Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 15*x^16 + 21*x^15 - 189*x^14 + 447*x^13 - 277*x^12 - 819*x^11 + 2771*x^10 - 6116*x^9 + 8916*x^8 - 5025*x^7 + 13003*x^6 - 24345*x^5 + 9437*x^4 - 22214*x^3 + 36967*x^2 + 19033*x + 93919)
gp: K = bnfinit(y^18 - 6*y^17 + 15*y^16 + 21*y^15 - 189*y^14 + 447*y^13 - 277*y^12 - 819*y^11 + 2771*y^10 - 6116*y^9 + 8916*y^8 - 5025*y^7 + 13003*y^6 - 24345*y^5 + 9437*y^4 - 22214*y^3 + 36967*y^2 + 19033*y + 93919, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 15*x^16 + 21*x^15 - 189*x^14 + 447*x^13 - 277*x^12 - 819*x^11 + 2771*x^10 - 6116*x^9 + 8916*x^8 - 5025*x^7 + 13003*x^6 - 24345*x^5 + 9437*x^4 - 22214*x^3 + 36967*x^2 + 19033*x + 93919);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 6*x^17 + 15*x^16 + 21*x^15 - 189*x^14 + 447*x^13 - 277*x^12 - 819*x^11 + 2771*x^10 - 6116*x^9 + 8916*x^8 - 5025*x^7 + 13003*x^6 - 24345*x^5 + 9437*x^4 - 22214*x^3 + 36967*x^2 + 19033*x + 93919)
\( x^{18} - 6 x^{17} + 15 x^{16} + 21 x^{15} - 189 x^{14} + 447 x^{13} - 277 x^{12} - 819 x^{11} + \cdots + 93919 \)
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: |
\(-3601944964195995655415181312\)
\(\medspace = -\,2^{12}\cdot 3^{9}\cdot 7^{6}\cdot 11^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(33.96\) | 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: | $2^{2/3}3^{1/2}7^{2/3}11^{5/6}\approx 74.21184385155381$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(11\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11}a^{12}-\frac{1}{11}a^{11}+\frac{1}{11}a^{10}-\frac{5}{11}a^{9}+\frac{2}{11}a^{8}-\frac{4}{11}a^{7}-\frac{2}{11}a^{5}-\frac{2}{11}a^{3}+\frac{3}{11}a^{2}+\frac{2}{11}a-\frac{1}{11}$, $\frac{1}{11}a^{13}-\frac{4}{11}a^{10}-\frac{3}{11}a^{9}-\frac{2}{11}a^{8}-\frac{4}{11}a^{7}-\frac{2}{11}a^{6}-\frac{2}{11}a^{5}-\frac{2}{11}a^{4}+\frac{1}{11}a^{3}+\frac{5}{11}a^{2}+\frac{1}{11}a-\frac{1}{11}$, $\frac{1}{11}a^{14}-\frac{4}{11}a^{11}-\frac{3}{11}a^{10}-\frac{2}{11}a^{9}-\frac{4}{11}a^{8}-\frac{2}{11}a^{7}-\frac{2}{11}a^{6}-\frac{2}{11}a^{5}+\frac{1}{11}a^{4}+\frac{5}{11}a^{3}+\frac{1}{11}a^{2}-\frac{1}{11}a$, $\frac{1}{8452235}a^{15}-\frac{1}{1690447}a^{14}+\frac{4529}{768385}a^{13}+\frac{271694}{8452235}a^{12}+\frac{541571}{8452235}a^{11}-\frac{29063}{153677}a^{10}-\frac{3229287}{8452235}a^{9}+\frac{214778}{8452235}a^{8}-\frac{3857079}{8452235}a^{7}+\frac{873474}{8452235}a^{6}+\frac{700671}{8452235}a^{5}-\frac{344294}{768385}a^{4}-\frac{1788081}{8452235}a^{3}-\frac{3952543}{8452235}a^{2}-\frac{681966}{8452235}a+\frac{1098264}{8452235}$, $\frac{1}{8452235}a^{16}+\frac{49794}{8452235}a^{14}-\frac{247596}{8452235}a^{13}+\frac{363271}{8452235}a^{12}+\frac{48112}{153677}a^{11}-\frac{1232607}{8452235}a^{10}+\frac{2509583}{8452235}a^{9}+\frac{4132276}{8452235}a^{8}-\frac{739066}{8452235}a^{7}-\frac{1847424}{8452235}a^{6}-\frac{4125804}{8452235}a^{5}-\frac{2283011}{8452235}a^{4}-\frac{2135558}{8452235}a^{3}-\frac{3540211}{8452235}a^{2}+\frac{2298744}{8452235}a-\frac{131152}{1690447}$, $\frac{1}{10\!\cdots\!95}a^{17}-\frac{43\!\cdots\!88}{10\!\cdots\!95}a^{16}+\frac{34\!\cdots\!11}{95\!\cdots\!45}a^{15}-\frac{46\!\cdots\!38}{10\!\cdots\!95}a^{14}+\frac{21\!\cdots\!47}{10\!\cdots\!95}a^{13}+\frac{55\!\cdots\!54}{20\!\cdots\!39}a^{12}+\frac{66\!\cdots\!01}{20\!\cdots\!39}a^{11}-\frac{15\!\cdots\!51}{10\!\cdots\!95}a^{10}+\frac{39\!\cdots\!38}{10\!\cdots\!95}a^{9}+\frac{16\!\cdots\!57}{10\!\cdots\!95}a^{8}-\frac{45\!\cdots\!84}{95\!\cdots\!45}a^{7}+\frac{46\!\cdots\!81}{10\!\cdots\!95}a^{6}+\frac{32\!\cdots\!13}{10\!\cdots\!95}a^{5}+\frac{35\!\cdots\!57}{10\!\cdots\!95}a^{4}+\frac{16\!\cdots\!81}{10\!\cdots\!95}a^{3}+\frac{15\!\cdots\!56}{10\!\cdots\!95}a^{2}-\frac{34\!\cdots\!24}{10\!\cdots\!95}a-\frac{15\!\cdots\!07}{10\!\cdots\!95}$
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: |
\( -\frac{106653444}{5300511057865} a^{17} + \frac{508448312}{5300511057865} a^{16} - \frac{952207734}{5300511057865} a^{15} - \frac{3587155718}{5300511057865} a^{14} + \frac{16497731652}{5300511057865} a^{13} - \frac{5666231459}{1060102211573} a^{12} - \frac{1516513718}{1060102211573} a^{11} + \frac{95654295484}{5300511057865} a^{10} - \frac{217008678992}{5300511057865} a^{9} + \frac{469510631727}{5300511057865} a^{8} - \frac{428908118444}{5300511057865} a^{7} - \frac{161193888474}{5300511057865} a^{6} - \frac{465211707542}{5300511057865} a^{5} + \frac{29518691767}{5300511057865} a^{4} + \frac{1120656169666}{5300511057865} a^{3} + \frac{1009733201106}{5300511057865} a^{2} - \frac{77774608534}{5300511057865} a + \frac{2815400394673}{5300511057865} \)
(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: |
$\frac{239806644601956}{30\!\cdots\!11}a^{17}-\frac{15\!\cdots\!89}{30\!\cdots\!11}a^{16}+\frac{60\!\cdots\!56}{30\!\cdots\!11}a^{15}-\frac{34\!\cdots\!46}{30\!\cdots\!11}a^{14}-\frac{39\!\cdots\!79}{30\!\cdots\!11}a^{13}+\frac{19\!\cdots\!07}{30\!\cdots\!11}a^{12}-\frac{33\!\cdots\!34}{30\!\cdots\!11}a^{11}+\frac{62\!\cdots\!64}{30\!\cdots\!11}a^{10}+\frac{12\!\cdots\!15}{30\!\cdots\!11}a^{9}-\frac{32\!\cdots\!03}{30\!\cdots\!11}a^{8}+\frac{48\!\cdots\!22}{30\!\cdots\!11}a^{7}-\frac{60\!\cdots\!92}{30\!\cdots\!11}a^{6}+\frac{41\!\cdots\!73}{30\!\cdots\!11}a^{5}-\frac{17\!\cdots\!99}{30\!\cdots\!11}a^{4}+\frac{15\!\cdots\!24}{30\!\cdots\!11}a^{3}-\frac{54\!\cdots\!56}{30\!\cdots\!11}a^{2}-\frac{85\!\cdots\!82}{30\!\cdots\!11}a-\frac{27\!\cdots\!92}{30\!\cdots\!11}$, $\frac{29\!\cdots\!12}{10\!\cdots\!95}a^{17}-\frac{15\!\cdots\!21}{10\!\cdots\!95}a^{16}+\frac{39\!\cdots\!82}{10\!\cdots\!95}a^{15}+\frac{59\!\cdots\!04}{10\!\cdots\!95}a^{14}-\frac{46\!\cdots\!91}{10\!\cdots\!95}a^{13}+\frac{24\!\cdots\!80}{20\!\cdots\!39}a^{12}-\frac{19\!\cdots\!92}{20\!\cdots\!39}a^{11}-\frac{16\!\cdots\!32}{10\!\cdots\!95}a^{10}+\frac{86\!\cdots\!31}{10\!\cdots\!95}a^{9}-\frac{20\!\cdots\!96}{10\!\cdots\!95}a^{8}+\frac{25\!\cdots\!82}{10\!\cdots\!95}a^{7}-\frac{17\!\cdots\!58}{10\!\cdots\!95}a^{6}+\frac{23\!\cdots\!91}{10\!\cdots\!95}a^{5}-\frac{65\!\cdots\!36}{10\!\cdots\!95}a^{4}+\frac{30\!\cdots\!42}{10\!\cdots\!95}a^{3}-\frac{38\!\cdots\!78}{10\!\cdots\!95}a^{2}-\frac{14\!\cdots\!28}{10\!\cdots\!95}a-\frac{44\!\cdots\!84}{10\!\cdots\!95}$, $\frac{31\!\cdots\!92}{95\!\cdots\!45}a^{17}-\frac{16\!\cdots\!61}{10\!\cdots\!95}a^{16}-\frac{49\!\cdots\!28}{20\!\cdots\!39}a^{15}+\frac{34\!\cdots\!44}{10\!\cdots\!95}a^{14}-\frac{80\!\cdots\!99}{10\!\cdots\!95}a^{13}-\frac{15\!\cdots\!08}{10\!\cdots\!95}a^{12}+\frac{77\!\cdots\!23}{10\!\cdots\!95}a^{11}-\frac{93\!\cdots\!77}{10\!\cdots\!95}a^{10}-\frac{30\!\cdots\!29}{20\!\cdots\!39}a^{9}+\frac{36\!\cdots\!48}{10\!\cdots\!95}a^{8}-\frac{82\!\cdots\!72}{20\!\cdots\!39}a^{7}+\frac{10\!\cdots\!29}{10\!\cdots\!95}a^{6}+\frac{45\!\cdots\!34}{10\!\cdots\!95}a^{5}-\frac{26\!\cdots\!13}{10\!\cdots\!95}a^{4}-\frac{64\!\cdots\!06}{10\!\cdots\!95}a^{3}+\frac{17\!\cdots\!83}{10\!\cdots\!95}a^{2}+\frac{76\!\cdots\!44}{10\!\cdots\!95}a+\frac{72\!\cdots\!33}{10\!\cdots\!95}$, $\frac{32\!\cdots\!19}{10\!\cdots\!95}a^{17}-\frac{71\!\cdots\!18}{10\!\cdots\!95}a^{16}+\frac{29\!\cdots\!62}{10\!\cdots\!95}a^{15}+\frac{14\!\cdots\!09}{10\!\cdots\!95}a^{14}-\frac{15\!\cdots\!44}{10\!\cdots\!95}a^{13}-\frac{41\!\cdots\!59}{10\!\cdots\!95}a^{12}+\frac{11\!\cdots\!38}{10\!\cdots\!95}a^{11}-\frac{14\!\cdots\!57}{10\!\cdots\!95}a^{10}+\frac{70\!\cdots\!98}{10\!\cdots\!95}a^{9}-\frac{28\!\cdots\!24}{95\!\cdots\!45}a^{8}-\frac{40\!\cdots\!02}{95\!\cdots\!45}a^{7}-\frac{12\!\cdots\!48}{19\!\cdots\!49}a^{6}+\frac{60\!\cdots\!04}{10\!\cdots\!95}a^{5}+\frac{49\!\cdots\!32}{95\!\cdots\!45}a^{4}+\frac{48\!\cdots\!89}{10\!\cdots\!95}a^{3}+\frac{11\!\cdots\!51}{10\!\cdots\!95}a^{2}-\frac{35\!\cdots\!93}{10\!\cdots\!95}a-\frac{23\!\cdots\!51}{10\!\cdots\!95}$, $\frac{19\!\cdots\!72}{95\!\cdots\!45}a^{17}-\frac{15\!\cdots\!52}{10\!\cdots\!95}a^{16}+\frac{39\!\cdots\!94}{10\!\cdots\!95}a^{15}+\frac{60\!\cdots\!11}{19\!\cdots\!49}a^{14}-\frac{51\!\cdots\!57}{10\!\cdots\!95}a^{13}+\frac{12\!\cdots\!77}{10\!\cdots\!95}a^{12}-\frac{10\!\cdots\!03}{10\!\cdots\!95}a^{11}-\frac{68\!\cdots\!78}{20\!\cdots\!39}a^{10}+\frac{88\!\cdots\!94}{10\!\cdots\!95}a^{9}-\frac{14\!\cdots\!81}{95\!\cdots\!45}a^{8}+\frac{23\!\cdots\!50}{20\!\cdots\!39}a^{7}-\frac{56\!\cdots\!91}{10\!\cdots\!95}a^{6}+\frac{44\!\cdots\!17}{10\!\cdots\!95}a^{5}-\frac{34\!\cdots\!38}{10\!\cdots\!95}a^{4}-\frac{68\!\cdots\!92}{10\!\cdots\!95}a^{3}-\frac{14\!\cdots\!58}{10\!\cdots\!95}a^{2}-\frac{14\!\cdots\!54}{10\!\cdots\!95}a-\frac{22\!\cdots\!51}{10\!\cdots\!95}$, $\frac{74\!\cdots\!93}{10\!\cdots\!95}a^{17}-\frac{93\!\cdots\!76}{20\!\cdots\!39}a^{16}+\frac{88\!\cdots\!42}{10\!\cdots\!95}a^{15}+\frac{11\!\cdots\!07}{10\!\cdots\!95}a^{14}-\frac{12\!\cdots\!82}{10\!\cdots\!95}a^{13}+\frac{41\!\cdots\!33}{20\!\cdots\!39}a^{12}-\frac{47\!\cdots\!46}{10\!\cdots\!95}a^{11}-\frac{20\!\cdots\!81}{10\!\cdots\!95}a^{10}+\frac{52\!\cdots\!63}{10\!\cdots\!95}a^{9}-\frac{61\!\cdots\!08}{10\!\cdots\!95}a^{8}+\frac{60\!\cdots\!98}{10\!\cdots\!95}a^{7}-\frac{10\!\cdots\!47}{10\!\cdots\!95}a^{6}+\frac{41\!\cdots\!77}{10\!\cdots\!95}a^{5}-\frac{11\!\cdots\!44}{10\!\cdots\!95}a^{4}-\frac{42\!\cdots\!83}{10\!\cdots\!95}a^{3}-\frac{24\!\cdots\!58}{10\!\cdots\!95}a^{2}-\frac{50\!\cdots\!66}{20\!\cdots\!39}a-\frac{38\!\cdots\!59}{20\!\cdots\!39}$, $\frac{14\!\cdots\!13}{10\!\cdots\!95}a^{17}-\frac{77\!\cdots\!59}{10\!\cdots\!95}a^{16}+\frac{27\!\cdots\!28}{20\!\cdots\!39}a^{15}+\frac{51\!\cdots\!26}{95\!\cdots\!45}a^{14}-\frac{27\!\cdots\!26}{10\!\cdots\!95}a^{13}+\frac{36\!\cdots\!38}{10\!\cdots\!95}a^{12}+\frac{51\!\cdots\!17}{10\!\cdots\!95}a^{11}-\frac{20\!\cdots\!33}{10\!\cdots\!95}a^{10}+\frac{41\!\cdots\!12}{20\!\cdots\!39}a^{9}-\frac{17\!\cdots\!23}{10\!\cdots\!95}a^{8}+\frac{51\!\cdots\!74}{20\!\cdots\!39}a^{7}+\frac{30\!\cdots\!36}{10\!\cdots\!95}a^{6}+\frac{14\!\cdots\!06}{10\!\cdots\!95}a^{5}-\frac{46\!\cdots\!07}{10\!\cdots\!95}a^{4}-\frac{11\!\cdots\!04}{10\!\cdots\!95}a^{3}+\frac{50\!\cdots\!52}{10\!\cdots\!95}a^{2}+\frac{60\!\cdots\!36}{10\!\cdots\!95}a-\frac{46\!\cdots\!38}{10\!\cdots\!95}$, $\frac{81\!\cdots\!13}{10\!\cdots\!95}a^{17}-\frac{26\!\cdots\!47}{10\!\cdots\!95}a^{16}-\frac{40\!\cdots\!30}{20\!\cdots\!39}a^{15}+\frac{51\!\cdots\!99}{10\!\cdots\!95}a^{14}-\frac{95\!\cdots\!78}{10\!\cdots\!95}a^{13}-\frac{25\!\cdots\!53}{20\!\cdots\!39}a^{12}+\frac{81\!\cdots\!27}{10\!\cdots\!95}a^{11}-\frac{84\!\cdots\!07}{10\!\cdots\!95}a^{10}-\frac{86\!\cdots\!44}{10\!\cdots\!95}a^{9}+\frac{17\!\cdots\!74}{10\!\cdots\!95}a^{8}-\frac{33\!\cdots\!87}{10\!\cdots\!95}a^{7}+\frac{11\!\cdots\!88}{10\!\cdots\!95}a^{6}+\frac{69\!\cdots\!33}{95\!\cdots\!45}a^{5}-\frac{71\!\cdots\!04}{10\!\cdots\!95}a^{4}-\frac{10\!\cdots\!07}{20\!\cdots\!39}a^{3}+\frac{74\!\cdots\!75}{20\!\cdots\!39}a^{2}+\frac{78\!\cdots\!14}{10\!\cdots\!95}a+\frac{97\!\cdots\!22}{95\!\cdots\!45}$
| 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: | \( 1305071.139073367 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 1305071.139073367 \cdot 3}{6\cdot\sqrt{3601944964195995655415181312}}\cr\approx \mathstrut & 0.165941292628272 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 15*x^16 + 21*x^15 - 189*x^14 + 447*x^13 - 277*x^12 - 819*x^11 + 2771*x^10 - 6116*x^9 + 8916*x^8 - 5025*x^7 + 13003*x^6 - 24345*x^5 + 9437*x^4 - 22214*x^3 + 36967*x^2 + 19033*x + 93919)
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 - 6*x^17 + 15*x^16 + 21*x^15 - 189*x^14 + 447*x^13 - 277*x^12 - 819*x^11 + 2771*x^10 - 6116*x^9 + 8916*x^8 - 5025*x^7 + 13003*x^6 - 24345*x^5 + 9437*x^4 - 22214*x^3 + 36967*x^2 + 19033*x + 93919, 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 - 6*x^17 + 15*x^16 + 21*x^15 - 189*x^14 + 447*x^13 - 277*x^12 - 819*x^11 + 2771*x^10 - 6116*x^9 + 8916*x^8 - 5025*x^7 + 13003*x^6 - 24345*x^5 + 9437*x^4 - 22214*x^3 + 36967*x^2 + 19033*x + 93919);
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 - 6*x^17 + 15*x^16 + 21*x^15 - 189*x^14 + 447*x^13 - 277*x^12 - 819*x^11 + 2771*x^10 - 6116*x^9 + 8916*x^8 - 5025*x^7 + 13003*x^6 - 24345*x^5 + 9437*x^4 - 22214*x^3 + 36967*x^2 + 19033*x + 93919);
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
| \(\Q(\sqrt{-3}) \), 3.1.44.1, 6.0.19370043.1, 6.0.52272.1 |
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: | 12.0.11622150774897594624.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | R | R | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.12.1 | $x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$ | $3$ | $6$ | $12$ | $S_3 \times C_3$ | $[\ ]_{3}^{6}$ |
|
\(3\)
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(7\)
| 7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 7.6.4.1 | $x^{6} + 14 x^{3} - 245$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 7.6.0.1 | $x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(11\)
| 11.6.4.2 | $x^{6} - 110 x^{3} - 16819$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
| 11.12.10.2 | $x^{12} - 198 x^{6} - 10043$ | $6$ | $2$ | $10$ | $C_6\times S_3$ | $[\ ]_{6}^{6}$ |