Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 25*x^16 + 97*x^15 + 324*x^14 - 1424*x^13 - 2893*x^12 + 12134*x^11 + 25059*x^10 - 36958*x^9 - 87068*x^8 + 95416*x^7 + 313237*x^6 + 151945*x^5 - 220562*x^4 - 362142*x^3 - 140268*x^2 + 158994*x + 138087)
gp: K = bnfinit(y^18 - 3*y^17 - 25*y^16 + 97*y^15 + 324*y^14 - 1424*y^13 - 2893*y^12 + 12134*y^11 + 25059*y^10 - 36958*y^9 - 87068*y^8 + 95416*y^7 + 313237*y^6 + 151945*y^5 - 220562*y^4 - 362142*y^3 - 140268*y^2 + 158994*y + 138087, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 - 25*x^16 + 97*x^15 + 324*x^14 - 1424*x^13 - 2893*x^12 + 12134*x^11 + 25059*x^10 - 36958*x^9 - 87068*x^8 + 95416*x^7 + 313237*x^6 + 151945*x^5 - 220562*x^4 - 362142*x^3 - 140268*x^2 + 158994*x + 138087);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 - 25*x^16 + 97*x^15 + 324*x^14 - 1424*x^13 - 2893*x^12 + 12134*x^11 + 25059*x^10 - 36958*x^9 - 87068*x^8 + 95416*x^7 + 313237*x^6 + 151945*x^5 - 220562*x^4 - 362142*x^3 - 140268*x^2 + 158994*x + 138087)
\( x^{18} - 3 x^{17} - 25 x^{16} + 97 x^{15} + 324 x^{14} - 1424 x^{13} - 2893 x^{12} + 12134 x^{11} + \cdots + 138087 \)
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: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-200930163501792205662554161152\)
\(\medspace = -\,2^{18}\cdot 3^{9}\cdot 7^{10}\cdot 13^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(42.46\) | 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: | $2^{3/2}3^{1/2}7^{5/6}13^{5/6}\approx 210.20358299362027$ | ||
| 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: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6}a^{12}+\frac{1}{3}a^{11}+\frac{1}{6}a^{10}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{2}$, $\frac{1}{6}a^{13}-\frac{1}{2}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{2}-\frac{1}{2}a$, $\frac{1}{6}a^{14}-\frac{1}{3}a^{11}-\frac{1}{6}a^{10}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{1842}a^{15}-\frac{74}{921}a^{14}+\frac{2}{921}a^{13}-\frac{25}{1842}a^{12}-\frac{279}{614}a^{11}-\frac{141}{614}a^{10}+\frac{328}{921}a^{9}-\frac{319}{921}a^{8}-\frac{21}{307}a^{7}+\frac{91}{307}a^{6}-\frac{22}{921}a^{5}+\frac{418}{921}a^{4}-\frac{173}{1842}a^{3}+\frac{334}{921}a^{2}+\frac{35}{614}a+\frac{45}{614}$, $\frac{1}{5526}a^{16}-\frac{103}{5526}a^{14}-\frac{47}{5526}a^{13}+\frac{125}{1842}a^{12}+\frac{343}{5526}a^{11}-\frac{2083}{5526}a^{10}+\frac{640}{2763}a^{9}+\frac{308}{921}a^{8}-\frac{1376}{2763}a^{7}-\frac{1063}{2763}a^{6}+\frac{539}{2763}a^{5}+\frac{1369}{5526}a^{4}+\frac{119}{2763}a^{3}-\frac{403}{2763}a^{2}-\frac{305}{614}a-\frac{401}{1842}$, $\frac{1}{11\!\cdots\!34}a^{17}-\frac{26\!\cdots\!31}{58\!\cdots\!17}a^{16}+\frac{13\!\cdots\!55}{58\!\cdots\!17}a^{15}-\frac{76\!\cdots\!08}{19\!\cdots\!39}a^{14}-\frac{45\!