Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 11*x^16 - 75*x^15 + 321*x^14 + 147*x^13 + 641*x^12 - 11589*x^11 + 15447*x^10 + 30186*x^9 - 55196*x^8 - 52869*x^7 + 188785*x^6 + 172251*x^5 + 49959*x^4 + 210708*x^3 + 285795*x^2 + 30537*x + 1161)
gp: K = bnfinit(y^18 - 11*y^16 - 75*y^15 + 321*y^14 + 147*y^13 + 641*y^12 - 11589*y^11 + 15447*y^10 + 30186*y^9 - 55196*y^8 - 52869*y^7 + 188785*y^6 + 172251*y^5 + 49959*y^4 + 210708*y^3 + 285795*y^2 + 30537*y + 1161, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 11*x^16 - 75*x^15 + 321*x^14 + 147*x^13 + 641*x^12 - 11589*x^11 + 15447*x^10 + 30186*x^9 - 55196*x^8 - 52869*x^7 + 188785*x^6 + 172251*x^5 + 49959*x^4 + 210708*x^3 + 285795*x^2 + 30537*x + 1161);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 11*x^16 - 75*x^15 + 321*x^14 + 147*x^13 + 641*x^12 - 11589*x^11 + 15447*x^10 + 30186*x^9 - 55196*x^8 - 52869*x^7 + 188785*x^6 + 172251*x^5 + 49959*x^4 + 210708*x^3 + 285795*x^2 + 30537*x + 1161)
\( x^{18} - 11 x^{16} - 75 x^{15} + 321 x^{14} + 147 x^{13} + 641 x^{12} - 11589 x^{11} + 15447 x^{10} + \cdots + 1161 \)
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: |
\(-154125661129436703526043328000000\)
\(\medspace = -\,2^{12}\cdot 3^{9}\cdot 5^{6}\cdot 13^{10}\cdot 31^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(61.41\) | 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: | $2^{2/3}3^{1/2}5^{1/2}13^{5/6}31^{2/3}\approx 514.3509537200081$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(13\), \(31\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{12}-\frac{1}{3}a^{8}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{9}a^{15}+\frac{1}{9}a^{13}-\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{2}{9}a^{9}+\frac{1}{3}a^{8}+\frac{1}{9}a^{5}-\frac{1}{3}a^{4}+\frac{4}{9}a^{3}+\frac{1}{3}a$, $\frac{1}{34371}a^{16}-\frac{493}{11457}a^{15}+\frac{1264}{34371}a^{14}+\frac{3892}{11457}a^{13}-\frac{926}{11457}a^{12}+\frac{655}{11457}a^{11}+\frac{1244}{34371}a^{10}+\frac{4481}{11457}a^{9}+\frac{4853}{11457}a^{8}-\frac{184}{3819}a^{7}+\frac{2539}{34371}a^{6}+\frac{199}{11457}a^{5}-\frac{6329}{34371}a^{4}+\frac{2195}{11457}a^{3}-\frac{1432}{11457}a^{2}+\frac{1310}{3819}a-\frac{443}{3819}$, $\frac{1}{74\!\cdots\!89}a^{17}-\frac{17\!\cdots\!81}{74\!\cdots\!89}a^{16}-\frac{36\!\cdots\!95}{74\!\cdots\!89}a^{15}+\frac{83\!\cdots\!16}{74\!\cdots\!89}a^{14}+\frac{56\!\cdots\!43}{43\!\cdots\!59}a^{13}+\frac{79\!\cdots\!60}{63\!\cdots\!17}a^{12}-\frac{34\!\cdots\!13}{74\!\cdots\!89}a^{11}+\frac{19\!\cdots\!42}{57\!\cdots\!53}a^{10}-\frac{28\!\cdots\!33}{82\!\cdots\!21}a^{9}+\frac{10\!\cdots\!19}{24\!\cdots\!63}a^{8}+\frac{26\!\cdots\!25}{74\!\cdots\!89}a^{7}-\frac{86\!\cdots\!26}{74\!\cdots\!89}a^{6}+\frac{41\!\cdots\!98}{74\!\cdots\!89}a^{5}-\frac{55\!\cdots\!38}{74\!\cdots\!89}a^{4}+\frac{78\!\cdots\!66}{27\!\cdots\!07}a^{3}+\frac{19\!\cdots\!10}{24\!\cdots\!63}a^{2}+\frac{14\!\cdots\!04}{82\!\cdots\!21}a-\frac{13\!\cdots\!76}{82\!\cdots\!21}$
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
$C_{3}$, 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: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{225870134970968454935051636}{2162915254159580529129852396249} a^{17} - \frac{7541670641543877254081904}{720971751386526843043284132083} a^{16} - \frac{828065039489618787578115514}{720971751386526843043284132083} a^{15} - \frac{5555717911027175180535250948}{720971751386526843043284132083} a^{14} + \frac{74151435637955055953987208520}{2162915254159580529129852396249} a^{13} + \frac{8568277340235851870479924991}{720971751386526843043284132083} a^{12} + \frac{140192991892965865228173653782}{2162915254159580529129852396249} a^{11} - \frac{873809216803819927729283219102}{720971751386526843043284132083} a^{10} + \frac{3736777842193481179988644136204}{2162915254159580529129852396249} a^{9} + \frac{2165522315529573061859647606695}{720971751386526843043284132083} a^{8} - \frac{13474389197839878758634537164092}{2162915254159580529129852396249} a^{7} - \frac{3243902532402500866976094027176}{720971751386526843043284132083} a^{6} + \frac{14378795291836331488648832424738}{720971751386526843043284132083} a^{5} + \frac{11229897221990829367184343345585}{720971751386526843043284132083} a^{4} + \frac{8687011358732154152369969950582}{2162915254159580529129852396249} a^{3} + \frac{16241055410062159964150738150082}{720971751386526843043284132083} a^{2} + \frac{19561212094396052456404847971206}{720971751386526843043284132083} a + \frac{1433259002026806861146308757962}{720971751386526843043284132083} \)
(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: |
$\frac{53\!\cdots\!59}{10\!\cdots\!29}a^{17}-\frac{19\!\cdots\!06}{30\!\cdots\!87}a^{16}-\frac{58\!\cdots\!09}{10\!\cdots\!29}a^{15}-\frac{11\!\cdots\!41}{30\!\cdots\!87}a^{14}+\frac{19\!\cdots\!11}{11\!\cdots\!81}a^{13}+\frac{57\!\cdots\!59}{10\!\cdots\!29}a^{12}+\frac{36\!\cdots\!34}{11\!\cdots\!81}a^{11}-\frac{18\!\cdots\!18}{30\!\cdots\!87}a^{10}+\frac{99\!\cdots\!79}{11\!\cdots\!81}a^{9}+\frac{15\!\cdots\!79}{10\!\cdots\!29}a^{8}-\frac{32\!\cdots\!59}{10\!\cdots\!29}a^{7}-\frac{68\!\cdots\!33}{30\!\cdots\!87}a^{6}+\frac{10\!\cdots\!24}{10\!\cdots\!29}a^{5}+\frac{23\!\cdots\!41}{30\!\cdots\!87}a^{4}+\frac{16\!\cdots\!49}{10\!\cdots\!29}a^{3}+\frac{11\!\cdots\!77}{10\!\cdots\!29}a^{2}+\frac{24\!\cdots\!95}{17\!\cdots\!97}a-\frac{48\!\cdots\!46}{33\!\cdots\!43}$, $\frac{20\!\cdots\!13}{57\!\cdots\!53}a^{17}-\frac{20\!\cdots\!65}{63\!\cdots\!17}a^{16}-\frac{23\!\cdots\!77}{57\!\cdots\!53}a^{15}-\frac{44\!\cdots\!79}{19\!\cdots\!51}a^{14}+\frac{15\!\cdots\!14}{11\!\cdots\!81}a^{13}-\frac{63\!\cdots\!62}{19\!\cdots\!51}a^{12}+\frac{53\!\cdots\!50}{57\!\cdots\!53}a^{11}-\frac{83\!\cdots\!42}{19\!\cdots\!51}a^{10}+\frac{19\!\cdots\!20}{21\!\cdots\!39}a^{9}+\frac{18\!\cdots\!86}{21\!\cdots\!39}a^{8}-\frac{19\!\cdots\!05}{57\!\cdots\!53}a^{7}-\frac{16\!\cdots\!86}{19\!\cdots\!51}a^{6}+\frac{57\!\cdots\!67}{57\!\cdots\!53}a^{5}+\frac{11\!\cdots\!11}{63\!\cdots\!17}a^{4}-\frac{18\!\cdots\!23}{19\!\cdots\!51}a^{3}+\frac{37\!\cdots\!88}{63\!\cdots\!17}a^{2}+\frac{19\!\cdots\!89}{21\!\cdots\!39}a+\frac{37\!\cdots\!45}{21\!\cdots\!39}$, $\frac{28\!\cdots\!49}{57\!\cdots\!53}a^{17}+\frac{29\!\cdots\!30}{57\!\cdots\!53}a^{16}-\frac{31\!\cdots\!14}{57\!\cdots\!53}a^{15}-\frac{21\!\cdots\!54}{57\!\cdots\!53}a^{14}+\frac{51\!\cdots\!88}{33\!\cdots\!43}a^{13}+\frac{59\!\cdots\!07}{63\!\cdots\!17}a^{12}+\frac{17\!\cdots\!45}{57\!\cdots\!53}a^{11}-\frac{32\!\cdots\!05}{57\!\cdots\!53}a^{10}+\frac{44\!\cdots\!85}{63\!\cdots\!17}a^{9}+\frac{31\!\cdots\!15}{19\!\cdots\!51}a^{8}-\frac{15\!\cdots\!56}{57\!\cdots\!53}a^{7}-\frac{17\!\cdots\!24}{57\!\cdots\!53}a^{6}+\frac{56\!\cdots\!67}{57\!\cdots\!53}a^{5}+\frac{52\!\cdots\!36}{57\!\cdots\!53}a^{4}+\frac{38\!\cdots\!61}{21\!\cdots\!39}a^{3}+\frac{20\!\cdots\!02}{19\!\cdots\!51}a^{2}+\frac{95\!\cdots\!93}{63\!\cdots\!17}a+\frac{61\!\cdots\!80}{63\!\cdots\!17}$, $\frac{73\!\cdots\!53}{57\!\cdots\!53}a^{17}-\frac{53\!\cdots\!91}{57\!\cdots\!53}a^{16}-\frac{80\!\cdots\!46}{57\!\cdots\!53}a^{15}-\frac{54\!\cdots\!50}{57\!\cdots\!53}a^{14}+\frac{41\!\cdots\!44}{10\!\cdots\!29}a^{13}+\frac{35\!\cdots\!20}{19\!\cdots\!51}a^{12}+\frac{46\!\cdots\!65}{57\!\cdots\!53}a^{11}-\frac{85\!\cdots\!09}{57\!\cdots\!53}a^{10}+\frac{37\!\cdots\!01}{19\!\cdots\!51}a^{9}+\frac{74\!\cdots\!22}{19\!\cdots\!51}a^{8}-\frac{41\!\cdots\!36}{57\!\cdots\!53}a^{7}-\frac{38\!\cdots\!50}{57\!\cdots\!53}a^{6}+\frac{14\!\cdots\!03}{57\!\cdots\!53}a^{5}+\frac{12\!\cdots\!62}{57\!\cdots\!53}a^{4}+\frac{10\!\cdots\!65}{19\!\cdots\!51}a^{3}+\frac{51\!\cdots\!46}{19\!\cdots\!51}a^{2}+\frac{76\!\cdots\!03}{21\!\cdots\!39}a+\frac{15\!\cdots\!43}{63\!\cdots\!17}$, $\frac{43\!\cdots\!71}{74\!\cdots\!89}a^{17}+\frac{26\!\cdots\!34}{74\!\cdots\!89}a^{16}-\frac{55\!\cdots\!52}{74\!\cdots\!89}a^{15}-\frac{34\!\cdots\!06}{74\!\cdots\!89}a^{14}+\frac{11\!\cdots\!58}{68\!\cdots\!83}a^{13}+\frac{16\!\cdots\!96}{63\!\cdots\!17}a^{12}+\frac{26\!\cdots\!11}{74\!\cdots\!89}a^{11}-\frac{35\!\cdots\!20}{57\!\cdots\!53}a^{10}+\frac{87\!\cdots\!78}{24\!\cdots\!63}a^{9}+\frac{88\!\cdots\!62}{24\!\cdots\!63}a^{8}-\frac{38\!\cdots\!06}{74\!\cdots\!89}a^{7}-\frac{26\!\cdots\!23}{74\!\cdots\!89}a^{6}+\frac{95\!\cdots\!08}{74\!\cdots\!89}a^{5}+\frac{87\!\cdots\!55}{74\!\cdots\!89}a^{4}+\frac{79\!\cdots\!19}{24\!\cdots\!63}a^{3}+\frac{31\!\cdots\!38}{24\!\cdots\!63}a^{2}+\frac{12\!\cdots\!25}{82\!\cdots\!21}a+\frac{47\!\cdots\!81}{82\!\cdots\!21}$, $\frac{87\!\cdots\!74}{74\!\cdots\!89}a^{17}-\frac{39\!\cdots\!20}{24\!\cdots\!63}a^{16}-\frac{96\!\cdots\!80}{74\!\cdots\!89}a^{15}-\frac{72\!\cdots\!26}{82\!\cdots\!21}a^{14}+\frac{49\!\cdots\!14}{13\!\cdots\!77}a^{13}+\frac{32\!\cdots\!03}{19\!\cdots\!51}a^{12}+\frac{56\!\cdots\!97}{74\!\cdots\!89}a^{11}-\frac{86\!\cdots\!14}{63\!\cdots\!17}a^{10}+\frac{45\!\cdots\!92}{24\!\cdots\!63}a^{9}+\frac{29\!\cdots\!09}{82\!\cdots\!21}a^{8}-\frac{48\!\cdots\!26}{74\!\cdots\!89}a^{7}-\frac{50\!\cdots\!11}{82\!\cdots\!21}a^{6}+\frac{16\!\cdots\!35}{74\!\cdots\!89}a^{5}+\frac{49\!\cdots\!96}{24\!\cdots\!63}a^{4}+\frac{45\!\cdots\!74}{82\!\cdots\!21}a^{3}+\frac{20\!\cdots\!32}{82\!\cdots\!21}a^{2}+\frac{27\!\cdots\!62}{82\!\cdots\!21}a+\frac{84\!\cdots\!86}{27\!\cdots\!07}$, $\frac{74\!\cdots\!72}{74\!\cdots\!89}a^{17}-\frac{37\!\cdots\!11}{74\!\cdots\!89}a^{16}-\frac{86\!\cdots\!54}{74\!\cdots\!89}a^{15}-\frac{51\!\cdots\!72}{74\!\cdots\!89}a^{14}+\frac{47\!\cdots\!76}{13\!\cdots\!77}a^{13}+\frac{84\!\cdots\!43}{19\!\cdots\!51}a^{12}+\frac{28\!\cdots\!19}{74\!\cdots\!89}a^{11}-\frac{69\!\cdots\!41}{57\!\cdots\!53}a^{10}+\frac{51\!\cdots\!74}{24\!\cdots\!63}a^{9}+\frac{75\!\cdots\!77}{24\!\cdots\!63}a^{8}-\frac{55\!\cdots\!62}{74\!\cdots\!89}a^{7}-\frac{40\!\cdots\!34}{74\!\cdots\!89}a^{6}+\frac{16\!\cdots\!46}{74\!\cdots\!89}a^{5}+\frac{13\!\cdots\!14}{74\!\cdots\!89}a^{4}-\frac{27\!\cdots\!10}{24\!\cdots\!63}a^{3}-\frac{32\!\cdots\!14}{24\!\cdots\!63}a^{2}-\frac{15\!\cdots\!32}{27\!\cdots\!07}a-\frac{10\!\cdots\!21}{82\!\cdots\!21}$, $\frac{70\!\cdots\!52}{24\!\cdots\!63}a^{17}-\frac{95\!\cdots\!57}{24\!\cdots\!63}a^{16}-\frac{78\!\cdots\!36}{24\!\cdots\!63}a^{15}-\frac{53\!\cdots\!56}{24\!\cdots\!63}a^{14}+\frac{12\!\cdots\!34}{13\!\cdots\!77}a^{13}+\frac{27\!\cdots\!86}{63\!\cdots\!17}a^{12}+\frac{43\!\cdots\!50}{24\!\cdots\!63}a^{11}-\frac{63\!\cdots\!28}{19\!\cdots\!51}a^{10}+\frac{10\!\cdots\!35}{24\!\cdots\!63}a^{9}+\frac{24\!\cdots\!14}{27\!\cdots\!07}a^{8}-\frac{40\!\cdots\!83}{24\!\cdots\!63}a^{7}-\frac{37\!\cdots\!59}{24\!\cdots\!63}a^{6}+\frac{13\!\cdots\!90}{24\!\cdots\!63}a^{5}+\frac{11\!\cdots\!06}{24\!\cdots\!63}a^{4}+\frac{25\!\cdots\!04}{24\!\cdots\!63}a^{3}+\frac{49\!\cdots\!44}{82\!\cdots\!21}a^{2}+\frac{65\!\cdots\!68}{82\!\cdots\!21}a+\frac{92\!\cdots\!14}{27\!\cdots\!07}$
| 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: | \( 1503257555.4600825 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 1503257555.4600825 \cdot 3}{6\cdot\sqrt{154125661129436703526043328000000}}\cr\approx \mathstrut & 0.924027164140094 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 11*x^16 - 75*x^15 + 321*x^14 + 147*x^13 + 641*x^12 - 11589*x^11 + 15447*x^10 + 30186*x^9 - 55196*x^8 - 52869*x^7 + 188785*x^6 + 172251*x^5 + 49959*x^4 + 210708*x^3 + 285795*x^2 + 30537*x + 1161)
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 - 11*x^16 - 75*x^15 + 321*x^14 + 147*x^13 + 641*x^12 - 11589*x^11 + 15447*x^10 + 30186*x^9 - 55196*x^8 - 52869*x^7 + 188785*x^6 + 172251*x^5 + 49959*x^4 + 210708*x^3 + 285795*x^2 + 30537*x + 1161, 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 - 11*x^16 - 75*x^15 + 321*x^14 + 147*x^13 + 641*x^12 - 11589*x^11 + 15447*x^10 + 30186*x^9 - 55196*x^8 - 52869*x^7 + 188785*x^6 + 172251*x^5 + 49959*x^4 + 210708*x^3 + 285795*x^2 + 30537*x + 1161);
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 - 11*x^16 - 75*x^15 + 321*x^14 + 147*x^13 + 641*x^12 - 11589*x^11 + 15447*x^10 + 30186*x^9 - 55196*x^8 - 52869*x^7 + 188785*x^6 + 172251*x^5 + 49959*x^4 + 210708*x^3 + 285795*x^2 + 30537*x + 1161);
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 $C_3\times S_3^2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.780.1, 6.0.4385043.1, 6.0.1825200.1 |
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 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: | 12.0.2196752419666077156000000.1 |
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 | R | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{9}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.12.1 | $x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$ | $3$ | $6$ | $12$ | $S_3 \times C_3$ | $[\ ]_{3}^{6}$ |
|
\(3\)
| 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 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(5\)
| 5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(13\)
| 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 13.3.2.3 | $x^{3} + 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 13.6.3.2 | $x^{6} + 338 x^{2} - 24167$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 13.6.5.6 | $x^{6} + 78$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
|
\(31\)
| 31.3.0.1 | $x^{3} + x + 28$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 31.3.2.3 | $x^{3} + 155$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 31.6.4.2 | $x^{6} - 899 x^{3} + 2883$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 31.6.0.1 | $x^{6} + 19 x^{3} + 16 x^{2} + 8 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |