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Properties

Label 18.0.217...368.1
Degree $18$
Signature $[0, 9]$
Discriminant $-2.171\times 10^{29}$
Root discriminant \(42.64\)
Ramified primes $2,3,139$
Class number $9$ (GRH)
Class group [3, 3] (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 11*x^16 - 18*x^15 + 68*x^14 + 70*x^13 + 533*x^12 + 1100*x^11 + 580*x^10 + 7210*x^9 + 16613*x^8 + 59948*x^7 + 60667*x^6 + 137522*x^5 + 142906*x^4 + 494826*x^3 + 675168*x^2 - 1634408*x + 1185076)
 
gp: K = bnfinit(y^18 + 11*y^16 - 18*y^15 + 68*y^14 + 70*y^13 + 533*y^12 + 1100*y^11 + 580*y^10 + 7210*y^9 + 16613*y^8 + 59948*y^7 + 60667*y^6 + 137522*y^5 + 142906*y^4 + 494826*y^3 + 675168*y^2 - 1634408*y + 1185076, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 11*x^16 - 18*x^15 + 68*x^14 + 70*x^13 + 533*x^12 + 1100*x^11 + 580*x^10 + 7210*x^9 + 16613*x^8 + 59948*x^7 + 60667*x^6 + 137522*x^5 + 142906*x^4 + 494826*x^3 + 675168*x^2 - 1634408*x + 1185076);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 11*x^16 - 18*x^15 + 68*x^14 + 70*x^13 + 533*x^12 + 1100*x^11 + 580*x^10 + 7210*x^9 + 16613*x^8 + 59948*x^7 + 60667*x^6 + 137522*x^5 + 142906*x^4 + 494826*x^3 + 675168*x^2 - 1634408*x + 1185076)
 

\( x^{18} + 11 x^{16} - 18 x^{15} + 68 x^{14} + 70 x^{13} + 533 x^{12} + 1100 x^{11} + 580 x^{10} + \cdots + 1185076 \) 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:   \(-217069718467413515899000762368\) \(\medspace = -\,2^{12}\cdot 3^{9}\cdot 139^{10}\) 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:  \(42.64\)
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:  $2^{2/3}3^{1/2}139^{5/6}\approx 167.9163394069419$
Ramified primes:   \(2\), \(3\), \(139\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Aut(K/\Q) }$:  $3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{15}-\frac{1}{4}a^{14}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{58\!\cdots\!12}a^{17}-\frac{12\!\cdots\!02}{14\!\cdots\!53}a^{16}+\frac{12\!\cdots\!45}{14\!\cdots\!53}a^{15}+\frac{18\!\cdots\!23}{58\!\cdots\!12}a^{14}-\frac{61\!\cdots\!77}{58\!\cdots\!12}a^{13}+\frac{79\!\cdots\!26}{14\!\cdots\!53}a^{12}+\frac{22\!\cdots\!68}{14\!\cdots\!53}a^{11}+\frac{43\!\cdots\!65}{58\!\cdots\!12}a^{10}-\frac{61\!\cdots\!49}{58\!\cdots\!12}a^{9}-\frac{26\!\cdots\!64}{14\!\cdots\!53}a^{8}+\frac{18\!\cdots\!47}{29\!\cdots\!06}a^{7}+\frac{40\!\cdots\!11}{58\!\cdots\!12}a^{6}-\frac{41\!\cdots\!27}{14\!\cdots\!53}a^{5}-\frac{19\!\cdots\!01}{29\!\cdots\!06}a^{4}+\frac{12\!\cdots\!39}{29\!\cdots\!06}a^{3}-\frac{31\!\cdots\!81}{14\!\cdots\!53}a^{2}-\frac{26\!\cdots\!37}{14\!\cdots\!53}a+\frac{30\!\cdots\!96}{14\!\cdots\!53}$ 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:  $2$

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:   \( -\frac{89098187975460870611104}{107957197730529933785371458301} a^{17} + \frac{107354931511173656679081}{215914395461059867570742916602} a^{16} - \frac{1617000367406799472857149}{215914395461059867570742916602} a^{15} + \frac{1939101879461298428870325}{215914395461059867570742916602} a^{14} - \frac{8893978148857156263341025}{215914395461059867570742916602} a^{13} - \frac{13810012277400510776668239}{107957197730529933785371458301} a^{12} - \frac{13244465693408543263017709}{107957197730529933785371458301} a^{11} - \frac{240179175628784724910979467}{215914395461059867570742916602} a^{10} - \frac{35919571221702292942087357}{215914395461059867570742916602} a^{9} - \frac{773216720024351424713533754}{107957197730529933785371458301} a^{8} - \frac{1209424028294137411012618987}{107957197730529933785371458301} a^{7} - \frac{8904840087239695559033884499}{215914395461059867570742916602} a^{6} - \frac{10526492263261481024715218185}{215914395461059867570742916602} a^{5} - \frac{30006825664018924686827042731}{215914395461059867570742916602} a^{4} - \frac{48041395390349649234031799145}{215914395461059867570742916602} a^{3} - \frac{15230617268125957204719029568}{107957197730529933785371458301} a^{2} - \frac{133825957177886006278538289386}{107957197730529933785371458301} a + \frac{139260876226920615793108233060}{107957197730529933785371458301} \)  (order $6$) 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:   $\frac{10\!\cdots\!75}{58\!\cdots\!12}a^{17}-\frac{19\!\cdots\!30}{14\!\cdots\!53}a^{16}+\frac{66\!\cdots\!59}{29\!\cdots\!06}a^{15}-\frac{16\!\cdots\!95}{58\!\cdots\!12}a^{14}+\frac{82\!\cdots\!79}{58\!\cdots\!12}a^{13}+\frac{30\!\cdots\!63}{29\!\cdots\!06}a^{12}+\frac{20\!\cdots\!49}{29\!\cdots\!06}a^{11}+\frac{15\!\cdots\!25}{58\!\cdots\!12}a^{10}+\frac{15\!\cdots\!09}{58\!\cdots\!12}a^{9}+\frac{41\!\cdots\!07}{29\!\cdots\!06}a^{8}+\frac{74\!\cdots\!95}{29\!\cdots\!06}a^{7}+\frac{65\!\cdots\!17}{58\!\cdots\!12}a^{6}+\frac{25\!\cdots\!43}{14\!\cdots\!53}a^{5}+\frac{13\!\cdots\!79}{29\!\cdots\!06}a^{4}+\frac{97\!\cdots\!51}{29\!\cdots\!06}a^{3}+\frac{11\!\cdots\!97}{14\!\cdots\!53}a^{2}+\frac{43\!\cdots\!60}{14\!\cdots\!53}a-\frac{26\!\cdots\!82}{14\!\cdots\!53}$, $\frac{35\!\cdots\!33}{58\!\cdots\!12}a^{17}-\frac{31\!\cdots\!19}{14\!\cdots\!53}a^{16}+\frac{15\!\cdots\!41}{29\!\cdots\!06}a^{15}-\frac{18\!\cdots\!63}{58\!\cdots\!12}a^{14}+\frac{54\!\cdots\!11}{58\!\cdots\!12}a^{13}+\frac{38\!\cdots\!77}{14\!\cdots\!53}a^{12}+\frac{18\!\cdots\!23}{14\!\cdots\!53}a^{11}-\frac{25\!\cdots\!11}{58\!\cdots\!12}a^{10}-\frac{16\!\cdots\!75}{58\!\cdots\!12}a^{9}+\frac{12\!\cdots\!10}{14\!\cdots\!53}a^{8}+\frac{23\!\cdots\!37}{14\!\cdots\!53}a^{7}+\frac{78\!\cdots\!77}{58\!\cdots\!12}a^{6}-\frac{25\!\cdots\!73}{29\!\cdots\!06}a^{5}-\frac{58\!\cdots\!39}{14\!\cdots\!53}a^{4}+\frac{68\!\cdots\!55}{14\!\cdots\!53}a^{3}+\frac{24\!\cdots\!86}{14\!\cdots\!53}a^{2}+\frac{11\!\cdots\!48}{14\!\cdots\!53}a-\frac{59\!\cdots\!87}{14\!\cdots\!53}$, $\frac{91\!\cdots\!25}{58\!\cdots\!12}a^{17}-\frac{26\!\cdots\!07}{58\!\cdots\!12}a^{16}+\frac{20\!\cdots\!51}{58\!\cdots\!12}a^{15}-\frac{14\!\cdots\!98}{14\!\cdots\!53}a^{14}+\frac{21\!\cdots\!15}{58\!\cdots\!12}a^{13}-\frac{40\!\cdots\!99}{58\!\cdots\!12}a^{12}+\frac{13\!\cdots\!47}{58\!\cdots\!12}a^{11}-\frac{25\!\cdots\!55}{14\!\cdots\!53}a^{10}+\frac{38\!\cdots\!03}{58\!\cdots\!12}a^{9}+\frac{42\!\cdots\!85}{58\!\cdots\!12}a^{8}+\frac{17\!\cdots\!81}{58\!\cdots\!12}a^{7}+\frac{11\!\cdots\!77}{14\!\cdots\!53}a^{6}+\frac{10\!\cdots\!08}{14\!\cdots\!53}a^{5}+\frac{48\!\cdots\!96}{14\!\cdots\!53}a^{4}-\frac{79\!\cdots\!79}{29\!\cdots\!06}a^{3}+\frac{28\!\cdots\!76}{14\!\cdots\!53}a^{2}-\frac{74\!\cdots\!31}{14\!\cdots\!53}a-\frac{10\!\cdots\!51}{14\!\cdots\!53}$, $\frac{21\!\cdots\!09}{29\!\cdots\!06}a^{17}+\frac{79\!\cdots\!29}{58\!\cdots\!12}a^{16}+\frac{43\!\cdots\!45}{58\!\cdots\!12}a^{15}-\frac{49\!\cdots\!51}{58\!\cdots\!12}a^{14}+\frac{11\!\cdots\!97}{29\!\cdots\!06}a^{13}+\frac{49\!\cdots\!19}{58\!\cdots\!12}a^{12}+\frac{19\!\cdots\!01}{58\!\cdots\!12}a^{11}+\frac{52\!\cdots\!41}{58\!\cdots\!12}a^{10}+\frac{12\!\cdots\!10}{14\!\cdots\!53}a^{9}+\frac{30\!\cdots\!79}{58\!\cdots\!12}a^{8}+\frac{77\!\cdots\!79}{58\!\cdots\!12}a^{7}+\frac{25\!\cdots\!69}{58\!\cdots\!12}a^{6}+\frac{16\!\cdots\!31}{29\!\cdots\!06}a^{5}+\frac{35\!\cdots\!49}{29\!\cdots\!06}a^{4}+\frac{22\!\cdots\!17}{14\!\cdots\!53}a^{3}+\frac{42\!\cdots\!56}{14\!\cdots\!53}a^{2}+\frac{91\!\cdots\!33}{14\!\cdots\!53}a-\frac{86\!\cdots\!93}{14\!\cdots\!53}$, $\frac{26\!\cdots\!37}{58\!\cdots\!12}a^{17}-\frac{17\!\cdots\!81}{29\!\cdots\!06}a^{16}+\frac{65\!\cdots\!71}{14\!\cdots\!53}a^{15}-\frac{30\!\cdots\!81}{58\!\cdots\!12}a^{14}+\frac{12\!\cdots\!49}{58\!\cdots\!12}a^{13}+\frac{13\!\cdots\!31}{29\!\cdots\!06}a^{12}+\frac{46\!\cdots\!45}{29\!\cdots\!06}a^{11}+\frac{30\!\cdots\!55}{58\!\cdots\!12}a^{10}+\frac{26\!\cdots\!39}{58\!\cdots\!12}a^{9}+\frac{77\!\cdots\!63}{29\!\cdots\!06}a^{8}+\frac{18\!\cdots\!95}{29\!\cdots\!06}a^{7}+\frac{13\!\cdots\!63}{58\!\cdots\!12}a^{6}+\frac{44\!\cdots\!48}{14\!\cdots\!53}a^{5}+\frac{82\!\cdots\!15}{14\!\cdots\!53}a^{4}+\frac{37\!\cdots\!99}{14\!\cdots\!53}a^{3}-\frac{24\!\cdots\!84}{14\!\cdots\!53}a^{2}+\frac{35\!\cdots\!35}{14\!\cdots\!53}a-\frac{37\!\cdots\!08}{14\!\cdots\!53}$, $\frac{38\!\cdots\!25}{58\!\cdots\!12}a^{17}-\frac{24\!\cdots\!57}{29\!\cdots\!06}a^{16}+\frac{57\!\cdots\!99}{29\!\cdots\!06}a^{15}-\frac{31\!\cdots\!95}{58\!\cdots\!12}a^{14}+\frac{83\!\cdots\!97}{58\!\cdots\!12}a^{13}-\frac{94\!\cdots\!05}{29\!\cdots\!06}a^{12}-\frac{11\!\cdots\!96}{14\!\cdots\!53}a^{11}+\frac{42\!\cdots\!67}{58\!\cdots\!12}a^{10}-\frac{30\!\cdots\!39}{58\!\cdots\!12}a^{9}+\frac{47\!\cdots\!01}{29\!\cdots\!06}a^{8}-\frac{10\!\cdots\!63}{29\!\cdots\!06}a^{7}-\frac{25\!\cdots\!15}{58\!\cdots\!12}a^{6}-\frac{21\!\cdots\!79}{29\!\cdots\!06}a^{5}+\frac{67\!\cdots\!39}{29\!\cdots\!06}a^{4}-\frac{52\!\cdots\!18}{14\!\cdots\!53}a^{3}-\frac{16\!\cdots\!17}{14\!\cdots\!53}a^{2}+\frac{29\!\cdots\!66}{14\!\cdots\!53}a-\frac{13\!\cdots\!33}{14\!\cdots\!53}$, $\frac{22\!\cdots\!39}{58\!\cdots\!12}a^{17}+\frac{26\!\cdots\!27}{29\!\cdots\!06}a^{16}+\frac{30\!\cdots\!69}{14\!\cdots\!53}a^{15}+\frac{17\!\cdots\!21}{58\!\cdots\!12}a^{14}-\frac{37\!\cdots\!21}{58\!\cdots\!12}a^{13}+\frac{35\!\cdots\!53}{29\!\cdots\!06}a^{12}+\frac{44\!\cdots\!81}{29\!\cdots\!06}a^{11}+\frac{12\!\cdots\!37}{58\!\cdots\!12}a^{10}+\frac{17\!\cdots\!17}{58\!\cdots\!12}a^{9}+\frac{72\!\cdots\!25}{29\!\cdots\!06}a^{8}+\frac{29\!\cdots\!15}{29\!\cdots\!06}a^{7}+\frac{10\!\cdots\!97}{58\!\cdots\!12}a^{6}+\frac{15\!\cdots\!76}{14\!\cdots\!53}a^{5}+\frac{23\!\cdots\!47}{14\!\cdots\!53}a^{4}+\frac{24\!\cdots\!98}{14\!\cdots\!53}a^{3}+\frac{63\!\cdots\!26}{14\!\cdots\!53}a^{2}-\frac{39\!\cdots\!39}{14\!\cdots\!53}a+\frac{33\!\cdots\!34}{14\!\cdots\!53}$, $\frac{30\!\cdots\!99}{29\!\cdots\!06}a^{17}+\frac{29\!\cdots\!05}{29\!\cdots\!06}a^{16}+\frac{24\!\cdots\!58}{14\!\cdots\!53}a^{15}-\frac{11\!\cdots\!65}{29\!\cdots\!06}a^{14}+\frac{16\!\cdots\!18}{14\!\cdots\!53}a^{13}+\frac{67\!\cdots\!65}{29\!\cdots\!06}a^{12}+\frac{41\!\cdots\!50}{14\!\cdots\!53}a^{11}+\frac{36\!\cdots\!54}{14\!\cdots\!53}a^{10}+\frac{13\!\cdots\!52}{14\!\cdots\!53}a^{9}+\frac{81\!\cdots\!73}{14\!\cdots\!53}a^{8}+\frac{56\!\cdots\!03}{29\!\cdots\!06}a^{7}+\frac{43\!\cdots\!96}{14\!\cdots\!53}a^{6}+\frac{28\!\cdots\!81}{14\!\cdots\!53}a^{5}+\frac{60\!\cdots\!37}{14\!\cdots\!53}a^{4}+\frac{83\!\cdots\!05}{29\!\cdots\!06}a^{3}+\frac{93\!\cdots\!33}{14\!\cdots\!53}a^{2}-\frac{65\!\cdots\!51}{14\!\cdots\!53}a+\frac{40\!\cdots\!87}{14\!\cdots\!53}$ 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:  \( 9350988.404859575 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 9350988.404859575 \cdot 9}{6\cdot\sqrt{217069718467413515899000762368}}\cr\approx \mathstrut & 0.459481437671540 \end{aligned}\] (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 - 18*x^15 + 68*x^14 + 70*x^13 + 533*x^12 + 1100*x^11 + 580*x^10 + 7210*x^9 + 16613*x^8 + 59948*x^7 + 60667*x^6 + 137522*x^5 + 142906*x^4 + 494826*x^3 + 675168*x^2 - 1634408*x + 1185076)
 
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 - 18*x^15 + 68*x^14 + 70*x^13 + 533*x^12 + 1100*x^11 + 580*x^10 + 7210*x^9 + 16613*x^8 + 59948*x^7 + 60667*x^6 + 137522*x^5 + 142906*x^4 + 494826*x^3 + 675168*x^2 - 1634408*x + 1185076, 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 - 18*x^15 + 68*x^14 + 70*x^13 + 533*x^12 + 1100*x^11 + 580*x^10 + 7210*x^9 + 16613*x^8 + 59948*x^7 + 60667*x^6 + 137522*x^5 + 142906*x^4 + 494826*x^3 + 675168*x^2 - 1634408*x + 1185076);
 
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 - 18*x^15 + 68*x^14 + 70*x^13 + 533*x^12 + 1100*x^11 + 580*x^10 + 7210*x^9 + 16613*x^8 + 59948*x^7 + 60667*x^6 + 137522*x^5 + 142906*x^4 + 494826*x^3 + 675168*x^2 - 1634408*x + 1185076);
 
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

\(\Q(\sqrt{-3}) \), 3.1.139.1, 6.0.8346672.1, 6.0.521667.1

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: 12.0.26006738789713255282944.1

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 ${\href{/padicField/5.6.0.1}{6} }^{3}$ ${\href{/padicField/7.3.0.1}{3} }^{6}$ ${\href{/padicField/11.6.0.1}{6} }^{3}$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }^{3}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{9}$ ${\href{/padicField/29.6.0.1}{6} }^{3}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{9}$ ${\href{/padicField/41.6.0.1}{6} }^{3}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }^{3}$ ${\href{/padicField/59.2.0.1}{2} }^{9}$

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
\(2\) Copy content Toggle raw display 2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
\(3\) Copy content Toggle raw display 3.6.3.2$x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.6.2$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
\(139\) Copy content Toggle raw display 139.3.2.2$x^{3} + 417$$3$$1$$2$$C_3$$[\ ]_{3}$
139.3.0.1$x^{3} + 6 x + 137$$1$$3$$0$$C_3$$[\ ]^{3}$
139.6.3.2$x^{6} + 429 x^{4} + 274 x^{3} + 57999 x^{2} - 112614 x + 2477540$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
139.6.5.4$x^{6} + 556$$6$$1$$5$$C_6$$[\ ]_{6}$