdots\!78}a^{3}+\frac{18\!\cdots\!79}{12\!\cdots\!78}a^{2}-\frac{34\!\cdots\!85}{61\!\cdots\!39}a+\frac{14\!\cdots\!21}{61\!\cdots\!39}$, $\frac{20\!\cdots\!31}{12\!\cdots\!24}a^{17}-\frac{21\!\cdots\!93}{64\!\cdots\!62}a^{16}-\frac{71\!\cdots\!89}{12\!\cdots\!24}a^{15}+\frac{14\!\cdots\!79}{12\!\cdots\!24}a^{14}+\frac{43\!\cdots\!79}{64\!\cdots\!62}a^{13}-\frac{18\!\cdots\!55}{12\!\cdots\!24}a^{12}-\frac{37\!\cdots\!31}{99\!\cdots\!48}a^{11}+\frac{27\!\cdots\!19}{32\!\cdots\!81}a^{10}+\frac{50\!\cdots\!67}{56\!\cdots\!88}a^{9}-\frac{32\!\cdots\!51}{12\!\cdots\!24}a^{8}-\frac{37\!\cdots\!63}{64\!\cdots\!62}a^{7}+\frac{33\!\cdots\!99}{99\!\cdots\!48}a^{6}-\frac{45\!\cdots\!77}{64\!\cdots\!62}a^{5}-\frac{10\!\cdots\!51}{64\!\cdots\!62}a^{4}+\frac{45\!\cdots\!47}{64\!\cdots\!62}a^{3}+\frac{74\!\cdots\!44}{32\!\cdots\!81}a^{2}-\frac{39\!\cdots\!67}{32\!\cdots\!81}a+\frac{27\!\cdots\!07}{24\!\cdots\!37}$, $\frac{17\!\cdots\!07}{24\!\cdots\!56}a^{17}-\frac{16\!\cdots\!39}{12\!\cdots\!24}a^{16}-\frac{61\!\cdots\!03}{24\!\cdots\!56}a^{15}+\frac{56\!\cdots\!98}{12\!\cdots\!19}a^{14}+\frac{76\!\cdots\!05}{24\!\cdots\!56}a^{13}-\frac{10\!\cdots\!51}{18\!\cdots\!12}a^{12}-\frac{10\!\cdots\!26}{61\!\cdots\!39}a^{11}+\frac{79\!\cdots\!89}{24\!\cdots\!56}a^{10}+\frac{49\!\cdots\!65}{10\!\cdots\!72}a^{9}-\frac{57\!\cdots\!64}{61\!\cdots\!39}a^{8}-\frac{10\!\cdots\!57}{24\!\cdots\!56}a^{7}+\frac{30\!\cdots\!95}{24\!\cdots\!56}a^{6}-\frac{12\!\cdots\!37}{24\!\cdots\!56}a^{5}-\frac{57\!\cdots\!23}{94\!\cdots\!06}a^{4}+\frac{14\!\cdots\!35}{94\!\cdots\!06}a^{3}+\frac{91\!\cdots\!61}{12\!\cdots\!78}a^{2}-\frac{15\!\cdots\!16}{61\!\cdots\!39}a+\frac{36\!\cdots\!57}{61\!\cdots\!39}$, $\frac{178573692240358}{61\!\cdots\!39}a^{17}-\frac{16\!\cdots\!53}{12\!\cdots\!24}a^{16}+\frac{16\!\cdots\!27}{61\!\cdots\!39}a^{15}+\frac{11\!\cdots\!93}{24\!\cdots\!56}a^{14}-\frac{23\!\cdots\!69}{24\!\cdots\!56}a^{13}-\frac{99\!\cdots\!57}{16\!\cdots\!47}a^{12}+\frac{29\!\cdots\!75}{24\!\cdots\!56}a^{11}+\frac{89\!\cdots\!09}{24\!\cdots\!56}a^{10}-\frac{37\!\cdots\!63}{53\!\cdots\!86}a^{9}-\frac{26\!\cdots\!01}{24\!\cdots\!56}a^{8}+\frac{48\!\cdots\!21}{24\!\cdots\!56}a^{7}+\frac{95\!\cdots\!89}{61\!\cdots\!39}a^{6}-\frac{62\!\cdots\!51}{24\!\cdots\!56}a^{5}-\frac{54\!\cdots\!24}{61\!\cdots\!39}a^{4}+\frac{15\!\cdots\!91}{12\!\cdots\!78}a^{3}+\frac{21\!\cdots\!43}{12\!\cdots\!78}a^{2}-\frac{33\!\cdots\!85}{21\!\cdots\!91}a+\frac{18\!\cdots\!98}{61\!\cdots\!39}$, $\frac{18\!\cdots\!35}{85\!\cdots\!64}a^{17}-\frac{17\!\cdots\!69}{44\!\cdots\!56}a^{16}-\frac{64\!\cdots\!61}{85\!\cdots\!64}a^{15}+\frac{29\!\cdots\!83}{21\!\cdots\!91}a^{14}+\frac{81\!\cdots\!75}{85\!\cdots\!64}a^{13}-\frac{14\!\cdots\!85}{85\!\cdots\!64}a^{12}-\frac{11\!\cdots\!67}{21\!\cdots\!91}a^{11}+\frac{87\!\cdots\!23}{85\!\cdots\!64}a^{10}+\frac{52\!\cdots\!87}{37\!\cdots\!68}a^{9}-\frac{63\!\cdots\!08}{21\!\cdots\!91}a^{8}-\frac{11\!\cdots\!39}{85\!\cdots\!64}a^{7}+\frac{33\!\cdots\!65}{85\!\cdots\!64}a^{6}-\frac{11\!\cdots\!53}{85\!\cdots\!64}a^{5}-\frac{84\!\cdots\!23}{42\!\cdots\!82}a^{4}+\frac{20\!\cdots\!75}{42\!\cdots\!82}a^{3}+\frac{11\!\cdots\!33}{42\!\cdots\!82}a^{2}-\frac{20\!\cdots\!26}{21\!\cdots\!91}a+\frac{70\!\cdots\!75}{21\!\cdots\!91}$
| 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: | \( 156226776.16 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 156226776.16 \cdot 1}{2\cdot\sqrt{4031470892249635921400320000}}\cr\approx \mathstrut & 0.32250290647 \end{aligned}\] (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 + 106*x^15 + 365*x^14 - 1333*x^13 - 1544*x^12 + 7717*x^11 + 893*x^10 - 21464*x^9 + 9947*x^8 + 25981*x^7 - 22611*x^6 - 8420*x^5 + 13270*x^4 - 1416*x^3 - 1968*x^2 + 512*x - 4)
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 + 106*x^15 + 365*x^14 - 1333*x^13 - 1544*x^12 + 7717*x^11 + 893*x^10 - 21464*x^9 + 9947*x^8 + 25981*x^7 - 22611*x^6 - 8420*x^5 + 13270*x^4 - 1416*x^3 - 1968*x^2 + 512*x - 4, 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 + 106*x^15 + 365*x^14 - 1333*x^13 - 1544*x^12 + 7717*x^11 + 893*x^10 - 21464*x^9 + 9947*x^8 + 25981*x^7 - 22611*x^6 - 8420*x^5 + 13270*x^4 - 1416*x^3 - 1968*x^2 + 512*x - 4);
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 + 106*x^15 + 365*x^14 - 1333*x^13 - 1544*x^12 + 7717*x^11 + 893*x^10 - 21464*x^9 + 9947*x^8 + 25981*x^7 - 22611*x^6 - 8420*x^5 + 13270*x^4 - 1416*x^3 - 1968*x^2 + 512*x - 4);
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
$C_3^3:S_3$ (as 18T88):
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 162 |
| The 13 conjugacy class representatives for $C_3^3:S_3$ |
| Character table for $C_3^3:S_3$ |
Intermediate fields
| \(\Q(\sqrt{37}) \), 3.3.148.1 x3, 6.6.810448.1, 9.9.10438327105600.1 x3 |
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 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 siblings: | data not computed |
| Minimal sibling: | 9.9.10438327105600.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.9.0.1}{9} }^{2}$ | R | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.9.0.1}{9} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{9}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.2.0.1}{2} }^{9}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/41.3.0.1}{3} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{6}$ | ${\href{/padicField/53.9.0.1}{9} }^{2}$ | R |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 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 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(5\)
| 5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(37\)
| 37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 37.6.3.1 | $x^{6} + 2775 x^{5} + 2566998 x^{4} + 791680745 x^{3} + 110476893 x^{2} + 4842384300 x + 27887445532$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 37.6.3.1 | $x^{6} + 2775 x^{5} + 2566998 x^{4} + 791680745 x^{3} + 110476893 x^{2} + 4842384300 x + 27887445532$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(59\)
| 59.6.4.1 | $x^{6} + 174 x^{5} + 10098 x^{4} + 195926 x^{3} + 30462 x^{2} + 595416 x + 11494565$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 59.6.0.1 | $x^{6} + 2 x^{4} + 18 x^{3} + 38 x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 59.6.0.1 | $x^{6} + 2 x^{4} + 18 x^{3} + 38 x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |