Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 27*x^16 - 42*x^15 - 12*x^14 + 242*x^13 - 833*x^12 + 2227*x^11 - 5196*x^10 + 10127*x^9 - 15321*x^8 + 14901*x^7 - 1829*x^6 - 22543*x^5 + 46651*x^4 - 53722*x^3 + 41301*x^2 - 20001*x + 4181)
gp: K = bnfinit(y^18 - 8*y^17 + 27*y^16 - 42*y^15 - 12*y^14 + 242*y^13 - 833*y^12 + 2227*y^11 - 5196*y^10 + 10127*y^9 - 15321*y^8 + 14901*y^7 - 1829*y^6 - 22543*y^5 + 46651*y^4 - 53722*y^3 + 41301*y^2 - 20001*y + 4181, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 8*x^17 + 27*x^16 - 42*x^15 - 12*x^14 + 242*x^13 - 833*x^12 + 2227*x^11 - 5196*x^10 + 10127*x^9 - 15321*x^8 + 14901*x^7 - 1829*x^6 - 22543*x^5 + 46651*x^4 - 53722*x^3 + 41301*x^2 - 20001*x + 4181);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 8*x^17 + 27*x^16 - 42*x^15 - 12*x^14 + 242*x^13 - 833*x^12 + 2227*x^11 - 5196*x^10 + 10127*x^9 - 15321*x^8 + 14901*x^7 - 1829*x^6 - 22543*x^5 + 46651*x^4 - 53722*x^3 + 41301*x^2 - 20001*x + 4181)
\( x^{18} - 8 x^{17} + 27 x^{16} - 42 x^{15} - 12 x^{14} + 242 x^{13} - 833 x^{12} + 2227 x^{11} + \cdots + 4181 \)
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: |
\(416191250312479879983411857\)
\(\medspace = 19^{16}\cdot 113^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(30.12\) | 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: | $19^{8/9}113^{1/2}\approx 145.61599739544846$ | ||
| Ramified primes: |
\(19\), \(113\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{113}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{37}a^{16}+\frac{12}{37}a^{15}+\frac{9}{37}a^{14}+\frac{2}{37}a^{13}-\frac{15}{37}a^{11}+\frac{14}{37}a^{10}+\frac{13}{37}a^{9}-\frac{1}{37}a^{8}-\frac{18}{37}a^{7}+\frac{6}{37}a^{6}+\frac{18}{37}a^{5}+\frac{17}{37}a^{4}+\frac{15}{37}a^{3}+\frac{15}{37}a^{2}-\frac{16}{37}a$, $\frac{1}{10\!\cdots\!19}a^{17}+\frac{60\!\cdots\!98}{10\!\cdots\!19}a^{16}-\frac{34\!\cdots\!72}{92\!\cdots\!63}a^{15}+\frac{35\!\cdots\!80}{10\!\cdots\!19}a^{14}-\frac{24\!\cdots\!61}{10\!\cdots\!19}a^{13}-\frac{37\!\cdots\!81}{10\!\cdots\!19}a^{12}-\frac{51\!\cdots\!85}{10\!\cdots\!19}a^{11}+\frac{29\!\cdots\!17}{10\!\cdots\!19}a^{10}+\frac{33\!\cdots\!87}{69\!\cdots\!69}a^{9}+\frac{21\!\cdots\!51}{10\!\cdots\!19}a^{8}+\frac{52\!\cdots\!51}{10\!\cdots\!19}a^{7}+\frac{36\!\cdots\!81}{10\!\cdots\!19}a^{6}+\frac{13\!\cdots\!15}{10\!\cdots\!19}a^{5}+\frac{14\!\cdots\!28}{10\!\cdots\!19}a^{4}+\frac{63\!\cdots\!25}{10\!\cdots\!19}a^{3}+\frac{20\!\cdots\!01}{69\!\cdots\!69}a^{2}-\frac{30\!\cdots\!90}{92\!\cdots\!63}a-\frac{16\!\cdots\!22}{25\!\cdots\!99}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: |
$\frac{16\!\cdots\!88}{10\!\cdots\!19}a^{17}-\frac{13\!\cdots\!74}{10\!\cdots\!19}a^{16}+\frac{39\!\cdots\!48}{92\!\cdots\!63}a^{15}-\frac{59\!\cdots\!16}{10\!\cdots\!19}a^{14}-\frac{53\!\cdots\!38}{10\!\cdots\!19}a^{13}+\frac{43\!\cdots\!25}{10\!\cdots\!19}a^{12}-\frac{13\!\cdots\!01}{10\!\cdots\!19}a^{11}+\frac{34\!\cdots\!90}{10\!\cdots\!19}a^{10}-\frac{51\!\cdots\!36}{69\!\cdots\!69}a^{9}+\frac{14\!\cdots\!41}{10\!\cdots\!19}a^{8}-\frac{20\!\cdots\!37}{10\!\cdots\!19}a^{7}+\frac{15\!\cdots\!27}{10\!\cdots\!19}a^{6}+\frac{81\!\cdots\!73}{10\!\cdots\!19}a^{5}-\frac{42\!\cdots\!01}{10\!\cdots\!19}a^{4}+\frac{65\!\cdots\!57}{10\!\cdots\!19}a^{3}-\frac{39\!\cdots\!19}{69\!\cdots\!69}a^{2}+\frac{31\!\cdots\!38}{92\!\cdots\!63}a-\frac{21\!\cdots\!00}{25\!\cdots\!99}$, $\frac{16\!\cdots\!02}{10\!\cdots\!19}a^{17}-\frac{12\!\cdots\!33}{10\!\cdots\!19}a^{16}+\frac{32\!\cdots\!40}{92\!\cdots\!63}a^{15}-\frac{41\!\cdots\!14}{10\!\cdots\!19}a^{14}-\frac{66\!\cdots\!50}{10\!\cdots\!19}a^{13}+\frac{38\!\cdots\!91}{10\!\cdots\!19}a^{12}-\frac{10\!\cdots\!57}{10\!\cdots\!19}a^{11}+\frac{27\!\cdots\!22}{10\!\cdots\!19}a^{10}-\frac{41\!\cdots\!00}{69\!\cdots\!69}a^{9}+\frac{11\!\cdots\!07}{10\!\cdots\!19}a^{8}-\frac{14\!\cdots\!81}{10\!\cdots\!19}a^{7}+\frac{86\!\cdots\!15}{10\!\cdots\!19}a^{6}+\frac{11\!\cdots\!97}{10\!\cdots\!19}a^{5}-\frac{35\!\cdots\!80}{10\!\cdots\!19}a^{4}+\frac{46\!\cdots\!79}{10\!\cdots\!19}a^{3}-\frac{23\!\cdots\!07}{69\!\cdots\!69}a^{2}+\frac{15\!\cdots\!87}{92\!\cdots\!63}a-\frac{10\!\cdots\!47}{25\!\cdots\!99}$, $\frac{47\!\cdots\!56}{10\!\cdots\!19}a^{17}-\frac{22\!\cdots\!83}{10\!\cdots\!19}a^{16}+\frac{25\!\cdots\!54}{92\!\cdots\!63}a^{15}+\frac{26\!\cdots\!82}{10\!\cdots\!19}a^{14}-\frac{14\!\cdots\!27}{10\!\cdots\!19}a^{13}+\frac{39\!\cdots\!49}{10\!\cdots\!19}a^{12}-\frac{13\!\cdots\!41}{10\!\cdots\!19}a^{11}+\frac{34\!\cdots\!85}{10\!\cdots\!19}a^{10}-\frac{43\!\cdots\!83}{69\!\cdots\!69}a^{9}+\frac{98\!\cdots\!82}{10\!\cdots\!19}a^{8}-\frac{10\!\cdots\!94}{10\!\cdots\!19}a^{7}+\frac{28\!\cdots\!39}{10\!\cdots\!19}a^{6}+\frac{33\!\cdots\!20}{10\!\cdots\!19}a^{5}-\frac{62\!\cdots\!85}{10\!\cdots\!19}a^{4}+\frac{38\!\cdots\!65}{10\!\cdots\!19}a^{3}-\frac{23\!\cdots\!83}{69\!\cdots\!69}a^{2}+\frac{42\!\cdots\!44}{92\!\cdots\!63}a-\frac{84\!\cdots\!80}{25\!\cdots\!99}$, $\frac{10\!\cdots\!97}{10\!\cdots\!19}a^{17}-\frac{69\!\cdots\!56}{10\!\cdots\!19}a^{16}+\frac{16\!\cdots\!47}{92\!\cdots\!63}a^{15}-\frac{19\!\cdots\!18}{10\!\cdots\!19}a^{14}-\frac{31\!\cdots\!24}{10\!\cdots\!19}a^{13}+\frac{19\!\cdots\!79}{10\!\cdots\!19}a^{12}-\frac{60\!\cdots\!42}{10\!\cdots\!19}a^{11}+\frac{15\!\cdots\!97}{10\!\cdots\!19}a^{10}-\frac{23\!\cdots\!88}{69\!\cdots\!69}a^{9}+\frac{63\!\cdots\!60}{10\!\cdots\!19}a^{8}-\frac{87\!\cdots\!79}{10\!\cdots\!19}a^{7}+\frac{17\!\cdots\!47}{28\!\cdots\!87}a^{6}+\frac{68\!\cdots\!16}{28\!\cdots\!87}a^{5}-\frac{16\!\cdots\!23}{10\!\cdots\!19}a^{4}+\frac{28\!\cdots\!12}{10\!\cdots\!19}a^{3}-\frac{18\!\cdots\!41}{69\!\cdots\!69}a^{2}+\frac{17\!\cdots\!92}{92\!\cdots\!63}a-\frac{14\!\cdots\!21}{25\!\cdots\!99}$, $\frac{10\!\cdots\!81}{10\!\cdots\!19}a^{17}-\frac{80\!\cdots\!94}{10\!\cdots\!19}a^{16}+\frac{21\!\cdots\!08}{92\!\cdots\!63}a^{15}-\frac{30\!\cdots\!01}{10\!\cdots\!19}a^{14}-\frac{32\!\cdots\!26}{10\!\cdots\!19}a^{13}+\frac{24\!\cdots\!61}{10\!\cdots\!19}a^{12}-\frac{75\!\cdots\!45}{10\!\cdots\!19}a^{11}+\frac{19\!\cdots\!09}{10\!\cdots\!19}a^{10}-\frac{29\!\cdots\!57}{69\!\cdots\!69}a^{9}+\frac{82\!\cdots\!38}{10\!\cdots\!19}a^{8}-\frac{11\!\cdots\!04}{10\!\cdots\!19}a^{7}+\frac{89\!\cdots\!52}{10\!\cdots\!19}a^{6}+\frac{39\!\cdots\!36}{10\!\cdots\!19}a^{5}-\frac{22\!\cdots\!64}{10\!\cdots\!19}a^{4}+\frac{36\!\cdots\!65}{10\!\cdots\!19}a^{3}-\frac{23\!\cdots\!63}{69\!\cdots\!69}a^{2}+\frac{19\!\cdots\!68}{92\!\cdots\!63}a-\frac{16\!\cdots\!37}{25\!\cdots\!99}$, $\frac{12\!\cdots\!81}{10\!\cdots\!19}a^{17}-\frac{17\!\cdots\!34}{10\!\cdots\!19}a^{16}+\frac{70\!\cdots\!26}{92\!\cdots\!63}a^{15}-\frac{13\!\cdots\!77}{10\!\cdots\!19}a^{14}-\frac{40\!\cdots\!48}{10\!\cdots\!19}a^{13}+\frac{72\!\cdots\!88}{10\!\cdots\!19}a^{12}-\frac{21\!\cdots\!25}{10\!\cdots\!19}a^{11}+\frac{15\!\cdots\!20}{28\!\cdots\!87}a^{10}-\frac{89\!\cdots\!92}{69\!\cdots\!69}a^{9}+\frac{26\!\cdots\!10}{10\!\cdots\!19}a^{8}-\frac{38\!\cdots\!08}{10\!\cdots\!19}a^{7}+\frac{33\!\cdots\!86}{10\!\cdots\!19}a^{6}+\frac{13\!\cdots\!11}{10\!\cdots\!19}a^{5}-\frac{86\!\cdots\!28}{10\!\cdots\!19}a^{4}+\frac{11\!\cdots\!48}{10\!\cdots\!19}a^{3}-\frac{75\!\cdots\!86}{69\!\cdots\!69}a^{2}+\frac{48\!\cdots\!00}{92\!\cdots\!63}a+\frac{92\!\cdots\!41}{25\!\cdots\!99}$, $\frac{28\!\cdots\!81}{10\!\cdots\!19}a^{17}-\frac{16\!\cdots\!32}{10\!\cdots\!19}a^{16}+\frac{34\!\cdots\!60}{92\!\cdots\!63}a^{15}-\frac{42\!\cdots\!91}{10\!\cdots\!19}a^{14}-\frac{30\!\cdots\!49}{10\!\cdots\!19}a^{13}+\frac{35\!\cdots\!32}{10\!\cdots\!19}a^{12}-\frac{14\!\cdots\!69}{10\!\cdots\!19}a^{11}+\frac{39\!\cdots\!11}{10\!\cdots\!19}a^{10}-\frac{57\!\cdots\!45}{69\!\cdots\!69}a^{9}+\frac{16\!\cdots\!14}{10\!\cdots\!19}a^{8}-\frac{24\!\cdots\!29}{10\!\cdots\!19}a^{7}+\frac{25\!\cdots\!84}{10\!\cdots\!19}a^{6}-\frac{10\!\cdots\!08}{10\!\cdots\!19}a^{5}-\frac{24\!\cdots\!37}{10\!\cdots\!19}a^{4}+\frac{22\!\cdots\!70}{28\!\cdots\!87}a^{3}-\frac{71\!\cdots\!08}{69\!\cdots\!69}a^{2}+\frac{83\!\cdots\!82}{92\!\cdots\!63}a-\frac{91\!\cdots\!55}{25\!\cdots\!99}$, $\frac{47\!\cdots\!09}{10\!\cdots\!19}a^{17}-\frac{35\!\cdots\!93}{10\!\cdots\!19}a^{16}+\frac{94\!\cdots\!33}{92\!\cdots\!63}a^{15}-\frac{13\!\cdots\!46}{10\!\cdots\!19}a^{14}-\frac{14\!\cdots\!55}{10\!\cdots\!19}a^{13}+\frac{11\!\cdots\!02}{10\!\cdots\!19}a^{12}-\frac{33\!\cdots\!72}{10\!\cdots\!19}a^{11}+\frac{84\!\cdots\!67}{10\!\cdots\!19}a^{10}-\frac{12\!\cdots\!96}{69\!\cdots\!69}a^{9}+\frac{35\!\cdots\!87}{10\!\cdots\!19}a^{8}-\frac{48\!\cdots\!70}{10\!\cdots\!19}a^{7}+\frac{35\!\cdots\!71}{10\!\cdots\!19}a^{6}+\frac{23\!\cdots\!18}{10\!\cdots\!19}a^{5}-\frac{10\!\cdots\!43}{10\!\cdots\!19}a^{4}+\frac{15\!\cdots\!97}{10\!\cdots\!19}a^{3}-\frac{95\!\cdots\!90}{69\!\cdots\!69}a^{2}+\frac{78\!\cdots\!42}{92\!\cdots\!63}a-\frac{42\!\cdots\!67}{25\!\cdots\!99}$, $\frac{49\!\cdots\!00}{10\!\cdots\!19}a^{17}-\frac{33\!\cdots\!02}{10\!\cdots\!19}a^{16}+\frac{84\!\cdots\!83}{92\!\cdots\!63}a^{15}-\frac{97\!\cdots\!96}{10\!\cdots\!19}a^{14}-\frac{19\!\cdots\!43}{10\!\cdots\!19}a^{13}+\frac{10\!\cdots\!10}{10\!\cdots\!19}a^{12}-\frac{29\!\cdots\!16}{10\!\cdots\!19}a^{11}+\frac{74\!\cdots\!27}{10\!\cdots\!19}a^{10}-\frac{10\!\cdots\!84}{69\!\cdots\!69}a^{9}+\frac{29\!\cdots\!65}{10\!\cdots\!19}a^{8}-\frac{38\!\cdots\!55}{10\!\cdots\!19}a^{7}+\frac{21\!\cdots\!23}{10\!\cdots\!19}a^{6}+\frac{28\!\cdots\!02}{10\!\cdots\!19}a^{5}-\frac{91\!\cdots\!93}{10\!\cdots\!19}a^{4}+\frac{11\!\cdots\!29}{10\!\cdots\!19}a^{3}-\frac{64\!\cdots\!56}{69\!\cdots\!69}a^{2}+\frac{46\!\cdots\!30}{92\!\cdots\!63}a-\frac{28\!\cdots\!17}{25\!\cdots\!99}$, $\frac{17\!\cdots\!59}{10\!\cdots\!19}a^{17}-\frac{13\!\cdots\!34}{10\!\cdots\!19}a^{16}+\frac{36\!\cdots\!60}{92\!\cdots\!63}a^{15}-\frac{53\!\cdots\!84}{10\!\cdots\!19}a^{14}-\frac{50\!\cdots\!08}{10\!\cdots\!19}a^{13}+\frac{40\!\cdots\!51}{10\!\cdots\!19}a^{12}-\frac{12\!\cdots\!76}{10\!\cdots\!19}a^{11}+\frac{33\!\cdots\!23}{10\!\cdots\!19}a^{10}-\frac{50\!\cdots\!92}{69\!\cdots\!69}a^{9}+\frac{14\!\cdots\!84}{10\!\cdots\!19}a^{8}-\frac{19\!\cdots\!23}{10\!\cdots\!19}a^{7}+\frac{16\!\cdots\!99}{10\!\cdots\!19}a^{6}+\frac{54\!\cdots\!48}{10\!\cdots\!19}a^{5}-\frac{37\!\cdots\!88}{10\!\cdots\!19}a^{4}+\frac{64\!\cdots\!90}{10\!\cdots\!19}a^{3}-\frac{42\!\cdots\!87}{69\!\cdots\!69}a^{2}+\frac{36\!\cdots\!36}{92\!\cdots\!63}a-\frac{34\!\cdots\!45}{25\!\cdots\!99}$, $\frac{29\!\cdots\!30}{10\!\cdots\!19}a^{17}-\frac{21\!\cdots\!39}{10\!\cdots\!19}a^{16}+\frac{57\!\cdots\!97}{92\!\cdots\!63}a^{15}-\frac{82\!\cdots\!72}{10\!\cdots\!19}a^{14}-\frac{84\!\cdots\!33}{10\!\cdots\!19}a^{13}+\frac{64\!\cdots\!94}{10\!\cdots\!19}a^{12}-\frac{20\!\cdots\!82}{10\!\cdots\!19}a^{11}+\frac{52\!\cdots\!04}{10\!\cdots\!19}a^{10}-\frac{79\!\cdots\!14}{69\!\cdots\!69}a^{9}+\frac{22\!\cdots\!97}{10\!\cdots\!19}a^{8}-\frac{31\!\cdots\!69}{10\!\cdots\!19}a^{7}+\frac{24\!\cdots\!25}{10\!\cdots\!19}a^{6}+\frac{86\!\cdots\!09}{10\!\cdots\!19}a^{5}-\frac{58\!\cdots\!14}{10\!\cdots\!19}a^{4}+\frac{99\!\cdots\!03}{10\!\cdots\!19}a^{3}-\frac{65\!\cdots\!53}{69\!\cdots\!69}a^{2}+\frac{57\!\cdots\!96}{92\!\cdots\!63}a-\frac{55\!\cdots\!31}{25\!\cdots\!99}$
| 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: | \( 1701924.22156 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 1701924.22156 \cdot 1}{2\cdot\sqrt{416191250312479879983411857}}\cr\approx \mathstrut & 0.164256640010 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 27*x^16 - 42*x^15 - 12*x^14 + 242*x^13 - 833*x^12 + 2227*x^11 - 5196*x^10 + 10127*x^9 - 15321*x^8 + 14901*x^7 - 1829*x^6 - 22543*x^5 + 46651*x^4 - 53722*x^3 + 41301*x^2 - 20001*x + 4181)
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 - 8*x^17 + 27*x^16 - 42*x^15 - 12*x^14 + 242*x^13 - 833*x^12 + 2227*x^11 - 5196*x^10 + 10127*x^9 - 15321*x^8 + 14901*x^7 - 1829*x^6 - 22543*x^5 + 46651*x^4 - 53722*x^3 + 41301*x^2 - 20001*x + 4181, 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 - 8*x^17 + 27*x^16 - 42*x^15 - 12*x^14 + 242*x^13 - 833*x^12 + 2227*x^11 - 5196*x^10 + 10127*x^9 - 15321*x^8 + 14901*x^7 - 1829*x^6 - 22543*x^5 + 46651*x^4 - 53722*x^3 + 41301*x^2 - 20001*x + 4181);
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 - 8*x^17 + 27*x^16 - 42*x^15 - 12*x^14 + 242*x^13 - 833*x^12 + 2227*x^11 - 5196*x^10 + 10127*x^9 - 15321*x^8 + 14901*x^7 - 1829*x^6 - 22543*x^5 + 46651*x^4 - 53722*x^3 + 41301*x^2 - 20001*x + 4181);
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
$C_2^2:C_{18}$ (as 18T26):
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 72 |
| The 24 conjugacy class representatives for $C_2^2:C_{18}$ |
| Character table for $C_2^2:C_{18}$ is not computed |
Intermediate fields
| 3.3.361.1, 6.2.14726273.1, \(\Q(\zeta_{19})^+\) |
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 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 | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | $18$ | $18$ | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | $18$ | R | $18$ | $18$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{12}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | $18$ | $18$ | ${\href{/padicField/53.9.0.1}{9} }^{2}$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(19\)
| 19.18.16.1 | $x^{18} + 162 x^{17} + 11682 x^{16} + 492480 x^{15} + 13390416 x^{14} + 243982368 x^{13} + 2990277024 x^{12} + 23974071552 x^{11} + 116854153056 x^{10} + 292311592166 x^{9} + 233708309190 x^{8} + 95896505088 x^{7} + 23931351696 x^{6} + 4148844336 x^{5} + 4813362864 x^{4} + 52323118080 x^{3} + 400888193472 x^{2} + 1792784840544 x + 3563298115785$ | $9$ | $2$ | $16$ | $C_{18}$ | $[\ ]_{9}^{2}$ |
|
\(113\)
| $\Q_{113}$ | $x + 110$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{113}$ | $x + 110$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{113}$ | $x + 110$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{113}$ | $x + 110$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{113}$ | $x + 110$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{113}$ | $x + 110$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 113.2.1.2 | $x^{2} + 339$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 113.2.1.2 | $x^{2} + 339$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 113.2.1.2 | $x^{2} + 339$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |