Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 10*x^16 + 91*x^15 + 151*x^14 + 34*x^13 + 3415*x^12 + 14395*x^11 + 29125*x^10 - 21192*x^9 - 97073*x^8 - 90513*x^7 + 153009*x^6 + 344700*x^5 - 49059*x^4 - 421497*x^3 - 92718*x^2 + 222021*x + 111051)
gp: K = bnfinit(y^18 - 3*y^17 + 10*y^16 + 91*y^15 + 151*y^14 + 34*y^13 + 3415*y^12 + 14395*y^11 + 29125*y^10 - 21192*y^9 - 97073*y^8 - 90513*y^7 + 153009*y^6 + 344700*y^5 - 49059*y^4 - 421497*y^3 - 92718*y^2 + 222021*y + 111051, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 + 10*x^16 + 91*x^15 + 151*x^14 + 34*x^13 + 3415*x^12 + 14395*x^11 + 29125*x^10 - 21192*x^9 - 97073*x^8 - 90513*x^7 + 153009*x^6 + 344700*x^5 - 49059*x^4 - 421497*x^3 - 92718*x^2 + 222021*x + 111051);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 + 10*x^16 + 91*x^15 + 151*x^14 + 34*x^13 + 3415*x^12 + 14395*x^11 + 29125*x^10 - 21192*x^9 - 97073*x^8 - 90513*x^7 + 153009*x^6 + 344700*x^5 - 49059*x^4 - 421497*x^3 - 92718*x^2 + 222021*x + 111051)
\( x^{18} - 3 x^{17} + 10 x^{16} + 91 x^{15} + 151 x^{14} + 34 x^{13} + 3415 x^{12} + 14395 x^{11} + \cdots + 111051 \)
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: |
\(-127192258415404127890787986833408\)
\(\medspace = -\,2^{20}\cdot 3^{9}\cdot 151^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(60.75\) | 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: | $2^{4/3}3^{1/2}151^{5/6}\approx 285.5933413976806$ | ||
| Ramified primes: |
\(2\), \(3\), \(151\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}$, $\frac{1}{9}a^{8}-\frac{1}{9}a^{7}+\frac{1}{9}a^{6}+\frac{2}{9}a^{5}+\frac{4}{9}a^{4}-\frac{1}{9}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{9}a^{9}-\frac{1}{3}a^{5}-\frac{4}{9}a^{3}-\frac{1}{3}a$, $\frac{1}{9}a^{10}-\frac{1}{9}a^{4}$, $\frac{1}{27}a^{11}+\frac{1}{27}a^{10}-\frac{1}{27}a^{9}+\frac{1}{27}a^{8}+\frac{2}{27}a^{7}+\frac{4}{27}a^{6}+\frac{7}{27}a^{5}+\frac{2}{9}a^{4}-\frac{1}{9}a^{3}+\frac{1}{3}a$, $\frac{1}{54}a^{12}-\frac{1}{54}a^{11}-\frac{1}{18}a^{10}-\frac{1}{18}a^{8}+\frac{1}{18}a^{7}-\frac{2}{27}a^{6}-\frac{5}{54}a^{5}-\frac{1}{2}a^{4}-\frac{1}{9}a^{3}-\frac{1}{6}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{54}a^{13}+\frac{1}{54}a^{10}-\frac{1}{54}a^{9}-\frac{1}{27}a^{8}-\frac{5}{54}a^{7}+\frac{1}{54}a^{6}+\frac{1}{27}a^{5}+\frac{7}{18}a^{4}-\frac{1}{6}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{2}$, $\frac{1}{486}a^{14}-\frac{1}{486}a^{13}+\frac{1}{486}a^{12}-\frac{1}{243}a^{11}+\frac{23}{486}a^{10}-\frac{5}{486}a^{9}+\frac{8}{243}a^{8}-\frac{55}{486}a^{7}-\frac{5}{486}a^{6}+\frac{32}{81}a^{5}-\frac{7}{162}a^{4}-\frac{17}{54}a^{3}-\frac{23}{54}a^{2}+\frac{4}{9}a-\frac{1}{3}$, $\frac{1}{486}a^{15}-\frac{1}{486}a^{12}+\frac{1}{162}a^{11}-\frac{25}{486}a^{9}-\frac{1}{162}a^{8}+\frac{2}{81}a^{7}+\frac{7}{486}a^{6}-\frac{1}{54}a^{5}-\frac{38}{81}a^{4}+\frac{1}{27}a^{3}+\frac{19}{54}a^{2}-\frac{2}{9}a-\frac{1}{3}$, $\frac{1}{3402}a^{16}-\frac{1}{3402}a^{14}+\frac{1}{189}a^{13}+\frac{10}{1701}a^{12}-\frac{17}{1701}a^{11}+\frac{1}{567}a^{10}+\frac{82}{1701}a^{9}+\frac{79}{1701}a^{8}+\frac{184}{1701}a^{7}-\frac{11}{1701}a^{6}+\frac{260}{567}a^{5}-\frac{113}{1134}a^{4}+\frac{8}{21}a^{3}-\frac{43}{378}a^{2}-\frac{13}{63}a-\frac{8}{21}$, $\frac{1}{20\!\cdots\!42}a^{17}-\frac{23\!\cdots\!89}{15\!\cdots\!34}a^{16}+\frac{88\!\cdots\!77}{20\!\cdots\!42}a^{15}-\frac{22\!\cdots\!39}{34\!\cdots\!07}a^{14}-\frac{74\!\cdots\!94}{14\!\cdots\!03}a^{13}-\frac{37\!\cdots\!70}{10\!\cdots\!21}a^{12}+\frac{10\!\cdots\!29}{68\!\cdots\!14}a^{11}+\frac{11\!\cdots\!15}{20\!\cdots\!42}a^{10}-\frac{58\!\cdots\!69}{10\!\cdots\!21}a^{9}+\frac{65\!\cdots\!45}{20\!\cdots\!42}a^{8}+\frac{42\!\cdots\!05}{36\!\cdots\!26}a^{7}-\frac{27\!\cdots\!82}{34\!\cdots\!07}a^{6}+\frac{91\!\cdots\!22}{11\!\cdots\!69}a^{5}+\frac{76\!\cdots\!38}{57\!\cdots\!31}a^{4}+\frac{22\!\cdots\!71}{36\!\cdots\!26}a^{3}-\frac{57\!\cdots\!65}{25\!\cdots\!82}a^{2}-\frac{21\!\cdots\!79}{65\!\cdots\!38}a-\frac{70\!\cdots\!05}{14\!\cdots\!49}$
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
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: |
\( -\frac{297315354830625725}{2182365564981978672063} a^{17} + \frac{1214305770600668876}{2182365564981978672063} a^{16} - \frac{8473984410150345367}{4364731129963957344126} a^{15} - \frac{45362029987017806209}{4364731129963957344126} a^{14} - \frac{39000447423334776965}{4364731129963957344126} a^{13} + \frac{14869738806380916730}{2182365564981978672063} a^{12} - \frac{2055594753431207547001}{4364731129963957344126} a^{11} - \frac{6331028965883345065769}{4364731129963957344126} a^{10} - \frac{5041295140694853050192}{2182365564981978672063} a^{9} + \frac{8210584184673992140823}{1454910376654652448042} a^{8} + \frac{10994171243113242672469}{1454910376654652448042} a^{7} + \frac{800586997084220703217}{242485062775775408007} a^{6} - \frac{12287494082943167480429}{484970125551550816014} a^{5} - \frac{3300969313035205355117}{161656708517183605338} a^{4} + \frac{5287346800635609458101}{161656708517183605338} a^{3} + \frac{669343280074647520630}{26942784752863934223} a^{2} - \frac{481667236537586183171}{26942784752863934223} a - \frac{109867102990607432974}{8980928250954644741} \)
(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: |
$\frac{55\!\cdots\!82}{43\!\cdots\!99}a^{17}-\frac{49\!\cdots\!25}{86\!\cdots\!98}a^{16}+\frac{63\!\cdots\!05}{32\!\cdots\!74}a^{15}+\frac{40\!\cdots\!75}{43\!\cdots\!99}a^{14}+\frac{29\!\cdots\!46}{61\!\cdots\!57}a^{13}-\frac{34\!\cdots\!09}{28\!\cdots\!66}a^{12}+\frac{19\!\cdots\!50}{43\!\cdots\!99}a^{11}+\frac{10\!\cdots\!75}{86\!\cdots\!98}a^{10}+\frac{46\!\cdots\!35}{28\!\cdots\!66}a^{9}-\frac{28\!\cdots\!20}{43\!\cdots\!99}a^{8}-\frac{78\!\cdots\!63}{12\!\cdots\!14}a^{7}-\frac{16\!\cdots\!27}{28\!\cdots\!66}a^{6}+\frac{39\!\cdots\!82}{14\!\cdots\!33}a^{5}+\frac{36\!\cdots\!71}{24\!\cdots\!89}a^{4}-\frac{38\!\cdots\!17}{98\!\cdots\!39}a^{3}-\frac{32\!\cdots\!31}{16\!\cdots\!37}a^{2}+\frac{22\!\cdots\!39}{10\!\cdots\!58}a+\frac{19\!\cdots\!80}{17\!\cdots\!93}$, $\frac{40\!\cdots\!67}{41\!\cdots\!58}a^{17}-\frac{12\!\cdots\!85}{32\!\cdots\!66}a^{16}+\frac{28\!\cdots\!98}{20\!\cdots\!29}a^{15}+\frac{10\!\cdots\!01}{13\!\cdots\!86}a^{14}+\frac{27\!\cdots\!31}{41\!\cdots\!58}a^{13}-\frac{89\!\cdots\!70}{20\!\cdots\!29}a^{12}+\frac{23\!\cdots\!76}{69\!\cdots\!43}a^{11}+\frac{21\!\cdots\!56}{20\!\cdots\!29}a^{10}+\frac{35\!\cdots\!56}{20\!\cdots\!29}a^{9}-\frac{81\!\cdots\!11}{20\!\cdots\!29}a^{8}-\frac{37\!\cdots\!69}{69\!\cdots\!43}a^{7}-\frac{18\!\cdots\!93}{69\!\cdots\!43}a^{6}+\frac{91\!\cdots\!81}{51\!\cdots\!18}a^{5}+\frac{35\!\cdots\!09}{23\!\cdots\!38}a^{4}-\frac{57\!\cdots\!08}{25\!\cdots\!09}a^{3}-\frac{27\!\cdots\!55}{15\!\cdots\!54}a^{2}+\frac{46\!\cdots\!11}{39\!\cdots\!86}a+\frac{79\!\cdots\!39}{86\!\cdots\!03}$, $\frac{11\!\cdots\!25}{13\!\cdots\!86}a^{17}-\frac{79\!\cdots\!17}{35\!\cdots\!74}a^{16}+\frac{56\!\cdots\!19}{69\!\cdots\!43}a^{15}+\frac{10\!\cdots\!41}{13\!\cdots\!86}a^{14}+\frac{66\!\cdots\!17}{46\!\cdots\!62}a^{13}+\frac{79\!\cdots\!59}{69\!\cdots\!43}a^{12}+\frac{19\!\cdots\!11}{69\!\cdots\!43}a^{11}+\frac{93\!\cdots\!67}{77\!\cdots\!27}a^{10}+\frac{19\!\cdots\!80}{69\!\cdots\!43}a^{9}-\frac{15\!\cdots\!87}{69\!\cdots\!43}a^{8}-\frac{47\!\cdots\!86}{77\!\cdots\!27}a^{7}-\frac{73\!\cdots\!34}{69\!\cdots\!43}a^{6}+\frac{54\!\cdots\!05}{15\!\cdots\!54}a^{5}+\frac{49\!\cdots\!83}{23\!\cdots\!38}a^{4}+\frac{66\!\cdots\!87}{77\!\cdots\!27}a^{3}-\frac{19\!\cdots\!43}{15\!\cdots\!54}a^{2}-\frac{43\!\cdots\!73}{39\!\cdots\!86}a-\frac{23\!\cdots\!63}{86\!\cdots\!03}$, $\frac{84\!\cdots\!35}{20\!\cdots\!42}a^{17}-\frac{11\!\cdots\!62}{79\!\cdots\!17}a^{16}+\frac{98\!\cdots\!09}{20\!\cdots\!42}a^{15}+\frac{13\!\cdots\!14}{38\!\cdots\!23}a^{14}+\frac{12\!\cdots\!79}{29\!\cdots\!06}a^{13}-\frac{12\!\cdots\!62}{10\!\cdots\!21}a^{12}+\frac{48\!\cdots\!11}{34\!\cdots\!07}a^{11}+\frac{10\!\cdots\!27}{20\!\cdots\!42}a^{10}+\frac{18\!\cdots\!25}{20\!\cdots\!42}a^{9}-\frac{14\!\cdots\!83}{10\!\cdots\!21}a^{8}-\frac{32\!\cdots\!45}{97\!\cdots\!02}a^{7}-\frac{13\!\cdots\!39}{84\!\cdots\!94}a^{6}+\frac{16\!\cdots\!93}{22\!\cdots\!38}a^{5}+\frac{39\!\cdots\!57}{38\!\cdots\!54}a^{4}-\frac{15\!\cdots\!90}{18\!\cdots\!63}a^{3}-\frac{48\!\cdots\!50}{38\!\cdots\!23}a^{2}+\frac{44\!\cdots\!26}{97\!\cdots\!57}a+\frac{57\!\cdots\!77}{84\!\cdots\!94}$, $\frac{12\!\cdots\!04}{14\!\cdots\!03}a^{17}-\frac{12\!\cdots\!87}{32\!\cdots\!66}a^{16}+\frac{19\!\cdots\!76}{14\!\cdots\!03}a^{15}+\frac{30\!\cdots\!65}{48\!\cdots\!01}a^{14}+\frac{12\!\cdots\!15}{29\!\cdots\!06}a^{13}-\frac{13\!\cdots\!99}{14\!\cdots\!03}a^{12}+\frac{27\!\cdots\!77}{97\!\cdots\!02}a^{11}+\frac{12\!\cdots\!35}{14\!\cdots\!03}a^{10}+\frac{34\!\cdots\!51}{29\!\cdots\!06}a^{9}-\frac{14\!\cdots\!75}{29\!\cdots\!06}a^{8}-\frac{35\!\cdots\!12}{48\!\cdots\!01}a^{7}-\frac{66\!\cdots\!33}{97\!\cdots\!02}a^{6}+\frac{56\!\cdots\!55}{32\!\cdots\!34}a^{5}+\frac{31\!\cdots\!31}{16\!\cdots\!66}a^{4}-\frac{14\!\cdots\!39}{10\!\cdots\!78}a^{3}-\frac{23\!\cdots\!25}{10\!\cdots\!78}a^{2}-\frac{16\!\cdots\!59}{27\!\cdots\!02}a-\frac{37\!\cdots\!15}{17\!\cdots\!06}$, $\frac{25\!\cdots\!57}{12\!\cdots\!97}a^{17}-\frac{26\!\cdots\!83}{25\!\cdots\!94}a^{16}+\frac{47\!\cdots\!92}{12\!\cdots\!97}a^{15}+\frac{50\!\cdots\!33}{43\!\cdots\!99}a^{14}-\frac{72\!\cdots\!77}{37\!\cdots\!42}a^{13}-\frac{40\!\cdots\!74}{12\!\cdots\!97}a^{12}+\frac{23\!\cdots\!27}{35\!\cdots\!86}a^{11}+\frac{17\!\cdots\!89}{12\!\cdots\!97}a^{10}+\frac{26\!\cdots\!87}{25\!\cdots\!94}a^{9}-\frac{34\!\cdots\!17}{25\!\cdots\!94}a^{8}-\frac{10\!\cdots\!39}{20\!\cdots\!19}a^{7}+\frac{42\!\cdots\!95}{86\!\cdots\!98}a^{6}+\frac{16\!\cdots\!29}{28\!\cdots\!66}a^{5}-\frac{47\!\cdots\!33}{14\!\cdots\!34}a^{4}-\frac{65\!\cdots\!29}{72\!\cdots\!14}a^{3}-\frac{34\!\cdots\!67}{32\!\cdots\!74}a^{2}+\frac{54\!\cdots\!59}{10\!\cdots\!58}a+\frac{23\!\cdots\!61}{35\!\cdots\!86}$, $\frac{16\!\cdots\!35}{34\!\cdots\!07}a^{17}-\frac{12\!\cdots\!01}{52\!\cdots\!78}a^{16}+\frac{28\!\cdots\!23}{34\!\cdots\!07}a^{15}+\frac{22\!\cdots\!15}{76\!\cdots\!46}a^{14}+\frac{53\!\cdots\!19}{86\!\cdots\!03}a^{13}-\frac{36\!\cdots\!98}{11\!\cdots\!69}a^{12}+\frac{18\!\cdots\!04}{11\!\cdots\!69}a^{11}+\frac{42\!\cdots\!58}{11\!\cdots\!69}a^{10}+\frac{18\!\cdots\!30}{38\!\cdots\!23}a^{9}-\frac{80\!\cdots\!41}{34\!\cdots\!07}a^{8}-\frac{37\!\cdots\!70}{48\!\cdots\!01}a^{7}-\frac{56\!\cdots\!02}{34\!\cdots\!07}a^{6}+\frac{10\!\cdots\!07}{11\!\cdots\!69}a^{5}-\frac{54\!\cdots\!93}{11\!\cdots\!62}a^{4}-\frac{67\!\cdots\!81}{54\!\cdots\!89}a^{3}+\frac{13\!\cdots\!15}{76\!\cdots\!46}a^{2}+\frac{63\!\cdots\!00}{97\!\cdots\!57}a-\frac{52\!\cdots\!31}{42\!\cdots\!47}$, $\frac{55\!\cdots\!80}{10\!\cdots\!21}a^{17}-\frac{13\!\cdots\!64}{79\!\cdots\!17}a^{16}+\frac{11\!\cdots\!25}{20\!\cdots\!42}a^{15}+\frac{61\!\cdots\!89}{12\!\cdots\!41}a^{14}+\frac{96\!\cdots\!58}{14\!\cdots\!03}a^{13}-\frac{69\!\cdots\!13}{10\!\cdots\!21}a^{12}+\frac{63\!\cdots\!91}{34\!\cdots\!07}a^{11}+\frac{15\!\cdots\!77}{20\!\cdots\!42}a^{10}+\frac{27\!\cdots\!19}{20\!\cdots\!42}a^{9}-\frac{15\!\cdots\!61}{10\!\cdots\!21}a^{8}-\frac{52\!\cdots\!29}{10\!\cdots\!78}a^{7}-\frac{15\!\cdots\!05}{68\!\cdots\!14}a^{6}+\frac{10\!\cdots\!35}{11\!\cdots\!69}a^{5}+\frac{18\!\cdots\!75}{11\!\cdots\!62}a^{4}-\frac{62\!\cdots\!52}{54\!\cdots\!89}a^{3}-\frac{75\!\cdots\!94}{38\!\cdots\!23}a^{2}+\frac{12\!\cdots\!91}{19\!\cdots\!14}a+\frac{94\!\cdots\!95}{84\!\cdots\!94}$
| 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: | \( 45444000566.5836 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 45444000566.5836 \cdot 1}{6\cdot\sqrt{127192258415404127890787986833408}}\cr\approx \mathstrut & 10.2497620214696 \end{aligned}\] (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 + 10*x^16 + 91*x^15 + 151*x^14 + 34*x^13 + 3415*x^12 + 14395*x^11 + 29125*x^10 - 21192*x^9 - 97073*x^8 - 90513*x^7 + 153009*x^6 + 344700*x^5 - 49059*x^4 - 421497*x^3 - 92718*x^2 + 222021*x + 111051)
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 + 10*x^16 + 91*x^15 + 151*x^14 + 34*x^13 + 3415*x^12 + 14395*x^11 + 29125*x^10 - 21192*x^9 - 97073*x^8 - 90513*x^7 + 153009*x^6 + 344700*x^5 - 49059*x^4 - 421497*x^3 - 92718*x^2 + 222021*x + 111051, 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 + 10*x^16 + 91*x^15 + 151*x^14 + 34*x^13 + 3415*x^12 + 14395*x^11 + 29125*x^10 - 21192*x^9 - 97073*x^8 - 90513*x^7 + 153009*x^6 + 344700*x^5 - 49059*x^4 - 421497*x^3 - 92718*x^2 + 222021*x + 111051);
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 + 10*x^16 + 91*x^15 + 151*x^14 + 34*x^13 + 3415*x^12 + 14395*x^11 + 29125*x^10 - 21192*x^9 - 97073*x^8 - 90513*x^7 + 153009*x^6 + 344700*x^5 - 49059*x^4 - 421497*x^3 - 92718*x^2 + 222021*x + 111051);
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
| \(\Q(\sqrt{-3}) \), 3.1.1812.1, 6.0.9850032.2, 6.0.9850032.3 |
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: | 12.0.12912877685616059713388544.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.4.2 | $x^{6} - 2 x^{3} + 4$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
| 2.12.16.3 | $x^{12} + 4 x^{11} + 8 x^{10} + 10 x^{9} + 8 x^{8} + 4 x^{7} + 2 x^{6} - 4 x^{5} - 4 x^{4} + 4 x^{3} + 4$ | $6$ | $2$ | $16$ | $C_6\times S_3$ | $[2]_{3}^{6}$ | |
|
\(3\)
| 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 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(151\)
| 151.3.2.2 | $x^{3} + 302$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 151.3.0.1 | $x^{3} + x + 145$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 151.6.3.1 | $x^{6} + 22801 x^{2} - 499227895$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 151.6.5.3 | $x^{6} + 906$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |