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Properties

Label 18.0.829...791.1
Degree $18$
Signature $[0, 9]$
Discriminant $-8.298\times 10^{25}$
Root discriminant \(27.54\)
Ramified primes $11,31$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_9$ (as 18T5)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 18*x^16 - 48*x^15 + 83*x^14 - 203*x^13 + 90*x^12 + 47*x^11 + 131*x^10 + 918*x^9 + 3164*x^8 + 2377*x^7 - 796*x^6 + 1065*x^5 + 9952*x^4 - 8374*x^3 - 16172*x^2 + 7750*x + 19127)
 
gp: K = bnfinit(y^18 - 3*y^17 + 18*y^16 - 48*y^15 + 83*y^14 - 203*y^13 + 90*y^12 + 47*y^11 + 131*y^10 + 918*y^9 + 3164*y^8 + 2377*y^7 - 796*y^6 + 1065*y^5 + 9952*y^4 - 8374*y^3 - 16172*y^2 + 7750*y + 19127, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 + 18*x^16 - 48*x^15 + 83*x^14 - 203*x^13 + 90*x^12 + 47*x^11 + 131*x^10 + 918*x^9 + 3164*x^8 + 2377*x^7 - 796*x^6 + 1065*x^5 + 9952*x^4 - 8374*x^3 - 16172*x^2 + 7750*x + 19127);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 + 18*x^16 - 48*x^15 + 83*x^14 - 203*x^13 + 90*x^12 + 47*x^11 + 131*x^10 + 918*x^9 + 3164*x^8 + 2377*x^7 - 796*x^6 + 1065*x^5 + 9952*x^4 - 8374*x^3 - 16172*x^2 + 7750*x + 19127)
 

\( x^{18} - 3 x^{17} + 18 x^{16} - 48 x^{15} + 83 x^{14} - 203 x^{13} + 90 x^{12} + 47 x^{11} + \cdots + 19127 \) Copy content Toggle raw display

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:   \(-82978860458831265178139791\) \(\medspace = -\,11^{12}\cdot 31^{9}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(27.54\)
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:  $11^{2/3}31^{1/2}\approx 27.538649401961177$
Ramified primes:   \(11\), \(31\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-31}) \)
$\card{ \Gal(K/\Q) }$:  $18$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{9}a^{14}-\frac{1}{9}a^{13}+\frac{1}{9}a^{11}-\frac{1}{9}a^{10}+\frac{1}{9}a^{8}-\frac{1}{9}a^{7}-\frac{2}{9}a^{6}-\frac{1}{9}a^{4}-\frac{2}{9}a^{3}-\frac{1}{9}a-\frac{2}{9}$, $\frac{1}{549}a^{15}-\frac{52}{549}a^{13}-\frac{56}{549}a^{12}+\frac{26}{183}a^{11}-\frac{85}{549}a^{10}-\frac{5}{549}a^{9}-\frac{14}{183}a^{8}+\frac{4}{61}a^{7}-\frac{251}{549}a^{6}+\frac{80}{549}a^{5}-\frac{35}{183}a^{4}-\frac{2}{9}a^{3}-\frac{58}{549}a^{2}+\frac{11}{183}a-\frac{128}{549}$, $\frac{1}{18568827}a^{16}+\frac{326}{18568827}a^{15}+\frac{873529}{18568827}a^{14}-\frac{73619}{2063203}a^{13}+\frac{33512}{304407}a^{12}-\frac{1572857}{18568827}a^{11}+\frac{49220}{6189609}a^{10}+\frac{2891924}{18568827}a^{9}-\frac{1218223}{18568827}a^{8}+\frac{1028111}{18568827}a^{7}-\frac{312036}{2063203}a^{6}-\frac{7111757}{18568827}a^{5}-\frac{358201}{18568827}a^{4}-\frac{2442073}{6189609}a^{3}+\frac{8427673}{18568827}a^{2}+\frac{6328034}{18568827}a-\frac{8316866}{18568827}$, $\frac{1}{65\!\cdots\!93}a^{17}+\frac{12\!\cdots\!18}{65\!\cdots\!93}a^{16}+\frac{45\!\cdots\!95}{65\!\cdots\!93}a^{15}+\frac{19\!\cdots\!44}{65\!\cdots\!93}a^{14}-\frac{61\!\cdots\!60}{21\!\cdots\!31}a^{13}-\frac{35\!\cdots\!66}{65\!\cdots\!93}a^{12}-\frac{18\!\cdots\!00}{65\!\cdots\!93}a^{11}-\frac{12\!\cdots\!56}{21\!\cdots\!31}a^{10}+\frac{55\!\cdots\!21}{65\!\cdots\!93}a^{9}-\frac{68\!\cdots\!85}{65\!\cdots\!93}a^{8}-\frac{55\!\cdots\!68}{65\!\cdots\!93}a^{7}+\frac{97\!\cdots\!36}{31\!\cdots\!51}a^{6}+\frac{99\!\cdots\!88}{65\!\cdots\!93}a^{5}+\frac{30\!\cdots\!10}{65\!\cdots\!93}a^{4}-\frac{12\!\cdots\!84}{21\!\cdots\!31}a^{3}+\frac{16\!\cdots\!42}{65\!\cdots\!93}a^{2}-\frac{10\!\cdots\!13}{65\!\cdots\!93}a-\frac{10\!\cdots\!09}{65\!\cdots\!93}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $3$

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:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $\frac{58\!\cdots\!51}{56\!\cdots\!33}a^{17}+\frac{18\!\cdots\!38}{56\!\cdots\!33}a^{16}-\frac{21\!\cdots\!92}{56\!\cdots\!33}a^{15}+\frac{48\!\cdots\!29}{56\!\cdots\!33}a^{14}-\frac{20\!\cdots\!11}{62\!\cdots\!37}a^{13}+\frac{41\!\cdots\!35}{56\!\cdots\!33}a^{12}-\frac{12\!\cdots\!88}{56\!\cdots\!33}a^{11}+\frac{53\!\cdots\!99}{18\!\cdots\!11}a^{10}-\frac{18\!\cdots\!90}{56\!\cdots\!33}a^{9}+\frac{26\!\cdots\!39}{56\!\cdots\!33}a^{8}+\frac{53\!\cdots\!17}{56\!\cdots\!33}a^{7}+\frac{31\!\cdots\!59}{18\!\cdots\!11}a^{6}+\frac{11\!\cdots\!70}{56\!\cdots\!33}a^{5}-\frac{21\!\cdots\!57}{56\!\cdots\!33}a^{4}+\frac{46\!\cdots\!66}{18\!\cdots\!11}a^{3}+\frac{10\!\cdots\!51}{56\!\cdots\!33}a^{2}-\frac{40\!\cdots\!88}{56\!\cdots\!33}a+\frac{28\!\cdots\!41}{56\!\cdots\!33}$, $\frac{93\!\cdots\!54}{52\!\cdots\!89}a^{17}-\frac{62\!\cdots\!39}{52\!\cdots\!89}a^{16}+\frac{29\!\cdots\!13}{52\!\cdots\!89}a^{15}-\frac{11\!\cdots\!81}{52\!\cdots\!89}a^{14}+\frac{95\!\cdots\!59}{17\!\cdots\!63}a^{13}-\frac{58\!\cdots\!13}{52\!\cdots\!89}a^{12}+\frac{98\!\cdots\!24}{52\!\cdots\!89}a^{11}-\frac{79\!\cdots\!06}{58\!\cdots\!21}a^{10}+\frac{50\!\cdots\!35}{52\!\cdots\!89}a^{9}+\frac{55\!\cdots\!13}{52\!\cdots\!89}a^{8}+\frac{22\!\cdots\!02}{52\!\cdots\!89}a^{7}-\frac{24\!\cdots\!51}{17\!\cdots\!63}a^{6}+\frac{28\!\cdots\!21}{52\!\cdots\!89}a^{5}+\frac{57\!\cdots\!91}{52\!\cdots\!89}a^{4}+\frac{22\!\cdots\!88}{95\!\cdots\!61}a^{3}-\frac{35\!\cdots\!97}{52\!\cdots\!89}a^{2}+\frac{29\!\cdots\!64}{52\!\cdots\!89}a+\frac{24\!\cdots\!37}{52\!\cdots\!89}$, $\frac{43\!\cdots\!86}{65\!\cdots\!93}a^{17}-\frac{16\!\cdots\!83}{65\!\cdots\!93}a^{16}+\frac{81\!\cdots\!79}{65\!\cdots\!93}a^{15}-\frac{26\!\cdots\!55}{65\!\cdots\!93}a^{14}+\frac{15\!\cdots\!37}{21\!\cdots\!31}a^{13}-\frac{12\!\cdots\!28}{65\!\cdots\!93}a^{12}+\frac{17\!\cdots\!96}{65\!\cdots\!93}a^{11}-\frac{32\!\cdots\!89}{21\!\cdots\!31}a^{10}+\frac{53\!\cdots\!14}{65\!\cdots\!93}a^{9}-\frac{97\!\cdots\!13}{65\!\cdots\!93}a^{8}+\frac{11\!\cdots\!57}{65\!\cdots\!93}a^{7}-\frac{68\!\cdots\!26}{24\!\cdots\!59}a^{6}-\frac{49\!\cdots\!29}{65\!\cdots\!93}a^{5}-\frac{62\!\cdots\!71}{65\!\cdots\!93}a^{4}+\frac{16\!\cdots\!85}{21\!\cdots\!31}a^{3}+\frac{76\!\cdots\!45}{65\!\cdots\!93}a^{2}+\frac{14\!\cdots\!79}{65\!\cdots\!93}a-\frac{58\!\cdots\!90}{65\!\cdots\!93}$, $\frac{19\!\cdots\!42}{65\!\cdots\!93}a^{17}-\frac{45\!\cdots\!19}{65\!\cdots\!93}a^{16}+\frac{31\!\cdots\!96}{65\!\cdots\!93}a^{15}-\frac{68\!\cdots\!78}{65\!\cdots\!93}a^{14}+\frac{29\!\cdots\!36}{21\!\cdots\!31}a^{13}-\frac{22\!\cdots\!14}{65\!\cdots\!93}a^{12}-\frac{34\!\cdots\!66}{65\!\cdots\!93}a^{11}+\frac{25\!\cdots\!12}{21\!\cdots\!31}a^{10}-\frac{11\!\cdots\!98}{65\!\cdots\!93}a^{9}+\frac{43\!\cdots\!11}{65\!\cdots\!93}a^{8}+\frac{47\!\cdots\!57}{65\!\cdots\!93}a^{7}+\frac{15\!\cdots\!18}{72\!\cdots\!77}a^{6}+\frac{82\!\cdots\!70}{65\!\cdots\!93}a^{5}+\frac{11\!\cdots\!04}{65\!\cdots\!93}a^{4}+\frac{49\!\cdots\!36}{21\!\cdots\!31}a^{3}-\frac{10\!\cdots\!31}{65\!\cdots\!93}a^{2}-\frac{43\!\cdots\!99}{65\!\cdots\!93}a-\frac{72\!\cdots\!80}{43\!\cdots\!57}$, $\frac{43\!\cdots\!57}{65\!\cdots\!93}a^{17}-\frac{22\!\cdots\!69}{65\!\cdots\!93}a^{16}+\frac{11\!\cdots\!82}{65\!\cdots\!93}a^{15}-\frac{43\!\cdots\!03}{65\!\cdots\!93}a^{14}+\frac{37\!\cdots\!72}{21\!\cdots\!31}a^{13}-\frac{27\!\cdots\!95}{65\!\cdots\!93}a^{12}+\frac{48\!\cdots\!41}{65\!\cdots\!93}a^{11}-\frac{22\!\cdots\!07}{21\!\cdots\!31}a^{10}+\frac{85\!\cdots\!36}{65\!\cdots\!93}a^{9}-\frac{53\!\cdots\!21}{65\!\cdots\!93}a^{8}+\frac{14\!\cdots\!17}{65\!\cdots\!93}a^{7}-\frac{80\!\cdots\!84}{48\!\cdots\!73}a^{6}-\frac{43\!\cdots\!42}{65\!\cdots\!93}a^{5}+\frac{24\!\cdots\!78}{65\!\cdots\!93}a^{4}-\frac{30\!\cdots\!11}{21\!\cdots\!31}a^{3}-\frac{47\!\cdots\!66}{65\!\cdots\!93}a^{2}+\frac{43\!\cdots\!24}{65\!\cdots\!93}a+\frac{18\!\cdots\!19}{10\!\cdots\!13}$, $\frac{21\!\cdots\!11}{65\!\cdots\!93}a^{17}-\frac{52\!\cdots\!77}{65\!\cdots\!93}a^{16}+\frac{35\!\cdots\!97}{65\!\cdots\!93}a^{15}-\frac{82\!\cdots\!10}{65\!\cdots\!93}a^{14}+\frac{43\!\cdots\!53}{21\!\cdots\!31}a^{13}-\frac{36\!\cdots\!16}{65\!\cdots\!93}a^{12}+\frac{32\!\cdots\!07}{65\!\cdots\!93}a^{11}+\frac{57\!\cdots\!61}{72\!\cdots\!77}a^{10}+\frac{47\!\cdots\!94}{65\!\cdots\!93}a^{9}+\frac{17\!\cdots\!84}{65\!\cdots\!93}a^{8}+\frac{76\!\cdots\!51}{65\!\cdots\!93}a^{7}+\frac{29\!\cdots\!41}{21\!\cdots\!31}a^{6}+\frac{33\!\cdots\!03}{65\!\cdots\!93}a^{5}+\frac{59\!\cdots\!88}{65\!\cdots\!93}a^{4}+\frac{11\!\cdots\!80}{24\!\cdots\!59}a^{3}+\frac{10\!\cdots\!22}{65\!\cdots\!93}a^{2}-\frac{38\!\cdots\!91}{65\!\cdots\!93}a-\frac{60\!\cdots\!47}{10\!\cdots\!13}$, $\frac{15\!\cdots\!59}{65\!\cdots\!93}a^{17}-\frac{27\!\cdots\!01}{65\!\cdots\!93}a^{16}+\frac{23\!\cdots\!91}{65\!\cdots\!93}a^{15}-\frac{43\!\cdots\!51}{65\!\cdots\!93}a^{14}+\frac{22\!\cdots\!18}{21\!\cdots\!31}a^{13}-\frac{22\!\cdots\!14}{65\!\cdots\!93}a^{12}-\frac{13\!\cdots\!26}{65\!\cdots\!93}a^{11}-\frac{28\!\cdots\!42}{21\!\cdots\!31}a^{10}+\frac{27\!\cdots\!13}{65\!\cdots\!93}a^{9}+\frac{16\!\cdots\!34}{65\!\cdots\!93}a^{8}+\frac{69\!\cdots\!81}{65\!\cdots\!93}a^{7}+\frac{13\!\cdots\!53}{72\!\cdots\!77}a^{6}+\frac{10\!\cdots\!18}{65\!\cdots\!93}a^{5}+\frac{10\!\cdots\!78}{65\!\cdots\!93}a^{4}+\frac{54\!\cdots\!71}{14\!\cdots\!19}a^{3}+\frac{75\!\cdots\!98}{65\!\cdots\!93}a^{2}-\frac{26\!\cdots\!10}{65\!\cdots\!93}a-\frac{24\!\cdots\!49}{65\!\cdots\!93}$, $\frac{41\!\cdots\!54}{65\!\cdots\!93}a^{17}-\frac{18\!\cdots\!82}{65\!\cdots\!93}a^{16}+\frac{99\!\cdots\!41}{65\!\cdots\!93}a^{15}-\frac{34\!\cdots\!48}{65\!\cdots\!93}a^{14}+\frac{26\!\cdots\!98}{21\!\cdots\!31}a^{13}-\frac{18\!\cdots\!75}{65\!\cdots\!93}a^{12}+\frac{27\!\cdots\!78}{65\!\cdots\!93}a^{11}-\frac{28\!\cdots\!36}{72\!\cdots\!77}a^{10}+\frac{20\!\cdots\!49}{65\!\cdots\!93}a^{9}+\frac{21\!\cdots\!37}{65\!\cdots\!93}a^{8}+\frac{43\!\cdots\!03}{65\!\cdots\!93}a^{7}+\frac{99\!\cdots\!53}{21\!\cdots\!31}a^{6}-\frac{42\!\cdots\!76}{65\!\cdots\!93}a^{5}+\frac{14\!\cdots\!85}{65\!\cdots\!93}a^{4}+\frac{62\!\cdots\!98}{24\!\cdots\!59}a^{3}-\frac{64\!\cdots\!32}{65\!\cdots\!93}a^{2}+\frac{54\!\cdots\!56}{65\!\cdots\!93}a-\frac{22\!\cdots\!85}{65\!\cdots\!93}$ Copy content Toggle raw display (assuming GRH)
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:  \( 605404.210476 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 605404.210476 \cdot 1}{2\cdot\sqrt{82978860458831265178139791}}\cr\approx \mathstrut & 0.507166458158 \end{aligned}\] (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 + 18*x^16 - 48*x^15 + 83*x^14 - 203*x^13 + 90*x^12 + 47*x^11 + 131*x^10 + 918*x^9 + 3164*x^8 + 2377*x^7 - 796*x^6 + 1065*x^5 + 9952*x^4 - 8374*x^3 - 16172*x^2 + 7750*x + 19127)
 
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 + 18*x^16 - 48*x^15 + 83*x^14 - 203*x^13 + 90*x^12 + 47*x^11 + 131*x^10 + 918*x^9 + 3164*x^8 + 2377*x^7 - 796*x^6 + 1065*x^5 + 9952*x^4 - 8374*x^3 - 16172*x^2 + 7750*x + 19127, 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 + 18*x^16 - 48*x^15 + 83*x^14 - 203*x^13 + 90*x^12 + 47*x^11 + 131*x^10 + 918*x^9 + 3164*x^8 + 2377*x^7 - 796*x^6 + 1065*x^5 + 9952*x^4 - 8374*x^3 - 16172*x^2 + 7750*x + 19127);
 
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 + 18*x^16 - 48*x^15 + 83*x^14 - 203*x^13 + 90*x^12 + 47*x^11 + 131*x^10 + 918*x^9 + 3164*x^8 + 2377*x^7 - 796*x^6 + 1065*x^5 + 9952*x^4 - 8374*x^3 - 16172*x^2 + 7750*x + 19127);
 
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

$D_9$ (as 18T5):

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 18
The 6 conjugacy class representatives for $D_9$
Character table for $D_9$

Intermediate fields

\(\Q(\sqrt{-31}) \), 3.1.31.1 x3, 6.0.29791.1, 9.1.1636073786281.1 x9

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 9 sibling: 9.1.1636073786281.1
Minimal sibling: 9.1.1636073786281.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.9.0.1}{9} }^{2}$ ${\href{/padicField/3.2.0.1}{2} }^{9}$ ${\href{/padicField/5.9.0.1}{9} }^{2}$ ${\href{/padicField/7.9.0.1}{9} }^{2}$ R ${\href{/padicField/13.2.0.1}{2} }^{9}$ ${\href{/padicField/17.2.0.1}{2} }^{9}$ ${\href{/padicField/19.9.0.1}{9} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{9}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ R ${\href{/padicField/37.2.0.1}{2} }^{9}$ ${\href{/padicField/41.9.0.1}{9} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{9}$ ${\href{/padicField/47.3.0.1}{3} }^{6}$ ${\href{/padicField/53.2.0.1}{2} }^{9}$ ${\href{/padicField/59.9.0.1}{9} }^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(11\) Copy content Toggle raw display 11.6.4.1$x^{6} + 21 x^{5} + 153 x^{4} + 449 x^{3} + 537 x^{2} + 1569 x + 3440$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 21 x^{5} + 153 x^{4} + 449 x^{3} + 537 x^{2} + 1569 x + 3440$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 21 x^{5} + 153 x^{4} + 449 x^{3} + 537 x^{2} + 1569 x + 3440$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
\(31\) Copy content Toggle raw display 31.2.1.2$x^{2} + 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 31$$2$$1$$1$$C_2$$[\ ]_{2}$