\cdots\!67}{11\!\cdots\!34}a^{13}-\frac{18\!\cdots\!57}{11\!\cdots\!34}a^{12}-\frac{29\!\cdots\!79}{19\!\cdots\!39}a^{11}-\frac{33\!\cdots\!11}{11\!\cdots\!34}a^{10}+\frac{14\!\cdots\!46}{58\!\cdots\!17}a^{9}+\frac{27\!\cdots\!32}{58\!\cdots\!17}a^{8}+\frac{34\!\cdots\!20}{19\!\cdots\!39}a^{7}-\frac{17\!\cdots\!32}{58\!\cdots\!17}a^{6}-\frac{47\!\cdots\!97}{11\!\cdots\!34}a^{5}+\frac{28\!\cdots\!39}{58\!\cdots\!17}a^{4}-\frac{16\!\cdots\!61}{11\!\cdots\!34}a^{3}+\frac{11\!\cdots\!95}{58\!\cdots\!17}a^{2}-\frac{62\!\cdots\!55}{19\!\cdots\!39}a+\frac{27\!\cdots\!77}{57\!\cdots\!34}$
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 $1$ (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: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{4483978576306547}{1115926554573558198} a^{17} + \frac{3557382330649714}{185987759095593033} a^{16} + \frac{74468977801064783}{1115926554573558198} a^{15} - \frac{282798671586213547}{557963277286779099} a^{14} - \frac{25383947323679686}{61995919698531011} a^{13} + \frac{7179652498751357887}{1115926554573558198} a^{12} + \frac{172938812805192796}{557963277286779099} a^{11} - \frac{54912094179860851537}{1115926554573558198} a^{10} - \frac{884223825364639105}{61995919698531011} a^{9} + \frac{96530931712142259139}{557963277286779099} a^{8} + \frac{25941886794083057876}{557963277286779099} a^{7} - \frac{258398468744761478959}{557963277286779099} a^{6} - \frac{499002537550884123773}{1115926554573558198} a^{5} + \frac{95627786631079824050}{557963277286779099} a^{4} + \frac{328559708883832918883}{557963277286779099} a^{3} + \frac{79592079321924536408}{185987759095593033} a^{2} - \frac{34094619062556010075}{185987759095593033} a - \frac{590831116390971609}{1850624468612866} \)
(order $6$)
| 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: |
$\frac{15\!\cdots\!87}{39\!\cdots\!74}a^{17}-\frac{31\!\cdots\!29}{13\!\cdots\!58}a^{16}-\frac{67\!\cdots\!02}{19\!\cdots\!87}a^{15}+\frac{71\!\cdots\!35}{13\!\cdots\!58}a^{14}-\frac{12\!\cdots\!77}{39\!\cdots\!74}a^{13}-\frac{38\!\cdots\!69}{66\!\cdots\!29}a^{12}+\frac{14\!\cdots\!46}{19\!\cdots\!87}a^{11}+\frac{75\!\cdots\!66}{19\!\cdots\!87}a^{10}-\frac{70\!\cdots\!62}{19\!\cdots\!87}a^{9}-\frac{78\!\cdots\!92}{66\!\cdots\!29}a^{8}+\frac{20\!\cdots\!35}{19\!\cdots\!87}a^{7}+\frac{58\!\cdots\!66}{19\!\cdots\!87}a^{6}+\frac{88\!\cdots\!97}{13\!\cdots\!58}a^{5}-\frac{28\!\cdots\!85}{13\!\cdots\!58}a^{4}-\frac{12\!\cdots\!37}{39\!\cdots\!74}a^{3}-\frac{12\!\cdots\!25}{19\!\cdots\!87}a^{2}+\frac{13\!\cdots\!92}{66\!\cdots\!29}a+\frac{34\!\cdots\!76}{99\!\cdots\!87}$, $\frac{74\!\cdots\!64}{59\!\cdots\!61}a^{17}-\frac{68\!\cdots\!43}{11\!\cdots\!22}a^{16}-\frac{13\!\cdots\!23}{59\!\cdots\!61}a^{15}+\frac{92\!\cdots\!64}{59\!\cdots\!61}a^{14}+\frac{93\!\cdots\!90}{59\!\cdots\!61}a^{13}-\frac{11\!\cdots\!21}{59\!\cdots\!61}a^{12}-\frac{25\!\cdots\!32}{59\!\cdots\!61}a^{11}+\frac{62\!\cdots\!97}{39\!\cdots\!74}a^{10}+\frac{39\!\cdots\!41}{59\!\cdots\!61}a^{9}-\frac{33\!\cdots\!80}{59\!\cdots\!61}a^{8}-\frac{12\!\cdots\!07}{59\!\cdots\!61}a^{7}+\frac{99\!\cdots\!73}{66\!\cdots\!29}a^{6}+\frac{30\!\cdots\!07}{19\!\cdots\!87}a^{5}-\frac{19\!\cdots\!29}{39\!\cdots\!74}a^{4}-\frac{38\!\cdots\!06}{19\!\cdots\!87}a^{3}-\frac{17\!\cdots\!29}{11\!\cdots\!22}a^{2}+\frac{11\!\cdots\!31}{19\!\cdots\!87}a+\frac{65\!\cdots\!41}{59\!\cdots\!22}$, $\frac{14\!\cdots\!64}{58\!\cdots\!17}a^{17}-\frac{70\!\cdots\!93}{58\!\cdots\!17}a^{16}-\frac{49\!\cdots\!93}{11\!\cdots\!34}a^{15}+\frac{18\!\cdots\!32}{58\!\cdots\!17}a^{14}+\frac{15\!\cdots\!81}{58\!\cdots\!17}a^{13}-\frac{47\!\cdots\!51}{11\!\cdots\!34}a^{12}-\frac{26\!\cdots\!31}{11\!\cdots\!34}a^{11}+\frac{12\!\cdots\!89}{38\!\cdots\!78}a^{10}+\frac{53\!\cdots\!96}{58\!\cdots\!17}a^{9}-\frac{64\!\cdots\!96}{58\!\cdots\!17}a^{8}-\frac{17\!\cdots\!79}{58\!\cdots\!17}a^{7}+\frac{57\!\cdots\!23}{19\!\cdots\!39}a^{6}+\frac{18\!\cdots\!64}{64\!\cdots\!13}a^{5}-\frac{20\!\cdots\!07}{19\!\cdots\!39}a^{4}-\frac{48\!\cdots\!77}{12\!\cdots\!26}a^{3}-\frac{15\!\cdots\!20}{58\!\cdots\!17}a^{2}+\frac{45\!\cdots\!01}{38\!\cdots\!78}a+\frac{11\!\cdots\!01}{57\!\cdots\!34}$, $\frac{12\!\cdots\!87}{11\!\cdots\!34}a^{17}-\frac{82\!\cdots\!23}{11\!\cdots\!34}a^{16}-\frac{22\!\cdots\!64}{58\!\cdots\!17}a^{15}+\frac{84\!\cdots\!99}{58\!\cdots\!17}a^{14}-\frac{10\!\cdots\!43}{58\!\cdots\!17}a^{13}-\frac{83\!\cdots\!16}{58\!\cdots\!17}a^{12}+\frac{34\!\cdots\!47}{11\!\cdots\!34}a^{11}+\frac{32\!\cdots\!21}{38\!\cdots\!78}a^{10}-\frac{87\!\cdots\!08}{58\!\cdots\!17}a^{9}-\frac{14\!\cdots\!90}{58\!\cdots\!17}a^{8}+\frac{24\!\cdots\!98}{58\!\cdots\!17}a^{7}+\frac{40\!\cdots\!53}{64\!\cdots\!13}a^{6}-\frac{79\!\cdots\!35}{38\!\cdots\!78}a^{5}-\frac{32\!\cdots\!01}{38\!\cdots\!78}a^{4}-\frac{10\!\cdots\!65}{12\!\cdots\!26}a^{3}+\frac{26\!\cdots\!35}{11\!\cdots\!34}a^{2}+\frac{27\!\cdots\!15}{38\!\cdots\!78}a+\frac{75\!\cdots\!99}{57\!\cdots\!34}$, $\frac{22\!\cdots\!25}{11\!\cdots\!34}a^{17}-\frac{55\!\cdots\!64}{58\!\cdots\!17}a^{16}-\frac{36\!\cdots\!19}{11\!\cdots\!34}a^{15}+\frac{96\!\cdots\!57}{38\!\cdots\!78}a^{14}+\frac{20\!\cdots\!75}{11\!\cdots\!34}a^{13}-\frac{36\!\cdots\!83}{11\!\cdots\!34}a^{12}+\frac{31\!\cdots\!99}{38\!\cdots\!78}a^{11}+\frac{13\!\cdots\!69}{58\!\cdots\!17}a^{10}+\frac{32\!\cdots\!97}{58\!\cdots\!17}a^{9}-\frac{47\!\cdots\!32}{58\!\cdots\!17}a^{8}-\frac{11\!\cdots\!17}{64\!\cdots\!13}a^{7}+\frac{12\!\cdots\!27}{58\!\cdots\!17}a^{6}+\frac{24\!\cdots\!01}{11\!\cdots\!34}a^{5}-\frac{48\!\cdots\!31}{58\!\cdots\!17}a^{4}-\frac{16\!\cdots\!76}{58\!\cdots\!17}a^{3}-\frac{22\!\cdots\!61}{11\!\cdots\!34}a^{2}+\frac{34\!\cdots\!13}{38\!\cdots\!78}a+\frac{42\!\cdots\!68}{28\!\cdots\!17}$, $\frac{18\!\cdots\!25}{64\!\cdots\!13}a^{17}-\frac{15\!\cdots\!03}{11\!\cdots\!34}a^{16}-\frac{18\!\cdots\!43}{38\!\cdots\!78}a^{15}+\frac{41\!\cdots\!03}{11\!\cdots\!34}a^{14}+\frac{17\!\cdots\!66}{58\!\cdots\!17}a^{13}-\frac{59\!\cdots\!47}{12\!\cdots\!26}a^{12}-\frac{42\!\cdots\!25}{11\!\cdots\!34}a^{11}+\frac{41\!\cdots\!37}{11\!\cdots\!34}a^{10}+\frac{62\!\cdots\!42}{58\!\cdots\!17}a^{9}-\frac{80\!\cdots\!14}{64\!\cdots\!13}a^{8}-\frac{19\!\cdots\!55}{58\!\cdots\!17}a^{7}+\frac{19\!\cdots\!85}{58\!\cdots\!17}a^{6}+\frac{18\!\cdots\!31}{58\!\cdots\!17}a^{5}-\frac{14\!\cdots\!19}{11\!\cdots\!34}a^{4}-\frac{48\!\cdots\!03}{11\!\cdots\!34}a^{3}-\frac{17\!\cdots\!68}{58\!\cdots\!17}a^{2}+\frac{17\!\cdots\!57}{12\!\cdots\!26}a+\frac{13\!\cdots\!91}{57\!\cdots\!34}$, $\frac{73\!\cdots\!90}{45\!\cdots\!59}a^{17}-\frac{12\!\cdots\!96}{15\!\cdots\!53}a^{16}-\frac{22\!\cdots\!53}{91\!\cdots\!18}a^{15}+\frac{18\!\cdots\!21}{91\!\cdots\!18}a^{14}+\frac{57\!\cdots\!74}{45\!\cdots\!59}a^{13}-\frac{38\!\cdots\!37}{15\!\cdots\!53}a^{12}+\frac{26\!\cdots\!99}{91\!\cdots\!18}a^{11}+\frac{17\!\cdots\!77}{91\!\cdots\!18}a^{10}+\frac{13\!\cdots\!89}{45\!\cdots\!59}a^{9}-\frac{10\!\cdots\!90}{15\!\cdots\!53}a^{8}-\frac{41\!\cdots\!21}{45\!\cdots\!59}a^{7}+\frac{79\!\cdots\!00}{45\!\cdots\!59}a^{6}+\frac{70\!\cdots\!28}{45\!\cdots\!59}a^{5}-\frac{96\!\cdots\!31}{15\!\cdots\!53}a^{4}-\frac{61\!\cdots\!45}{30\!\cdots\!06}a^{3}-\frac{44\!\cdots\!85}{30\!\cdots\!06}a^{2}+\frac{21\!\cdots\!57}{30\!\cdots\!06}a+\frac{51\!\cdots\!23}{45\!\cdots\!18}$, $\frac{28\!\cdots\!84}{13\!\cdots\!77}a^{17}-\frac{19\!\cdots\!38}{45\!\cdots\!59}a^{16}-\frac{94\!\cdots\!50}{13\!\cdots\!77}a^{15}+\frac{28\!\cdots\!82}{13\!\cdots\!77}a^{14}+\frac{45\!\cdots\!30}{45\!\cdots\!59}a^{13}-\frac{51\!\cdots\!52}{13\!\cdots\!77}a^{12}-\frac{12\!\cdots\!89}{13\!\cdots\!77}a^{11}+\frac{49\!\cdots\!12}{13\!\cdots\!77}a^{10}+\frac{29\!\cdots\!43}{45\!\cdots\!59}a^{9}-\frac{19\!\cdots\!59}{13\!\cdots\!77}a^{8}-\frac{27\!\cdots\!93}{13\!\cdots\!77}a^{7}+\frac{56\!\cdots\!20}{13\!\cdots\!77}a^{6}+\frac{91\!\cdots\!64}{13\!\cdots\!77}a^{5}-\frac{58\!\cdots\!40}{13\!\cdots\!77}a^{4}-\frac{84\!\cdots\!81}{13\!\cdots\!77}a^{3}-\frac{29\!\cdots\!87}{45\!\cdots\!59}a^{2}+\frac{24\!\cdots\!52}{45\!\cdots\!59}a+\frac{11\!\cdots\!58}{22\!\cdots\!59}$
| 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: | \( 187527338.95768923 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 187527338.95768923 \cdot 1}{6\cdot\sqrt{200930163501792205662554161152}}\cr\approx \mathstrut & 1.06416650926527 \end{aligned}\] (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 - 25*x^16 + 97*x^15 + 324*x^14 - 1424*x^13 - 2893*x^12 + 12134*x^11 + 25059*x^10 - 36958*x^9 - 87068*x^8 + 95416*x^7 + 313237*x^6 + 151945*x^5 - 220562*x^4 - 362142*x^3 - 140268*x^2 + 158994*x + 138087)
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 - 25*x^16 + 97*x^15 + 324*x^14 - 1424*x^13 - 2893*x^12 + 12134*x^11 + 25059*x^10 - 36958*x^9 - 87068*x^8 + 95416*x^7 + 313237*x^6 + 151945*x^5 - 220562*x^4 - 362142*x^3 - 140268*x^2 + 158994*x + 138087, 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 - 25*x^16 + 97*x^15 + 324*x^14 - 1424*x^13 - 2893*x^12 + 12134*x^11 + 25059*x^10 - 36958*x^9 - 87068*x^8 + 95416*x^7 + 313237*x^6 + 151945*x^5 - 220562*x^4 - 362142*x^3 - 140268*x^2 + 158994*x + 138087);
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 - 25*x^16 + 97*x^15 + 324*x^14 - 1424*x^13 - 2893*x^12 + 12134*x^11 + 25059*x^10 - 36958*x^9 - 87068*x^8 + 95416*x^7 + 313237*x^6 + 151945*x^5 - 220562*x^4 - 362142*x^3 - 140268*x^2 + 158994*x + 138087);
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
$C_3\times S_3^2$ (as 18T46):
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 27 conjugacy class representatives for $C_3\times S_3^2$ |
| Character table for $C_3\times S_3^2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.2184.1, 6.0.223587.1, 6.0.14309568.3 |
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 12 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.898666574987777490026496.6 |
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 | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.0.1 | $x^{6} + x^{4} + x^{3} + x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 2.12.18.15 | $x^{12} - 12 x^{11} - 56 x^{9} + 2372 x^{8} + 8992 x^{7} + 115648 x^{6} + 164160 x^{5} + 1305648 x^{4} - 282560 x^{3} + 5591296 x^{2} - 9640320 x + 8497856$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ | |
|
\(3\)
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 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 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(7\)
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 7.3.2.1 | $x^{3} + 14$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.6.3.2 | $x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 7.6.5.4 | $x^{6} + 14$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
|
\(13\)
| 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 13.3.2.1 | $x^{3} + 26$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 13.6.5.3 | $x^{6} + 39$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 13.6.3.1 | $x^{6} + 390 x^{5} + 50743 x^{4} + 2208202 x^{3} + 765301 x^{2} + 5017316 x + 24555184$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |