Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 21*x^16 + 54*x^15 + 228*x^14 - 504*x^13 - 1012*x^12 + 2040*x^11 + 4523*x^10 - 6703*x^9 - 1497*x^8 + 252*x^7 + 34702*x^6 - 21324*x^5 + 57104*x^4 - 90118*x^3 + 127507*x^2 + 22997*x + 160003)
gp: K = bnfinit(y^18 - 3*y^17 - 21*y^16 + 54*y^15 + 228*y^14 - 504*y^13 - 1012*y^12 + 2040*y^11 + 4523*y^10 - 6703*y^9 - 1497*y^8 + 252*y^7 + 34702*y^6 - 21324*y^5 + 57104*y^4 - 90118*y^3 + 127507*y^2 + 22997*y + 160003, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 - 21*x^16 + 54*x^15 + 228*x^14 - 504*x^13 - 1012*x^12 + 2040*x^11 + 4523*x^10 - 6703*x^9 - 1497*x^8 + 252*x^7 + 34702*x^6 - 21324*x^5 + 57104*x^4 - 90118*x^3 + 127507*x^2 + 22997*x + 160003);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 - 21*x^16 + 54*x^15 + 228*x^14 - 504*x^13 - 1012*x^12 + 2040*x^11 + 4523*x^10 - 6703*x^9 - 1497*x^8 + 252*x^7 + 34702*x^6 - 21324*x^5 + 57104*x^4 - 90118*x^3 + 127507*x^2 + 22997*x + 160003)
\( x^{18} - 3 x^{17} - 21 x^{16} + 54 x^{15} + 228 x^{14} - 504 x^{13} - 1012 x^{12} + 2040 x^{11} + \cdots + 160003 \)
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: |
\(-195416899593798138941703979008\)
\(\medspace = -\,2^{20}\cdot 3^{9}\cdot 79^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(42.39\) | 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: | $2^{4/3}3^{1/2}79^{5/6}\approx 166.45232445421618$ | ||
| Ramified primes: |
\(2\), \(3\), \(79\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{6}-\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{7}-\frac{1}{4}a^{3}-\frac{1}{4}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}$, $\frac{1}{327692}a^{15}+\frac{1467}{81923}a^{14}+\frac{383}{81923}a^{13}+\frac{1263}{327692}a^{12}+\frac{46535}{327692}a^{11}-\frac{4379}{19276}a^{10}-\frac{80905}{327692}a^{9}+\frac{7550}{81923}a^{8}-\frac{58577}{327692}a^{7}-\frac{22053}{327692}a^{6}-\frac{77839}{327692}a^{5}-\frac{25191}{81923}a^{4}-\frac{2862}{81923}a^{3}-\frac{12175}{327692}a^{2}+\frac{118733}{327692}a+\frac{755}{5372}$, $\frac{1}{327692}a^{16}-\frac{6058}{81923}a^{14}+\frac{23017}{327692}a^{13}+\frac{8321}{327692}a^{12}-\frac{10541}{327692}a^{11}-\frac{62817}{327692}a^{10}-\frac{11242}{81923}a^{9}+\frac{9195}{327692}a^{8}-\frac{41125}{327692}a^{7}+\frac{54671}{327692}a^{6}-\frac{2120}{4819}a^{5}-\frac{24255}{163846}a^{4}+\frac{151675}{327692}a^{3}-\frac{39069}{327692}a^{2}-\frac{5997}{327692}a-\frac{563}{2686}$, $\frac{1}{56\!\cdots\!52}a^{17}-\frac{52\!\cdots\!51}{56\!\cdots\!52}a^{16}+\frac{42\!\cdots\!83}{56\!\cdots\!52}a^{15}+\frac{39\!\cdots\!65}{56\!\cdots\!52}a^{14}-\frac{63\!\cdots\!61}{56\!\cdots\!52}a^{13}+\frac{31\!\cdots\!09}{28\!\cdots\!26}a^{12}-\frac{16\!\cdots\!48}{14\!\cdots\!13}a^{11}-\frac{12\!\cdots\!15}{56\!\cdots\!52}a^{10}-\frac{26\!\cdots\!31}{28\!\cdots\!26}a^{9}+\frac{13\!\cdots\!57}{17\!\cdots\!47}a^{8}+\frac{26\!\cdots\!08}{14\!\cdots\!13}a^{7}+\frac{18\!\cdots\!19}{56\!\cdots\!52}a^{6}-\frac{16\!\cdots\!17}{56\!\cdots\!52}a^{5}+\frac{16\!\cdots\!71}{56\!\cdots\!52}a^{4}-\frac{61\!\cdots\!43}{71\!\cdots\!88}a^{3}+\frac{68\!\cdots\!98}{14\!\cdots\!13}a^{2}+\frac{13\!\cdots\!41}{56\!\cdots\!52}a+\frac{16\!\cdots\!98}{54\!\cdots\!31}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: |
\( -\frac{364364098457571}{207400872804292820803} a^{17} + \frac{8470847911051943}{829603491217171283212} a^{16} + \frac{12655034244460129}{829603491217171283212} a^{15} - \frac{9114402344017391}{48800205365715957836} a^{14} + \frac{13541076340459015}{414801745608585641606} a^{13} + \frac{759692914170821617}{414801745608585641606} a^{12} - \frac{2238504867100437601}{829603491217171283212} a^{11} - \frac{5969365566453214405}{829603491217171283212} a^{10} + \frac{11203687387169003263}{829603491217171283212} a^{9} + \frac{12068981103106149881}{414801745608585641606} a^{8} - \frac{71593234695704845825}{829603491217171283212} a^{7} + \frac{2985105994737669467}{829603491217171283212} a^{6} + \frac{49542960340331085291}{829603491217171283212} a^{5} + \frac{9342765837580502723}{48800205365715957836} a^{4} - \frac{254093235546166278319}{414801745608585641606} a^{3} + \frac{156296732622930161005}{414801745608585641606} a^{2} - \frac{590252664696434040735}{829603491217171283212} a + \frac{3886921569721071737}{3400014308267095423} \)
(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: |
$\frac{94\!\cdots\!79}{14\!\cdots\!13}a^{17}+\frac{73\!\cdots\!77}{33\!\cdots\!56}a^{16}-\frac{33\!\cdots\!95}{14\!\cdots\!13}a^{15}-\frac{16\!\cdots\!87}{28\!\cdots\!26}a^{14}+\frac{16\!\cdots\!29}{56\!\cdots\!52}a^{13}+\frac{17\!\cdots\!17}{28\!\cdots\!26}a^{12}-\frac{93\!\cdots\!13}{56\!\cdots\!52}a^{11}-\frac{98\!\cdots\!07}{46\!\cdots\!66}a^{10}+\frac{43\!\cdots\!82}{14\!\cdots\!13}a^{9}+\frac{22\!\cdots\!35}{56\!\cdots\!52}a^{8}+\frac{13\!\cdots\!27}{56\!\cdots\!52}a^{7}+\frac{10\!\cdots\!75}{28\!\cdots\!26}a^{6}-\frac{16\!\cdots\!10}{14\!\cdots\!13}a^{5}+\frac{72\!\cdots\!87}{28\!\cdots\!26}a^{4}+\frac{17\!\cdots\!43}{56\!\cdots\!52}a^{3}+\frac{17\!\cdots\!73}{14\!\cdots\!13}a^{2}+\frac{16\!\cdots\!71}{56\!\cdots\!52}a-\frac{30\!\cdots\!85}{21\!\cdots\!24}$, $\frac{14\!\cdots\!83}{56\!\cdots\!52}a^{17}-\frac{54\!\cdots\!47}{28\!\cdots\!26}a^{16}+\frac{39\!\cdots\!59}{83\!\cdots\!89}a^{15}+\frac{21\!\cdots\!25}{56\!\cdots\!52}a^{14}-\frac{24\!\cdots\!89}{28\!\cdots\!26}a^{13}-\frac{53\!\cdots\!99}{14\!\cdots\!13}a^{12}+\frac{23\!\cdots\!67}{28\!\cdots\!26}a^{11}+\frac{64\!\cdots\!13}{56\!\cdots\!52}a^{10}-\frac{19\!\cdots\!83}{56\!\cdots\!52}a^{9}-\frac{19\!\cdots\!51}{56\!\cdots\!52}a^{8}+\frac{36\!\cdots\!23}{28\!\cdots\!26}a^{7}-\frac{92\!\cdots\!19}{56\!\cdots\!52}a^{6}-\frac{21\!\cdots\!93}{28\!\cdots\!26}a^{5}-\frac{17\!\cdots\!37}{56\!\cdots\!52}a^{4}+\frac{11\!\cdots\!11}{28\!\cdots\!26}a^{3}-\frac{21\!\cdots\!62}{14\!\cdots\!13}a^{2}+\frac{63\!\cdots\!15}{56\!\cdots\!52}a-\frac{45\!\cdots\!69}{21\!\cdots\!24}$, $\frac{37\!\cdots\!08}{14\!\cdots\!13}a^{17}-\frac{84\!\cdots\!19}{56\!\cdots\!52}a^{16}-\frac{17\!\cdots\!23}{56\!\cdots\!52}a^{15}+\frac{13\!\cdots\!09}{56\!\cdots\!52}a^{14}+\frac{26\!\cdots\!04}{14\!\cdots\!13}a^{13}-\frac{62\!\cdots\!79}{33\!\cdots\!56}a^{12}+\frac{63\!\cdots\!51}{56\!\cdots\!52}a^{11}+\frac{12\!\cdots\!11}{28\!\cdots\!26}a^{10}-\frac{18\!\cdots\!75}{56\!\cdots\!52}a^{9}+\frac{37\!\cdots\!13}{28\!\cdots\!26}a^{8}+\frac{32\!\cdots\!79}{56\!\cdots\!52}a^{7}-\frac{65\!\cdots\!49}{28\!\cdots\!26}a^{6}+\frac{22\!\cdots\!09}{56\!\cdots\!52}a^{5}-\frac{21\!\cdots\!15}{56\!\cdots\!52}a^{4}+\frac{80\!\cdots\!01}{28\!\cdots\!26}a^{3}-\frac{82\!\cdots\!43}{56\!\cdots\!52}a^{2}+\frac{60\!\cdots\!13}{56\!\cdots\!52}a-\frac{28\!\cdots\!41}{21\!\cdots\!24}$, $\frac{51\!\cdots\!73}{71\!\cdots\!88}a^{17}-\frac{23\!\cdots\!29}{71\!\cdots\!88}a^{16}-\frac{25\!\cdots\!87}{28\!\cdots\!26}a^{15}+\frac{31\!\cdots\!61}{56\!\cdots\!52}a^{14}+\frac{16\!\cdots\!03}{56\!\cdots\!52}a^{13}-\frac{72\!\cdots\!44}{14\!\cdots\!13}a^{12}+\frac{43\!\cdots\!19}{56\!\cdots\!52}a^{11}+\frac{92\!\cdots\!35}{56\!\cdots\!52}a^{10}-\frac{25\!\cdots\!81}{56\!\cdots\!52}a^{9}-\frac{88\!\cdots\!98}{14\!\cdots\!13}a^{8}+\frac{16\!\cdots\!79}{56\!\cdots\!52}a^{7}-\frac{19\!\cdots\!77}{56\!\cdots\!52}a^{6}-\frac{64\!\cdots\!48}{14\!\cdots\!13}a^{5}-\frac{41\!\cdots\!73}{56\!\cdots\!52}a^{4}+\frac{52\!\cdots\!85}{56\!\cdots\!52}a^{3}-\frac{23\!\cdots\!93}{28\!\cdots\!26}a^{2}+\frac{42\!\cdots\!64}{14\!\cdots\!13}a-\frac{34\!\cdots\!16}{54\!\cdots\!31}$, $\frac{62\!\cdots\!99}{33\!\cdots\!56}a^{17}-\frac{33\!\cdots\!91}{83\!\cdots\!89}a^{16}-\frac{14\!\cdots\!17}{33\!\cdots\!56}a^{15}+\frac{11\!\cdots\!85}{16\!\cdots\!78}a^{14}+\frac{17\!\cdots\!25}{33\!\cdots\!56}a^{13}-\frac{48\!\cdots\!34}{83\!\cdots\!89}a^{12}-\frac{23\!\cdots\!78}{83\!\cdots\!89}a^{11}+\frac{36\!\cdots\!39}{16\!\cdots\!78}a^{10}+\frac{19\!\cdots\!79}{16\!\cdots\!78}a^{9}-\frac{87\!\cdots\!11}{16\!\cdots\!78}a^{8}-\frac{11\!\cdots\!35}{83\!\cdots\!89}a^{7}-\frac{20\!\cdots\!97}{16\!\cdots\!78}a^{6}+\frac{18\!\cdots\!47}{33\!\cdots\!56}a^{5}-\frac{17\!\cdots\!13}{16\!\cdots\!78}a^{4}-\frac{28\!\cdots\!19}{33\!\cdots\!56}a^{3}-\frac{27\!\cdots\!20}{83\!\cdots\!89}a^{2}+\frac{19\!\cdots\!57}{33\!\cdots\!56}a+\frac{25\!\cdots\!81}{63\!\cdots\!86}$, $\frac{68\!\cdots\!43}{46\!\cdots\!66}a^{17}-\frac{71\!\cdots\!98}{23\!\cdots\!33}a^{16}-\frac{31\!\cdots\!63}{93\!\cdots\!32}a^{15}+\frac{29\!\cdots\!59}{54\!\cdots\!96}a^{14}+\frac{87\!\cdots\!60}{23\!\cdots\!33}a^{13}-\frac{47\!\cdots\!57}{93\!\cdots\!32}a^{12}-\frac{16\!\cdots\!41}{93\!\cdots\!32}a^{11}+\frac{60\!\cdots\!35}{23\!\cdots\!33}a^{10}+\frac{58\!\cdots\!09}{93\!\cdots\!32}a^{9}-\frac{94\!\cdots\!25}{93\!\cdots\!32}a^{8}+\frac{12\!\cdots\!71}{93\!\cdots\!32}a^{7}+\frac{60\!\cdots\!37}{23\!\cdots\!33}a^{6}+\frac{25\!\cdots\!71}{93\!\cdots\!32}a^{5}-\frac{23\!\cdots\!37}{54\!\cdots\!96}a^{4}+\frac{21\!\cdots\!01}{46\!\cdots\!66}a^{3}+\frac{36\!\cdots\!37}{93\!\cdots\!32}a^{2}+\frac{10\!\cdots\!29}{93\!\cdots\!32}a-\frac{10\!\cdots\!14}{54\!\cdots\!31}$, $\frac{95\!\cdots\!91}{56\!\cdots\!52}a^{17}-\frac{42\!\cdots\!95}{56\!\cdots\!52}a^{16}-\frac{64\!\cdots\!31}{16\!\cdots\!78}a^{15}+\frac{42\!\cdots\!57}{28\!\cdots\!26}a^{14}+\frac{72\!\cdots\!20}{14\!\cdots\!13}a^{13}-\frac{79\!\cdots\!67}{56\!\cdots\!52}a^{12}-\frac{94\!\cdots\!33}{28\!\cdots\!26}a^{11}+\frac{25\!\cdots\!71}{56\!\cdots\!52}a^{10}+\frac{86\!\cdots\!27}{56\!\cdots\!52}a^{9}-\frac{64\!\cdots\!49}{56\!\cdots\!52}a^{8}-\frac{86\!\cdots\!83}{28\!\cdots\!26}a^{7}-\frac{25\!\cdots\!39}{56\!\cdots\!52}a^{6}+\frac{39\!\cdots\!89}{28\!\cdots\!26}a^{5}-\frac{28\!\cdots\!21}{28\!\cdots\!26}a^{4}-\frac{16\!\cdots\!21}{16\!\cdots\!78}a^{3}-\frac{28\!\cdots\!59}{56\!\cdots\!52}a^{2}-\frac{20\!\cdots\!03}{56\!\cdots\!52}a-\frac{20\!\cdots\!94}{54\!\cdots\!31}$, $\frac{36\!\cdots\!67}{56\!\cdots\!52}a^{17}-\frac{92\!\cdots\!55}{56\!\cdots\!52}a^{16}-\frac{22\!\cdots\!72}{14\!\cdots\!13}a^{15}+\frac{89\!\cdots\!79}{28\!\cdots\!26}a^{14}+\frac{26\!\cdots\!75}{14\!\cdots\!13}a^{13}-\frac{88\!\cdots\!81}{28\!\cdots\!26}a^{12}-\frac{31\!\cdots\!79}{28\!\cdots\!26}a^{11}+\frac{44\!\cdots\!21}{28\!\cdots\!26}a^{10}+\frac{29\!\cdots\!71}{56\!\cdots\!52}a^{9}-\frac{19\!\cdots\!81}{33\!\cdots\!56}a^{8}-\frac{29\!\cdots\!75}{28\!\cdots\!26}a^{7}+\frac{87\!\cdots\!11}{83\!\cdots\!89}a^{6}+\frac{59\!\cdots\!41}{16\!\cdots\!78}a^{5}-\frac{60\!\cdots\!87}{28\!\cdots\!26}a^{4}-\frac{28\!\cdots\!02}{14\!\cdots\!13}a^{3}+\frac{68\!\cdots\!57}{14\!\cdots\!13}a^{2}+\frac{39\!\cdots\!77}{56\!\cdots\!52}a+\frac{11\!\cdots\!41}{21\!\cdots\!24}$
| 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: | \( 24722362.03581395 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 24722362.03581395 \cdot 3}{6\cdot\sqrt{195416899593798138941703979008}}\cr\approx \mathstrut & 0.426773772809410 \end{aligned}\] (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 - 21*x^16 + 54*x^15 + 228*x^14 - 504*x^13 - 1012*x^12 + 2040*x^11 + 4523*x^10 - 6703*x^9 - 1497*x^8 + 252*x^7 + 34702*x^6 - 21324*x^5 + 57104*x^4 - 90118*x^3 + 127507*x^2 + 22997*x + 160003)
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 - 21*x^16 + 54*x^15 + 228*x^14 - 504*x^13 - 1012*x^12 + 2040*x^11 + 4523*x^10 - 6703*x^9 - 1497*x^8 + 252*x^7 + 34702*x^6 - 21324*x^5 + 57104*x^4 - 90118*x^3 + 127507*x^2 + 22997*x + 160003, 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 - 21*x^16 + 54*x^15 + 228*x^14 - 504*x^13 - 1012*x^12 + 2040*x^11 + 4523*x^10 - 6703*x^9 - 1497*x^8 + 252*x^7 + 34702*x^6 - 21324*x^5 + 57104*x^4 - 90118*x^3 + 127507*x^2 + 22997*x + 160003);
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 - 21*x^16 + 54*x^15 + 228*x^14 - 504*x^13 - 1012*x^12 + 2040*x^11 + 4523*x^10 - 6703*x^9 - 1497*x^8 + 252*x^7 + 34702*x^6 - 21324*x^5 + 57104*x^4 - 90118*x^3 + 127507*x^2 + 22997*x + 160003);
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
| \(\Q(\sqrt{-3}) \), 3.3.316.1, 6.0.2696112.3, 6.0.2696112.1 |
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: | 12.0.72481002122240522256384.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.3.0.1}{3} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{9}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 | $x^{12} + 4 x^{11} + 8 x^{10} + 10 x^{9} + 8 x^{8} + 4 x^{7} + 2 x^{6} - 4 x^{5} - 4 x^{4} + 4 x^{3} + 4$ | $6$ | $2$ | $16$ | $C_6\times S_3$ | $[2]_{3}^{6}$ | |
|
\(3\)
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 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 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(79\)
| 79.3.2.2 | $x^{3} + 158$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 79.3.0.1 | $x^{3} + 9 x + 76$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 79.6.3.1 | $x^{6} + 56169 x^{2} - 37470964$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 79.6.5.3 | $x^{6} + 237$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |