Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 7*x^16 + 20*x^15 + 45*x^14 - 443*x^13 + 1134*x^12 - 1479*x^11 + 175*x^10 + 2090*x^9 - 2977*x^8 + 831*x^7 + 1899*x^6 - 2660*x^5 + 1110*x^4 - 640*x^3 + 280*x^2 - 480*x + 20)
gp: K = bnfinit(y^18 - 3*y^17 - 7*y^16 + 20*y^15 + 45*y^14 - 443*y^13 + 1134*y^12 - 1479*y^11 + 175*y^10 + 2090*y^9 - 2977*y^8 + 831*y^7 + 1899*y^6 - 2660*y^5 + 1110*y^4 - 640*y^3 + 280*y^2 - 480*y + 20, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 - 7*x^16 + 20*x^15 + 45*x^14 - 443*x^13 + 1134*x^12 - 1479*x^11 + 175*x^10 + 2090*x^9 - 2977*x^8 + 831*x^7 + 1899*x^6 - 2660*x^5 + 1110*x^4 - 640*x^3 + 280*x^2 - 480*x + 20);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 - 7*x^16 + 20*x^15 + 45*x^14 - 443*x^13 + 1134*x^12 - 1479*x^11 + 175*x^10 + 2090*x^9 - 2977*x^8 + 831*x^7 + 1899*x^6 - 2660*x^5 + 1110*x^4 - 640*x^3 + 280*x^2 - 480*x + 20)
\( x^{18} - 3 x^{17} - 7 x^{16} + 20 x^{15} + 45 x^{14} - 443 x^{13} + 1134 x^{12} - 1479 x^{11} + \cdots + 20 \)
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: | $[2, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(11351585665720125000000000000\)
\(\medspace = 2^{12}\cdot 3^{8}\cdot 5^{15}\cdot 7^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(36.19\) | 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: | $2^{2/3}3^{1/2}5^{5/6}7^{2/3}\approx 38.46989386174112$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6}a^{12}+\frac{1}{3}a^{10}-\frac{1}{2}a^{9}+\frac{1}{6}a^{6}-\frac{1}{2}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{6}a^{13}+\frac{1}{3}a^{11}-\frac{1}{2}a^{10}+\frac{1}{6}a^{7}-\frac{1}{2}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{12}a^{14}+\frac{1}{4}a^{11}-\frac{1}{3}a^{10}+\frac{1}{12}a^{8}-\frac{1}{2}a^{7}-\frac{1}{6}a^{6}+\frac{1}{4}a^{5}+\frac{1}{6}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{1}{3}$, $\frac{1}{12}a^{15}-\frac{1}{12}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{12}a^{9}-\frac{1}{2}a^{8}-\frac{1}{6}a^{7}-\frac{1}{12}a^{6}+\frac{1}{6}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{4399956}a^{16}-\frac{25661}{4399956}a^{15}+\frac{25}{199998}a^{14}-\frac{163391}{4399956}a^{13}-\frac{119257}{4399956}a^{12}-\frac{621829}{2199978}a^{11}+\frac{5861}{35772}a^{10}+\frac{949555}{4399956}a^{9}-\frac{1028561}{2199978}a^{8}-\frac{160391}{399996}a^{7}+\frac{499747}{1466652}a^{6}-\frac{814469}{2199978}a^{5}-\frac{755423}{2199978}a^{4}-\frac{34942}{1099989}a^{3}-\frac{157853}{1099989}a^{2}+\frac{120557}{1099989}a+\frac{299308}{1099989}$, $\frac{1}{14\!\cdots\!92}a^{17}+\frac{73441387685535}{16\!\cdots\!88}a^{16}-\frac{84\!\cdots\!37}{24\!\cdots\!82}a^{15}+\frac{11\!\cdots\!91}{82\!\cdots\!94}a^{14}-\frac{51\!\cdots\!21}{14\!\cdots\!92}a^{13}+\frac{47\!\cdots\!81}{74\!\cdots\!46}a^{12}-\frac{29\!\cdots\!39}{74\!\cdots\!46}a^{11}+\frac{25\!\cdots\!27}{14\!\cdots\!92}a^{10}+\frac{37\!\cdots\!13}{74\!\cdots\!46}a^{9}-\frac{10\!\cdots\!47}{12\!\cdots\!41}a^{8}-\frac{44\!\cdots\!81}{14\!\cdots\!92}a^{7}-\frac{64\!\cdots\!64}{37\!\cdots\!23}a^{6}-\frac{92\!\cdots\!45}{49\!\cdots\!64}a^{5}-\frac{90\!\cdots\!59}{24\!\cdots\!82}a^{4}+\frac{27\!\cdots\!99}{74\!\cdots\!46}a^{3}+\frac{67\!\cdots\!07}{74\!\cdots\!46}a^{2}+\frac{16\!\cdots\!30}{37\!\cdots\!23}a-\frac{49\!\cdots\!73}{37\!\cdots\!23}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $3$ |
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: | $9$ | 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: |
$\frac{69461791448375}{22\!\cdots\!28}a^{17}-\frac{192216753293351}{22\!\cdots\!28}a^{16}-\frac{276723993848659}{11\!\cdots\!14}a^{15}+\frac{649469856712823}{11\!\cdots\!14}a^{14}+\frac{405820837005901}{24\!\cdots\!92}a^{13}-\frac{836903797333197}{623207171677073}a^{12}+\frac{35\!\cdots\!65}{11\!\cdots\!14}a^{11}-\frac{77\!\cdots\!79}{22\!\cdots\!28}a^{10}-\frac{16\!\cdots\!51}{18\!\cdots\!19}a^{9}+\frac{37\!\cdots\!77}{56\!\cdots\!57}a^{8}-\frac{14\!\cdots\!91}{22\!\cdots\!28}a^{7}-\frac{10\!\cdots\!73}{11\!\cdots\!14}a^{6}+\frac{15\!\cdots\!19}{22\!\cdots\!28}a^{5}-\frac{56\!\cdots\!61}{11\!\cdots\!14}a^{4}-\frac{240325765460588}{623207171677073}a^{3}-\frac{12\!\cdots\!25}{12\!\cdots\!46}a^{2}+\frac{27\!\cdots\!90}{18\!\cdots\!19}a-\frac{38\!\cdots\!34}{56\!\cdots\!57}$, $\frac{57\!\cdots\!69}{14\!\cdots\!92}a^{17}-\frac{70\!\cdots\!21}{49\!\cdots\!64}a^{16}-\frac{50\!\cdots\!79}{24\!\cdots\!82}a^{15}+\frac{12\!\cdots\!34}{12\!\cdots\!41}a^{14}+\frac{19\!\cdots\!75}{14\!\cdots\!92}a^{13}-\frac{69\!\cdots\!34}{37\!\cdots\!23}a^{12}+\frac{20\!\cdots\!46}{37\!\cdots\!23}a^{11}-\frac{12\!\cdots\!97}{14\!\cdots\!92}a^{10}+\frac{10\!\cdots\!06}{37\!\cdots\!23}a^{9}+\frac{24\!\cdots\!85}{24\!\cdots\!82}a^{8}-\frac{25\!\cdots\!29}{14\!\cdots\!92}a^{7}+\frac{52\!\cdots\!13}{74\!\cdots\!46}a^{6}+\frac{15\!\cdots\!35}{16\!\cdots\!88}a^{5}-\frac{59\!\cdots\!00}{41\!\cdots\!47}a^{4}+\frac{24\!\cdots\!58}{37\!\cdots\!23}a^{3}-\frac{18\!\cdots\!21}{67\!\cdots\!86}a^{2}+\frac{98\!\cdots\!66}{37\!\cdots\!23}a-\frac{83\!\cdots\!43}{37\!\cdots\!23}$, $\frac{13\!\cdots\!45}{74\!\cdots\!46}a^{17}+\frac{43\!\cdots\!75}{24\!\cdots\!82}a^{16}-\frac{10\!\cdots\!81}{24\!\cdots\!82}a^{15}-\frac{12\!\cdots\!37}{49\!\cdots\!64}a^{14}+\frac{24\!\cdots\!85}{74\!\cdots\!46}a^{13}-\frac{25\!\cdots\!07}{74\!\cdots\!46}a^{12}-\frac{26\!\cdots\!33}{14\!\cdots\!92}a^{11}+\frac{47\!\cdots\!91}{74\!\cdots\!46}a^{10}-\frac{43\!\cdots\!49}{74\!\cdots\!46}a^{9}-\frac{47\!\cdots\!45}{49\!\cdots\!64}a^{8}+\frac{11\!\cdots\!86}{37\!\cdots\!23}a^{7}-\frac{78\!\cdots\!27}{37\!\cdots\!23}a^{6}-\frac{91\!\cdots\!19}{45\!\cdots\!24}a^{5}+\frac{37\!\cdots\!91}{82\!\cdots\!94}a^{4}-\frac{18\!\cdots\!55}{67\!\cdots\!86}a^{3}-\frac{17\!\cdots\!31}{74\!\cdots\!46}a^{2}+\frac{89\!\cdots\!93}{37\!\cdots\!23}a+\frac{16\!\cdots\!18}{37\!\cdots\!23}$, $\frac{65267021795086}{35\!\cdots\!33}a^{17}-\frac{17\!\cdots\!65}{42\!\cdots\!96}a^{16}-\frac{71\!\cdots\!37}{42\!\cdots\!96}a^{15}+\frac{28\!\cdots\!00}{10\!\cdots\!99}a^{14}+\frac{46\!\cdots\!85}{42\!\cdots\!96}a^{13}-\frac{31\!\cdots\!01}{42\!\cdots\!96}a^{12}+\frac{15\!\cdots\!50}{10\!\cdots\!99}a^{11}-\frac{18\!\cdots\!35}{14\!\cdots\!32}a^{10}-\frac{61\!\cdots\!85}{42\!\cdots\!96}a^{9}+\frac{44\!\cdots\!85}{10\!\cdots\!99}a^{8}-\frac{15\!\cdots\!29}{42\!\cdots\!96}a^{7}-\frac{44\!\cdots\!15}{46\!\cdots\!44}a^{6}+\frac{48\!\cdots\!10}{10\!\cdots\!99}a^{5}-\frac{45\!\cdots\!90}{10\!\cdots\!99}a^{4}-\frac{36\!\cdots\!95}{10\!\cdots\!99}a^{3}+\frac{19\!\cdots\!40}{10\!\cdots\!99}a^{2}-\frac{19\!\cdots\!30}{10\!\cdots\!99}a-\frac{12\!\cdots\!72}{10\!\cdots\!99}$, $\frac{93\!\cdots\!27}{74\!\cdots\!46}a^{17}-\frac{49\!\cdots\!23}{14\!\cdots\!92}a^{16}-\frac{15\!\cdots\!77}{14\!\cdots\!92}a^{15}+\frac{16\!\cdots\!93}{74\!\cdots\!46}a^{14}+\frac{10\!\cdots\!31}{14\!\cdots\!92}a^{13}-\frac{79\!\cdots\!17}{14\!\cdots\!92}a^{12}+\frac{29\!\cdots\!47}{24\!\cdots\!82}a^{11}-\frac{17\!\cdots\!29}{13\!\cdots\!72}a^{10}-\frac{40\!\cdots\!99}{13\!\cdots\!72}a^{9}+\frac{16\!\cdots\!29}{74\!\cdots\!46}a^{8}-\frac{36\!\cdots\!39}{16\!\cdots\!88}a^{7}-\frac{23\!\cdots\!63}{14\!\cdots\!92}a^{6}+\frac{61\!\cdots\!62}{37\!\cdots\!23}a^{5}-\frac{46\!\cdots\!68}{33\!\cdots\!93}a^{4}+\frac{15\!\cdots\!39}{37\!\cdots\!23}a^{3}-\frac{52\!\cdots\!43}{37\!\cdots\!23}a^{2}+\frac{35\!\cdots\!26}{37\!\cdots\!23}a-\frac{37\!\cdots\!26}{12\!\cdots\!41}$, $\frac{15\!\cdots\!91}{24\!\cdots\!82}a^{17}-\frac{13\!\cdots\!65}{74\!\cdots\!46}a^{16}-\frac{66\!\cdots\!57}{14\!\cdots\!92}a^{15}+\frac{18\!\cdots\!41}{14\!\cdots\!92}a^{14}+\frac{21\!\cdots\!87}{74\!\cdots\!46}a^{13}-\frac{40\!\cdots\!03}{14\!\cdots\!92}a^{12}+\frac{10\!\cdots\!55}{14\!\cdots\!92}a^{11}-\frac{20\!\cdots\!61}{24\!\cdots\!82}a^{10}-\frac{78\!\cdots\!01}{14\!\cdots\!92}a^{9}+\frac{20\!\cdots\!67}{14\!\cdots\!92}a^{8}-\frac{12\!\cdots\!49}{74\!\cdots\!46}a^{7}+\frac{57\!\cdots\!89}{49\!\cdots\!64}a^{6}+\frac{20\!\cdots\!59}{14\!\cdots\!92}a^{5}-\frac{49\!\cdots\!05}{37\!\cdots\!23}a^{4}+\frac{15\!\cdots\!75}{37\!\cdots\!23}a^{3}-\frac{20\!\cdots\!63}{74\!\cdots\!46}a^{2}+\frac{81\!\cdots\!95}{37\!\cdots\!23}a-\frac{74\!\cdots\!08}{37\!\cdots\!23}$, $\frac{15\!\cdots\!61}{49\!\cdots\!64}a^{17}-\frac{34\!\cdots\!62}{37\!\cdots\!23}a^{16}-\frac{85\!\cdots\!45}{37\!\cdots\!23}a^{15}+\frac{93\!\cdots\!63}{14\!\cdots\!92}a^{14}+\frac{10\!\cdots\!21}{74\!\cdots\!46}a^{13}-\frac{51\!\cdots\!09}{37\!\cdots\!23}a^{12}+\frac{51\!\cdots\!45}{14\!\cdots\!92}a^{11}-\frac{15\!\cdots\!48}{37\!\cdots\!77}a^{10}-\frac{22\!\cdots\!67}{67\!\cdots\!86}a^{9}+\frac{11\!\cdots\!91}{14\!\cdots\!92}a^{8}-\frac{35\!\cdots\!78}{37\!\cdots\!23}a^{7}+\frac{29\!\cdots\!95}{24\!\cdots\!82}a^{6}+\frac{30\!\cdots\!33}{37\!\cdots\!23}a^{5}-\frac{63\!\cdots\!45}{67\!\cdots\!86}a^{4}+\frac{93\!\cdots\!63}{37\!\cdots\!23}a^{3}-\frac{11\!\cdots\!36}{37\!\cdots\!23}a^{2}-\frac{24\!\cdots\!40}{37\!\cdots\!23}a-\frac{34\!\cdots\!83}{37\!\cdots\!23}$, $\frac{92\!\cdots\!69}{12\!\cdots\!28}a^{17}-\frac{29\!\cdots\!05}{12\!\cdots\!28}a^{16}-\frac{16\!\cdots\!69}{31\!\cdots\!07}a^{15}+\frac{10\!\cdots\!53}{62\!\cdots\!14}a^{14}+\frac{41\!\cdots\!91}{12\!\cdots\!28}a^{13}-\frac{21\!\cdots\!85}{62\!\cdots\!14}a^{12}+\frac{60\!\cdots\!39}{69\!\cdots\!46}a^{11}-\frac{12\!\cdots\!41}{12\!\cdots\!28}a^{10}-\frac{15\!\cdots\!97}{62\!\cdots\!14}a^{9}+\frac{70\!\cdots\!24}{31\!\cdots\!07}a^{8}-\frac{26\!\cdots\!39}{10\!\cdots\!36}a^{7}-\frac{27\!\cdots\!62}{31\!\cdots\!07}a^{6}+\frac{34\!\cdots\!53}{12\!\cdots\!28}a^{5}-\frac{18\!\cdots\!57}{62\!\cdots\!14}a^{4}+\frac{59\!\cdots\!44}{31\!\cdots\!07}a^{3}+\frac{46\!\cdots\!21}{62\!\cdots\!14}a^{2}-\frac{19\!\cdots\!48}{28\!\cdots\!37}a+\frac{94\!\cdots\!74}{34\!\cdots\!23}$, $\frac{82\!\cdots\!05}{51\!\cdots\!48}a^{17}-\frac{55\!\cdots\!55}{17\!\cdots\!16}a^{16}-\frac{66\!\cdots\!79}{42\!\cdots\!29}a^{15}+\frac{54\!\cdots\!21}{28\!\cdots\!86}a^{14}+\frac{52\!\cdots\!07}{51\!\cdots\!48}a^{13}-\frac{15\!\cdots\!05}{25\!\cdots\!74}a^{12}+\frac{29\!\cdots\!23}{25\!\cdots\!74}a^{11}-\frac{42\!\cdots\!85}{51\!\cdots\!48}a^{10}-\frac{32\!\cdots\!49}{25\!\cdots\!74}a^{9}+\frac{36\!\cdots\!61}{14\!\cdots\!43}a^{8}-\frac{64\!\cdots\!57}{51\!\cdots\!48}a^{7}-\frac{19\!\cdots\!70}{12\!\cdots\!87}a^{6}+\frac{10\!\cdots\!05}{57\!\cdots\!72}a^{5}-\frac{52\!\cdots\!83}{85\!\cdots\!58}a^{4}-\frac{62\!\cdots\!20}{12\!\cdots\!87}a^{3}-\frac{27\!\cdots\!33}{23\!\cdots\!34}a^{2}-\frac{17\!\cdots\!12}{12\!\cdots\!87}a+\frac{46\!\cdots\!46}{12\!\cdots\!87}$
| 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: | \( 7580353.718580581 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 7580353.718580581 \cdot 3}{2\cdot\sqrt{11351585665720125000000000000}}\cr\approx \mathstrut & 1.03693468856743 \end{aligned}\] (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 - 7*x^16 + 20*x^15 + 45*x^14 - 443*x^13 + 1134*x^12 - 1479*x^11 + 175*x^10 + 2090*x^9 - 2977*x^8 + 831*x^7 + 1899*x^6 - 2660*x^5 + 1110*x^4 - 640*x^3 + 280*x^2 - 480*x + 20)
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 - 7*x^16 + 20*x^15 + 45*x^14 - 443*x^13 + 1134*x^12 - 1479*x^11 + 175*x^10 + 2090*x^9 - 2977*x^8 + 831*x^7 + 1899*x^6 - 2660*x^5 + 1110*x^4 - 640*x^3 + 280*x^2 - 480*x + 20, 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 - 7*x^16 + 20*x^15 + 45*x^14 - 443*x^13 + 1134*x^12 - 1479*x^11 + 175*x^10 + 2090*x^9 - 2977*x^8 + 831*x^7 + 1899*x^6 - 2660*x^5 + 1110*x^4 - 640*x^3 + 280*x^2 - 480*x + 20);
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 - 7*x^16 + 20*x^15 + 45*x^14 - 443*x^13 + 1134*x^12 - 1479*x^11 + 175*x^10 + 2090*x^9 - 2977*x^8 + 831*x^7 + 1899*x^6 - 2660*x^5 + 1110*x^4 - 640*x^3 + 280*x^2 - 480*x + 20);
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
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 36 |
| The 12 conjugacy class representatives for $C_6:S_3$ |
| Character table for $C_6:S_3$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 3.1.300.1, 3.1.3675.1, 3.1.14700.1, 3.1.588.1, 6.2.450000.1, 6.2.1080450000.1, 6.2.67528125.1, 6.2.43218000.1, 9.1.9529569000000.1 |
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
| Galois closure: | data not computed |
| Degree 18 sibling: | 18.0.34054756997160375000000000000.1 |
| 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 | R | R | R | R | ${\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.2.0.1}{2} }^{9}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{9}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 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 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(3\)
| 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(5\)
| 5.6.5.1 | $x^{6} + 5$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ |
| 5.12.10.1 | $x^{12} + 20 x^{7} + 10 x^{6} + 50 x^{2} + 100 x + 25$ | $6$ | $2$ | $10$ | $D_6$ | $[\ ]_{6}^{2}$ | |
|
\(7\)
| 7.18.12.1 | $x^{18} + 3 x^{16} + 57 x^{15} + 15 x^{14} + 90 x^{13} + 424 x^{12} - 921 x^{11} - 3090 x^{10} - 6496 x^{9} - 10560 x^{8} + 6912 x^{7} + 28033 x^{6} + 33237 x^{5} + 188463 x^{4} - 139476 x^{3} + 351552 x^{2} - 514905 x + 582014$ | $3$ | $6$ | $12$ | $C_6 \times C_3$ | $[\ ]_{3}^{6}$ |