Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 39*x^16 + 124*x^15 + 588*x^14 - 2016*x^13 - 4354*x^12 + 16662*x^11 + 16569*x^10 - 76017*x^9 - 29151*x^8 + 193350*x^7 + 7892*x^6 - 257946*x^5 + 38376*x^4 + 147342*x^3 - 33795*x^2 - 14013*x + 1863)
gp: K = bnfinit(y^18 - 3*y^17 - 39*y^16 + 124*y^15 + 588*y^14 - 2016*y^13 - 4354*y^12 + 16662*y^11 + 16569*y^10 - 76017*y^9 - 29151*y^8 + 193350*y^7 + 7892*y^6 - 257946*y^5 + 38376*y^4 + 147342*y^3 - 33795*y^2 - 14013*y + 1863, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 - 39*x^16 + 124*x^15 + 588*x^14 - 2016*x^13 - 4354*x^12 + 16662*x^11 + 16569*x^10 - 76017*x^9 - 29151*x^8 + 193350*x^7 + 7892*x^6 - 257946*x^5 + 38376*x^4 + 147342*x^3 - 33795*x^2 - 14013*x + 1863);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 - 39*x^16 + 124*x^15 + 588*x^14 - 2016*x^13 - 4354*x^12 + 16662*x^11 + 16569*x^10 - 76017*x^9 - 29151*x^8 + 193350*x^7 + 7892*x^6 - 257946*x^5 + 38376*x^4 + 147342*x^3 - 33795*x^2 - 14013*x + 1863)
\( x^{18} - 3 x^{17} - 39 x^{16} + 124 x^{15} + 588 x^{14} - 2016 x^{13} - 4354 x^{12} + 16662 x^{11} + \cdots + 1863 \)
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: |
\(2096291596176378856933557232324608\)
\(\medspace = 2^{12}\cdot 3^{20}\cdot 7^{12}\cdot 13^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(70.99\) | 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: | $2^{2/3}3^{4/3}7^{2/3}13^{1/2}\approx 90.61888333825345$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{8}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{9}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{12}a^{15}+\frac{1}{12}a^{12}+\frac{1}{6}a^{9}+\frac{5}{12}a^{3}+\frac{1}{4}$, $\frac{1}{72}a^{16}+\frac{1}{12}a^{14}-\frac{1}{36}a^{13}+\frac{1}{12}a^{12}+\frac{1}{36}a^{10}-\frac{1}{8}a^{8}-\frac{1}{6}a^{7}-\frac{1}{4}a^{6}+\frac{1}{6}a^{5}+\frac{13}{36}a^{4}-\frac{1}{2}a^{3}-\frac{1}{12}a+\frac{1}{8}$, $\frac{1}{19\!\cdots\!84}a^{17}+\frac{14\!\cdots\!07}{64\!\cdots\!28}a^{16}-\frac{31\!\cdots\!21}{16\!\cdots\!82}a^{15}+\frac{37\!\cdots\!19}{96\!\cdots\!92}a^{14}-\frac{19\!\cdots\!49}{16\!\cdots\!82}a^{13}+\frac{50\!\cdots\!25}{10\!\cdots\!88}a^{12}-\frac{11\!\cdots\!23}{96\!\cdots\!92}a^{11}-\frac{84\!\cdots\!89}{32\!\cdots\!64}a^{10}-\frac{52\!\cdots\!49}{21\!\cdots\!76}a^{9}+\frac{11\!\cdots\!79}{64\!\cdots\!28}a^{8}+\frac{20\!\cdots\!71}{10\!\cdots\!88}a^{7}+\frac{43\!\cdots\!45}{32\!\cdots\!64}a^{6}-\frac{59\!\cdots\!37}{96\!\cdots\!92}a^{5}+\frac{26\!\cdots\!65}{16\!\cdots\!82}a^{4}-\frac{37\!\cdots\!87}{10\!\cdots\!88}a^{3}+\frac{80\!\cdots\!13}{16\!\cdots\!82}a^{2}-\frac{313763447725963}{11\!\cdots\!44}a-\frac{15\!\cdots\!37}{31\!\cdots\!04}$
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
$C_{3}$, which has order $3$ (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: |
$\frac{6737967075}{8685009358988}a^{17}+\frac{6202869309}{8685009358988}a^{16}-\frac{145474172189}{4342504679494}a^{15}-\frac{204488598321}{8685009358988}a^{14}+\frac{5156261449371}{8685009358988}a^{13}+\frac{2617406696103}{8685009358988}a^{12}-\frac{48405458157195}{8685009358988}a^{11}-\frac{4185148870536}{2171252339747}a^{10}+\frac{258662412595351}{8685009358988}a^{9}+\frac{14474745492411}{2171252339747}a^{8}-\frac{783960745066491}{8685009358988}a^{7}-\frac{27954457546014}{2171252339747}a^{6}+\frac{313567557678387}{2171252339747}a^{5}+\frac{121573193507775}{8685009358988}a^{4}-\frac{865917464919273}{8685009358988}a^{3}-\frac{67120822159371}{8685009358988}a^{2}+\frac{29302200032031}{2171252339747}a-\frac{8821295561815}{8685009358988}$, $\frac{13\!\cdots\!09}{96\!\cdots\!92}a^{17}-\frac{10\!\cdots\!57}{32\!\cdots\!64}a^{16}-\frac{84\!\cdots\!93}{16\!\cdots\!82}a^{15}+\frac{12\!\cdots\!87}{96\!\cdots\!92}a^{14}+\frac{23\!\cdots\!23}{32\!\cdots\!64}a^{13}-\frac{63\!\cdots\!85}{35\!\cdots\!96}a^{12}-\frac{50\!\cdots\!41}{96\!\cdots\!92}a^{11}+\frac{37\!\cdots\!99}{32\!\cdots\!64}a^{10}+\frac{35\!\cdots\!09}{17\!\cdots\!98}a^{9}-\frac{11\!\cdots\!69}{32\!\cdots\!64}a^{8}-\frac{52\!\cdots\!51}{10\!\cdots\!88}a^{7}+\frac{13\!\cdots\!33}{32\!\cdots\!64}a^{6}+\frac{82\!\cdots\!85}{96\!\cdots\!92}a^{5}+\frac{11\!\cdots\!03}{80\!\cdots\!91}a^{4}-\frac{96\!\cdots\!59}{10\!\cdots\!88}a^{3}-\frac{40\!\cdots\!19}{16\!\cdots\!82}a^{2}+\frac{19\!\cdots\!01}{599265645770172}a-\frac{26\!\cdots\!29}{777308337629426}$, $\frac{260234207693683}{64\!\cdots\!28}a^{17}-\frac{11\!\cdots\!33}{32\!\cdots\!64}a^{16}-\frac{486729286643855}{10\!\cdots\!88}a^{15}+\frac{40\!\cdots\!71}{32\!\cdots\!64}a^{14}-\frac{59\!\cdots\!39}{32\!\cdots\!64}a^{13}-\frac{18\!\cdots\!31}{10\!\cdots\!88}a^{12}+\frac{68\!\cdots\!33}{16\!\cdots\!82}a^{11}+\frac{16\!\cdots\!31}{16\!\cdots\!82}a^{10}-\frac{24\!\cdots\!29}{71\!\cdots\!92}a^{9}-\frac{27\!\cdots\!13}{10\!\cdots\!88}a^{8}+\frac{70\!\cdots\!17}{53\!\cdots\!94}a^{7}+\frac{91\!\cdots\!63}{53\!\cdots\!94}a^{6}-\frac{19\!\cdots\!19}{80\!\cdots\!91}a^{5}+\frac{20\!\cdots\!53}{80\!\cdots\!91}a^{4}+\frac{61\!\cdots\!83}{35\!\cdots\!96}a^{3}-\frac{25\!\cdots\!89}{10\!\cdots\!88}a^{2}-\frac{19\!\cdots\!31}{11\!\cdots\!44}a+\frac{564793645625400}{388654168814713}$, $\frac{11\!\cdots\!75}{96\!\cdots\!92}a^{17}-\frac{538023823918745}{16\!\cdots\!82}a^{16}-\frac{73\!\cdots\!35}{16\!\cdots\!82}a^{15}+\frac{12\!\cdots\!33}{96\!\cdots\!92}a^{14}+\frac{20\!\cdots\!07}{32\!\cdots\!64}a^{13}-\frac{18\!\cdots\!78}{89\!\cdots\!99}a^{12}-\frac{43\!\cdots\!81}{96\!\cdots\!92}a^{11}+\frac{51\!\cdots\!13}{32\!\cdots\!64}a^{10}+\frac{58\!\cdots\!15}{35\!\cdots\!96}a^{9}-\frac{21\!\cdots\!43}{32\!\cdots\!64}a^{8}-\frac{32\!\cdots\!09}{10\!\cdots\!88}a^{7}+\frac{53\!\cdots\!29}{32\!\cdots\!64}a^{6}+\frac{11\!\cdots\!75}{48\!\cdots\!46}a^{5}-\frac{69\!\cdots\!61}{32\!\cdots\!64}a^{4}+\frac{70\!\cdots\!29}{10\!\cdots\!88}a^{3}+\frac{20\!\cdots\!01}{16\!\cdots\!82}a^{2}-\frac{20\!\cdots\!34}{149816411442543}a-\frac{21\!\cdots\!09}{15\!\cdots\!52}$, $\frac{64\!\cdots\!69}{19\!\cdots\!84}a^{17}-\frac{371226574833305}{10\!\cdots\!88}a^{16}-\frac{42\!\cdots\!07}{32\!\cdots\!64}a^{15}+\frac{13\!\cdots\!19}{96\!\cdots\!92}a^{14}+\frac{65\!\cdots\!77}{32\!\cdots\!64}a^{13}-\frac{21\!\cdots\!97}{10\!\cdots\!88}a^{12}-\frac{15\!\cdots\!51}{96\!\cdots\!92}a^{11}+\frac{76\!\cdots\!93}{53\!\cdots\!94}a^{10}+\frac{50\!\cdots\!79}{71\!\cdots\!92}a^{9}-\frac{39\!\cdots\!41}{80\!\cdots\!91}a^{8}-\frac{60\!\cdots\!45}{35\!\cdots\!96}a^{7}+\frac{67\!\cdots\!53}{80\!\cdots\!91}a^{6}+\frac{49\!\cdots\!97}{24\!\cdots\!73}a^{5}-\frac{70\!\cdots\!43}{10\!\cdots\!88}a^{4}-\frac{55\!\cdots\!79}{53\!\cdots\!94}a^{3}+\frac{68\!\cdots\!69}{32\!\cdots\!64}a^{2}+\frac{11\!\cdots\!29}{11\!\cdots\!44}a+\frac{231427784763951}{15\!\cdots\!52}$, $\frac{51\!\cdots\!39}{48\!\cdots\!46}a^{17}-\frac{21\!\cdots\!37}{64\!\cdots\!28}a^{16}-\frac{68\!\cdots\!47}{16\!\cdots\!82}a^{15}+\frac{42\!\cdots\!48}{24\!\cdots\!73}a^{14}+\frac{21\!\cdots\!47}{32\!\cdots\!64}a^{13}-\frac{33\!\cdots\!07}{10\!\cdots\!88}a^{12}-\frac{52\!\cdots\!19}{96\!\cdots\!92}a^{11}+\frac{20\!\cdots\!70}{80\!\cdots\!91}a^{10}+\frac{65\!\cdots\!25}{26\!\cdots\!97}a^{9}-\frac{70\!\cdots\!07}{64\!\cdots\!28}a^{8}-\frac{65\!\cdots\!51}{10\!\cdots\!88}a^{7}+\frac{20\!\cdots\!06}{80\!\cdots\!91}a^{6}+\frac{38\!\cdots\!47}{48\!\cdots\!46}a^{5}-\frac{91\!\cdots\!45}{32\!\cdots\!64}a^{4}-\frac{48\!\cdots\!89}{10\!\cdots\!88}a^{3}+\frac{20\!\cdots\!83}{16\!\cdots\!82}a^{2}+\frac{26\!\cdots\!77}{599265645770172}a-\frac{22\!\cdots\!91}{31\!\cdots\!04}$, $\frac{12\!\cdots\!15}{32\!\cdots\!64}a^{17}-\frac{300270456706715}{16\!\cdots\!82}a^{16}-\frac{84\!\cdots\!61}{53\!\cdots\!94}a^{15}+\frac{72\!\cdots\!76}{80\!\cdots\!91}a^{14}+\frac{38\!\cdots\!45}{16\!\cdots\!82}a^{13}-\frac{16\!\cdots\!77}{10\!\cdots\!88}a^{12}-\frac{15\!\cdots\!67}{80\!\cdots\!91}a^{11}+\frac{41\!\cdots\!43}{32\!\cdots\!64}a^{10}+\frac{87\!\cdots\!69}{10\!\cdots\!88}a^{9}-\frac{29\!\cdots\!17}{53\!\cdots\!94}a^{8}-\frac{51\!\cdots\!14}{26\!\cdots\!97}a^{7}+\frac{13\!\cdots\!21}{10\!\cdots\!88}a^{6}+\frac{18\!\cdots\!05}{80\!\cdots\!91}a^{5}-\frac{12\!\cdots\!02}{80\!\cdots\!91}a^{4}-\frac{59\!\cdots\!03}{53\!\cdots\!94}a^{3}+\frac{79\!\cdots\!93}{10\!\cdots\!88}a^{2}+\frac{43\!\cdots\!65}{599265645770172}a-\frac{10\!\cdots\!89}{15\!\cdots\!52}$, $\frac{195782039078113}{64\!\cdots\!28}a^{17}-\frac{27\!\cdots\!89}{64\!\cdots\!28}a^{16}-\frac{159362407244566}{26\!\cdots\!97}a^{15}+\frac{24\!\cdots\!63}{16\!\cdots\!82}a^{14}-\frac{74\!\cdots\!95}{16\!\cdots\!82}a^{13}-\frac{56\!\cdots\!08}{26\!\cdots\!97}a^{12}+\frac{58\!\cdots\!85}{32\!\cdots\!64}a^{11}+\frac{21\!\cdots\!61}{16\!\cdots\!82}a^{10}-\frac{31\!\cdots\!89}{21\!\cdots\!76}a^{9}-\frac{89\!\cdots\!27}{21\!\cdots\!76}a^{8}+\frac{50\!\cdots\!95}{10\!\cdots\!88}a^{7}+\frac{14\!\cdots\!83}{26\!\cdots\!97}a^{6}-\frac{81\!\cdots\!63}{16\!\cdots\!82}a^{5}-\frac{16\!\cdots\!75}{16\!\cdots\!82}a^{4}-\frac{27\!\cdots\!01}{10\!\cdots\!88}a^{3}-\frac{41\!\cdots\!12}{26\!\cdots\!97}a^{2}+\frac{63\!\cdots\!01}{11\!\cdots\!44}a-\frac{31\!\cdots\!67}{31\!\cdots\!04}$, $\frac{422346585891811}{48\!\cdots\!46}a^{17}-\frac{730521623968027}{16\!\cdots\!82}a^{16}-\frac{89\!\cdots\!73}{32\!\cdots\!64}a^{15}+\frac{40\!\cdots\!82}{24\!\cdots\!73}a^{14}+\frac{91\!\cdots\!45}{32\!\cdots\!64}a^{13}-\frac{64\!\cdots\!98}{26\!\cdots\!97}a^{12}-\frac{30\!\cdots\!17}{48\!\cdots\!46}a^{11}+\frac{25\!\cdots\!17}{16\!\cdots\!82}a^{10}-\frac{61\!\cdots\!31}{10\!\cdots\!88}a^{9}-\frac{87\!\cdots\!21}{16\!\cdots\!82}a^{8}+\frac{94\!\cdots\!01}{26\!\cdots\!97}a^{7}+\frac{70\!\cdots\!35}{80\!\cdots\!91}a^{6}-\frac{65\!\cdots\!89}{96\!\cdots\!92}a^{5}-\frac{83\!\cdots\!43}{16\!\cdots\!82}a^{4}+\frac{43\!\cdots\!33}{10\!\cdots\!88}a^{3}-\frac{40\!\cdots\!22}{80\!\cdots\!91}a^{2}+\frac{489062046848722}{149816411442543}a+\frac{44\!\cdots\!95}{777308337629426}$, $\frac{11\!\cdots\!61}{24\!\cdots\!73}a^{17}-\frac{706033946172671}{10\!\cdots\!88}a^{16}-\frac{62\!\cdots\!35}{32\!\cdots\!64}a^{15}+\frac{13\!\cdots\!87}{48\!\cdots\!46}a^{14}+\frac{24\!\cdots\!83}{80\!\cdots\!91}a^{13}-\frac{49\!\cdots\!03}{10\!\cdots\!88}a^{12}-\frac{12\!\cdots\!75}{48\!\cdots\!46}a^{11}+\frac{40\!\cdots\!97}{10\!\cdots\!88}a^{10}+\frac{12\!\cdots\!37}{10\!\cdots\!88}a^{9}-\frac{27\!\cdots\!15}{16\!\cdots\!82}a^{8}-\frac{53\!\cdots\!27}{17\!\cdots\!98}a^{7}+\frac{13\!\cdots\!77}{32\!\cdots\!64}a^{6}+\frac{40\!\cdots\!73}{96\!\cdots\!92}a^{5}-\frac{52\!\cdots\!97}{10\!\cdots\!88}a^{4}-\frac{97\!\cdots\!39}{35\!\cdots\!96}a^{3}+\frac{73\!\cdots\!03}{32\!\cdots\!64}a^{2}+\frac{27\!\cdots\!35}{599265645770172}a-\frac{13\!\cdots\!23}{15\!\cdots\!52}$, $\frac{257150700645313}{19\!\cdots\!84}a^{17}-\frac{75220581306812}{80\!\cdots\!91}a^{16}-\frac{36\!\cdots\!63}{32\!\cdots\!64}a^{15}+\frac{37\!\cdots\!05}{96\!\cdots\!92}a^{14}+\frac{95\!\cdots\!35}{32\!\cdots\!64}a^{13}-\frac{24\!\cdots\!97}{35\!\cdots\!96}a^{12}-\frac{34\!\cdots\!87}{96\!\cdots\!92}a^{11}+\frac{50\!\cdots\!77}{80\!\cdots\!91}a^{10}+\frac{16\!\cdots\!35}{71\!\cdots\!92}a^{9}-\frac{10\!\cdots\!49}{32\!\cdots\!64}a^{8}-\frac{84\!\cdots\!95}{10\!\cdots\!88}a^{7}+\frac{13\!\cdots\!01}{16\!\cdots\!82}a^{6}+\frac{65\!\cdots\!81}{48\!\cdots\!46}a^{5}-\frac{37\!\cdots\!81}{32\!\cdots\!64}a^{4}-\frac{53\!\cdots\!35}{53\!\cdots\!94}a^{3}+\frac{19\!\cdots\!45}{32\!\cdots\!64}a^{2}+\frac{18\!\cdots\!17}{11\!\cdots\!44}a-\frac{20\!\cdots\!59}{777308337629426}$, $\frac{10\!\cdots\!03}{19\!\cdots\!84}a^{17}-\frac{28\!\cdots\!45}{32\!\cdots\!64}a^{16}-\frac{70\!\cdots\!17}{32\!\cdots\!64}a^{15}+\frac{88\!\cdots\!75}{24\!\cdots\!73}a^{14}+\frac{96\!\cdots\!51}{26\!\cdots\!97}a^{13}-\frac{61\!\cdots\!85}{10\!\cdots\!88}a^{12}-\frac{73\!\cdots\!95}{24\!\cdots\!73}a^{11}+\frac{14\!\cdots\!21}{32\!\cdots\!64}a^{10}+\frac{31\!\cdots\!63}{21\!\cdots\!76}a^{9}-\frac{60\!\cdots\!59}{32\!\cdots\!64}a^{8}-\frac{21\!\cdots\!85}{53\!\cdots\!94}a^{7}+\frac{13\!\cdots\!07}{32\!\cdots\!64}a^{6}+\frac{14\!\cdots\!89}{24\!\cdots\!73}a^{5}-\frac{33\!\cdots\!15}{80\!\cdots\!91}a^{4}-\frac{44\!\cdots\!11}{10\!\cdots\!88}a^{3}+\frac{43\!\cdots\!99}{32\!\cdots\!64}a^{2}+\frac{19\!\cdots\!55}{399510430513448}a-\frac{20\!\cdots\!99}{388654168814713}$, $\frac{61\!\cdots\!95}{19\!\cdots\!84}a^{17}-\frac{21552305269489}{64\!\cdots\!28}a^{16}-\frac{42\!\cdots\!65}{32\!\cdots\!64}a^{15}+\frac{10\!\cdots\!91}{48\!\cdots\!46}a^{14}+\frac{70\!\cdots\!85}{32\!\cdots\!64}a^{13}-\frac{32\!\cdots\!13}{53\!\cdots\!94}a^{12}-\frac{18\!\cdots\!49}{96\!\cdots\!92}a^{11}+\frac{54\!\cdots\!07}{80\!\cdots\!91}a^{10}+\frac{19\!\cdots\!25}{21\!\cdots\!76}a^{9}-\frac{24\!\cdots\!01}{64\!\cdots\!28}a^{8}-\frac{26\!\cdots\!19}{10\!\cdots\!88}a^{7}+\frac{17\!\cdots\!29}{16\!\cdots\!82}a^{6}+\frac{34\!\cdots\!29}{96\!\cdots\!92}a^{5}-\frac{24\!\cdots\!67}{16\!\cdots\!82}a^{4}-\frac{59\!\cdots\!67}{26\!\cdots\!97}a^{3}+\frac{68\!\cdots\!42}{80\!\cdots\!91}a^{2}+\frac{31\!\cdots\!05}{11\!\cdots\!44}a-\frac{11\!\cdots\!71}{31\!\cdots\!04}$, $\frac{23\!\cdots\!21}{19\!\cdots\!84}a^{17}-\frac{35\!\cdots\!01}{32\!\cdots\!64}a^{16}-\frac{63\!\cdots\!85}{32\!\cdots\!64}a^{15}+\frac{39\!\cdots\!59}{96\!\cdots\!92}a^{14}-\frac{17\!\cdots\!39}{53\!\cdots\!94}a^{13}-\frac{58\!\cdots\!45}{10\!\cdots\!88}a^{12}+\frac{48\!\cdots\!61}{48\!\cdots\!46}a^{11}+\frac{10\!\cdots\!93}{32\!\cdots\!64}a^{10}-\frac{18\!\cdots\!85}{21\!\cdots\!76}a^{9}-\frac{28\!\cdots\!91}{32\!\cdots\!64}a^{8}+\frac{92\!\cdots\!01}{26\!\cdots\!97}a^{7}+\frac{12\!\cdots\!09}{32\!\cdots\!64}a^{6}-\frac{61\!\cdots\!53}{96\!\cdots\!92}a^{5}+\frac{31\!\cdots\!19}{16\!\cdots\!82}a^{4}+\frac{50\!\cdots\!01}{10\!\cdots\!88}a^{3}-\frac{17\!\cdots\!02}{80\!\cdots\!91}a^{2}-\frac{28\!\cdots\!05}{399510430513448}a+\frac{27\!\cdots\!68}{388654168814713}$, $\frac{496151854527493}{24\!\cdots\!73}a^{17}-\frac{94662565955581}{32\!\cdots\!64}a^{16}-\frac{14\!\cdots\!79}{16\!\cdots\!82}a^{15}+\frac{30\!\cdots\!31}{96\!\cdots\!92}a^{14}+\frac{12\!\cdots\!09}{80\!\cdots\!91}a^{13}-\frac{88\!\cdots\!01}{10\!\cdots\!88}a^{12}-\frac{13\!\cdots\!75}{96\!\cdots\!92}a^{11}+\frac{30\!\cdots\!75}{32\!\cdots\!64}a^{10}+\frac{71\!\cdots\!45}{10\!\cdots\!88}a^{9}-\frac{91\!\cdots\!01}{16\!\cdots\!82}a^{8}-\frac{19\!\cdots\!11}{10\!\cdots\!88}a^{7}+\frac{56\!\cdots\!01}{32\!\cdots\!64}a^{6}+\frac{23\!\cdots\!29}{96\!\cdots\!92}a^{5}-\frac{85\!\cdots\!89}{32\!\cdots\!64}a^{4}-\frac{12\!\cdots\!37}{10\!\cdots\!88}a^{3}+\frac{26\!\cdots\!91}{16\!\cdots\!82}a^{2}-\frac{74\!\cdots\!25}{599265645770172}a-\frac{28\!\cdots\!99}{15\!\cdots\!52}$, $\frac{82\!\cdots\!11}{19\!\cdots\!84}a^{17}+\frac{12\!\cdots\!19}{32\!\cdots\!64}a^{16}-\frac{53\!\cdots\!87}{32\!\cdots\!64}a^{15}-\frac{56\!\cdots\!41}{48\!\cdots\!46}a^{14}+\frac{13\!\cdots\!25}{53\!\cdots\!94}a^{13}+\frac{46\!\cdots\!31}{35\!\cdots\!96}a^{12}-\frac{10\!\cdots\!97}{48\!\cdots\!46}a^{11}-\frac{21\!\cdots\!17}{32\!\cdots\!64}a^{10}+\frac{19\!\cdots\!11}{21\!\cdots\!76}a^{9}+\frac{40\!\cdots\!37}{32\!\cdots\!64}a^{8}-\frac{12\!\cdots\!95}{53\!\cdots\!94}a^{7}+\frac{34\!\cdots\!59}{32\!\cdots\!64}a^{6}+\frac{72\!\cdots\!01}{24\!\cdots\!73}a^{5}-\frac{10\!\cdots\!11}{16\!\cdots\!82}a^{4}-\frac{61\!\cdots\!33}{35\!\cdots\!96}a^{3}+\frac{18\!\cdots\!61}{32\!\cdots\!64}a^{2}+\frac{90\!\cdots\!03}{399510430513448}a-\frac{21\!\cdots\!27}{777308337629426}$, $\frac{34\!\cdots\!01}{19\!\cdots\!84}a^{17}+\frac{17\!\cdots\!81}{32\!\cdots\!64}a^{16}-\frac{32\!\cdots\!23}{32\!\cdots\!64}a^{15}-\frac{17\!\cdots\!95}{96\!\cdots\!92}a^{14}+\frac{81\!\cdots\!95}{35\!\cdots\!96}a^{13}+\frac{22\!\cdots\!11}{10\!\cdots\!88}a^{12}-\frac{24\!\cdots\!55}{96\!\cdots\!92}a^{11}-\frac{14\!\cdots\!45}{16\!\cdots\!82}a^{10}+\frac{33\!\cdots\!29}{21\!\cdots\!76}a^{9}+\frac{12\!\cdots\!33}{16\!\cdots\!82}a^{8}-\frac{54\!\cdots\!97}{10\!\cdots\!88}a^{7}+\frac{72\!\cdots\!17}{80\!\cdots\!91}a^{6}+\frac{40\!\cdots\!07}{48\!\cdots\!46}a^{5}-\frac{65\!\cdots\!37}{32\!\cdots\!64}a^{4}-\frac{15\!\cdots\!40}{26\!\cdots\!97}a^{3}+\frac{41\!\cdots\!53}{32\!\cdots\!64}a^{2}+\frac{26\!\cdots\!27}{399510430513448}a-\frac{13\!\cdots\!41}{15\!\cdots\!52}$
| 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: | \( 127268889201 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 127268889201 \cdot 3}{2\cdot\sqrt{2096291596176378856933557232324608}}\cr\approx \mathstrut & 1.09301868943459 \end{aligned}\] (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 - 39*x^16 + 124*x^15 + 588*x^14 - 2016*x^13 - 4354*x^12 + 16662*x^11 + 16569*x^10 - 76017*x^9 - 29151*x^8 + 193350*x^7 + 7892*x^6 - 257946*x^5 + 38376*x^4 + 147342*x^3 - 33795*x^2 - 14013*x + 1863)
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 - 39*x^16 + 124*x^15 + 588*x^14 - 2016*x^13 - 4354*x^12 + 16662*x^11 + 16569*x^10 - 76017*x^9 - 29151*x^8 + 193350*x^7 + 7892*x^6 - 257946*x^5 + 38376*x^4 + 147342*x^3 - 33795*x^2 - 14013*x + 1863, 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 - 39*x^16 + 124*x^15 + 588*x^14 - 2016*x^13 - 4354*x^12 + 16662*x^11 + 16569*x^10 - 76017*x^9 - 29151*x^8 + 193350*x^7 + 7892*x^6 - 257946*x^5 + 38376*x^4 + 147342*x^3 - 33795*x^2 - 14013*x + 1863);
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 - 39*x^16 + 124*x^15 + 588*x^14 - 2016*x^13 - 4354*x^12 + 16662*x^11 + 16569*x^10 - 76017*x^9 - 29151*x^8 + 193350*x^7 + 7892*x^6 - 257946*x^5 + 38376*x^4 + 147342*x^3 - 33795*x^2 - 14013*x + 1863);
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
$C_3^2:C_6$ (as 18T22):
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 54 |
| The 10 conjugacy class representatives for $C_3^2:C_6$ |
| Character table for $C_3^2:C_6$ |
Intermediate fields
| \(\Q(\sqrt{13}) \), 6.6.6836396112.1 |
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 9 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 sibling: | data not computed |
| Minimal sibling: | 9.9.2332386934323264.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{5}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{6}$ | ${\href{/padicField/31.2.0.1}{2} }^{9}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{9}$ | ${\href{/padicField/53.3.0.1}{3} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{9}$ |
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 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(3\)
| 3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
| 3.3.4.3 | $x^{3} + 6 x^{2} + 12$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
| 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 3.9.12.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$ | $3$ | $3$ | $12$ | $C_3^2$ | $[2]^{3}$ | |
|
\(7\)
| 7.6.4.1 | $x^{6} + 14 x^{3} - 245$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 7.6.4.1 | $x^{6} + 14 x^{3} - 245$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 7.6.4.1 | $x^{6} + 14 x^{3} - 245$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
|
\(13\)
| 13.6.3.1 | $x^{6} + 390 x^{5} + 50743 x^{4} + 2208202 x^{3} + 765301 x^{2} + 5017316 x + 24555184$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 13.6.3.1 | $x^{6} + 390 x^{5} + 50743 x^{4} + 2208202 x^{3} + 765301 x^{2} + 5017316 x + 24555184$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 13.6.3.1 | $x^{6} + 390 x^{5} + 50743 x^{4} + 2208202 x^{3} + 765301 x^{2} + 5017316 x + 24555184$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |