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Properties

Label 18.0.154...167.1
Degree $18$
Signature $[0, 9]$
Discriminant $-1.549\times 10^{25}$
Root discriminant \(25.09\)
Ramified primes $7,47$
Class number $5$
Class group [5]
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 13*x^15 + 83*x^14 - 101*x^13 + 27*x^12 - 399*x^11 + 1430*x^10 - 743*x^9 - 2237*x^8 + 4368*x^7 - 3469*x^6 + 3460*x^5 + 1340*x^4 - 4333*x^3 + 6206*x^2 - 3917*x + 13411)
 
gp: K = bnfinit(y^18 - 2*y^17 - 13*y^15 + 83*y^14 - 101*y^13 + 27*y^12 - 399*y^11 + 1430*y^10 - 743*y^9 - 2237*y^8 + 4368*y^7 - 3469*y^6 + 3460*y^5 + 1340*y^4 - 4333*y^3 + 6206*y^2 - 3917*y + 13411, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 2*x^17 - 13*x^15 + 83*x^14 - 101*x^13 + 27*x^12 - 399*x^11 + 1430*x^10 - 743*x^9 - 2237*x^8 + 4368*x^7 - 3469*x^6 + 3460*x^5 + 1340*x^4 - 4333*x^3 + 6206*x^2 - 3917*x + 13411);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 2*x^17 - 13*x^15 + 83*x^14 - 101*x^13 + 27*x^12 - 399*x^11 + 1430*x^10 - 743*x^9 - 2237*x^8 + 4368*x^7 - 3469*x^6 + 3460*x^5 + 1340*x^4 - 4333*x^3 + 6206*x^2 - 3917*x + 13411)
 

\( x^{18} - 2 x^{17} - 13 x^{15} + 83 x^{14} - 101 x^{13} + 27 x^{12} - 399 x^{11} + 1430 x^{10} + \cdots + 13411 \) 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:   \(-15490206293606403634785167\) \(\medspace = -\,7^{12}\cdot 47^{9}\) 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:  \(25.09\)
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:  $7^{2/3}47^{1/2}\approx 25.086936025192795$
Ramified primes:   \(7\), \(47\) 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:  \(\Q(\sqrt{-47}) \)
$\card{ \Gal(K/\Q) }$:  $18$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{5}+\frac{1}{4}a^{3}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{8}a^{5}-\frac{1}{8}a^{4}+\frac{3}{8}a^{3}-\frac{1}{2}a^{2}+\frac{3}{8}a+\frac{3}{8}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{11}-\frac{1}{8}a^{10}-\frac{1}{4}a^{7}+\frac{1}{8}a^{6}-\frac{1}{8}a^{5}+\frac{1}{8}a^{4}-\frac{3}{8}a^{2}-\frac{1}{8}a-\frac{1}{4}$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{7}+\frac{1}{8}a^{6}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}-\frac{1}{2}a^{3}-\frac{3}{8}a^{2}+\frac{1}{8}a+\frac{1}{8}$, $\frac{1}{96}a^{15}+\frac{5}{96}a^{14}+\frac{1}{96}a^{13}-\frac{1}{32}a^{12}+\frac{5}{96}a^{11}-\frac{7}{96}a^{10}+\frac{3}{32}a^{9}-\frac{1}{96}a^{8}+\frac{1}{48}a^{7}+\frac{11}{48}a^{6}-\frac{5}{24}a^{5}+\frac{5}{48}a^{4}-\frac{17}{96}a^{3}+\frac{15}{32}a^{2}+\frac{9}{32}a-\frac{23}{96}$, $\frac{1}{96}a^{16}+\frac{1}{24}a^{13}-\frac{1}{24}a^{12}+\frac{1}{24}a^{11}+\frac{1}{12}a^{10}+\frac{1}{48}a^{9}+\frac{7}{96}a^{8}-\frac{1}{8}a^{7}-\frac{11}{48}a^{6}+\frac{1}{48}a^{5}+\frac{17}{96}a^{4}+\frac{17}{48}a^{3}+\frac{1}{16}a^{2}-\frac{13}{48}a+\frac{43}{96}$, $\frac{1}{15\!\cdots\!84}a^{17}+\frac{72\!\cdots\!23}{15\!\cdots\!84}a^{16}+\frac{18\!\cdots\!71}{39\!\cdots\!96}a^{15}-\frac{15\!\cdots\!11}{39\!\cdots\!96}a^{14}+\frac{96\!\cdots\!67}{19\!\cdots\!48}a^{13}+\frac{10\!\cdots\!15}{33\!\cdots\!58}a^{12}-\frac{29\!\cdots\!51}{49\!\cdots\!37}a^{11}-\frac{63\!\cdots\!59}{79\!\cdots\!92}a^{10}+\frac{15\!\cdots\!03}{52\!\cdots\!28}a^{9}+\frac{23\!\cdots\!53}{15\!\cdots\!84}a^{8}-\frac{19\!\cdots\!13}{26\!\cdots\!64}a^{7}+\frac{12\!\cdots\!14}{16\!\cdots\!79}a^{6}+\frac{53\!\cdots\!41}{52\!\cdots\!28}a^{5}+\frac{85\!\cdots\!29}{15\!\cdots\!84}a^{4}-\frac{34\!\cdots\!53}{13\!\cdots\!32}a^{3}+\frac{18\!\cdots\!11}{39\!\cdots\!96}a^{2}-\frac{57\!\cdots\!31}{15\!\cdots\!84}a-\frac{18\!\cdots\!13}{52\!\cdots\!28}$ 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

$C_{5}$, which has order $5$

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:   $\frac{27\!\cdots\!27}{18\!\cdots\!16}a^{17}-\frac{410048629984691}{11\!\cdots\!51}a^{16}-\frac{58\!\cdots\!65}{36\!\cdots\!32}a^{15}-\frac{89\!\cdots\!97}{36\!\cdots\!32}a^{14}+\frac{27\!\cdots\!79}{36\!\cdots\!32}a^{13}+\frac{14\!\cdots\!63}{36\!\cdots\!32}a^{12}+\frac{14\!\cdots\!15}{36\!\cdots\!32}a^{11}-\frac{18\!\cdots\!81}{36\!\cdots\!32}a^{10}+\frac{26\!\cdots\!41}{36\!\cdots\!32}a^{9}+\frac{17\!\cdots\!37}{36\!\cdots\!32}a^{8}-\frac{39\!\cdots\!75}{18\!\cdots\!16}a^{7}+\frac{85\!\cdots\!71}{18\!\cdots\!16}a^{6}-\frac{53\!\cdots\!95}{18\!\cdots\!16}a^{5}-\frac{89\!\cdots\!31}{18\!\cdots\!16}a^{4}+\frac{31\!\cdots\!57}{36\!\cdots\!32}a^{3}-\frac{10\!\cdots\!97}{36\!\cdots\!32}a^{2}+\frac{24\!\cdots\!87}{36\!\cdots\!32}a-\frac{82\!\cdots\!81}{36\!\cdots\!32}$, $\frac{20638081489699}{18\!\cdots\!16}a^{17}+\frac{17\!\cdots\!49}{36\!\cdots\!32}a^{16}+\frac{66\!\cdots\!47}{90\!\cdots\!08}a^{15}+\frac{27\!\cdots\!61}{22\!\cdots\!02}a^{14}-\frac{66\!\cdots\!11}{90\!\cdots\!08}a^{13}+\frac{18\!\cdots\!63}{90\!\cdots\!08}a^{12}+\frac{43\!\cdots\!07}{11\!\cdots\!51}a^{11}+\frac{10\!\cdots\!13}{90\!\cdots\!08}a^{10}-\frac{10\!\cdots\!77}{45\!\cdots\!04}a^{9}+\frac{22\!\cdots\!47}{36\!\cdots\!32}a^{8}+\frac{58\!\cdots\!93}{90\!\cdots\!08}a^{7}+\frac{30\!\cdots\!87}{18\!\cdots\!16}a^{6}+\frac{91\!\cdots\!49}{22\!\cdots\!02}a^{5}-\frac{83\!\cdots\!99}{36\!\cdots\!32}a^{4}+\frac{88\!\cdots\!75}{18\!\cdots\!16}a^{3}+\frac{58\!\cdots\!87}{18\!\cdots\!16}a^{2}+\frac{28\!\cdots\!17}{45\!\cdots\!04}a+\frac{36\!\cdots\!03}{36\!\cdots\!32}$, $\frac{13\!\cdots\!05}{99\!\cdots\!74}a^{17}-\frac{35\!\cdots\!55}{19\!\cdots\!48}a^{16}-\frac{22\!\cdots\!91}{15\!\cdots\!84}a^{15}-\frac{31\!\cdots\!83}{15\!\cdots\!84}a^{14}+\frac{13\!\cdots\!29}{15\!\cdots\!84}a^{13}-\frac{43\!\cdots\!41}{52\!\cdots\!28}a^{12}+\frac{11\!\cdots\!37}{15\!\cdots\!84}a^{11}-\frac{71\!\cdots\!51}{15\!\cdots\!84}a^{10}+\frac{67\!\cdots\!15}{52\!\cdots\!28}a^{9}-\frac{11\!\cdots\!89}{15\!\cdots\!84}a^{8}-\frac{83\!\cdots\!17}{26\!\cdots\!64}a^{7}+\frac{15\!\cdots\!53}{26\!\cdots\!64}a^{6}-\frac{41\!\cdots\!21}{13\!\cdots\!32}a^{5}-\frac{54\!\cdots\!59}{79\!\cdots\!92}a^{4}+\frac{27\!\cdots\!81}{52\!\cdots\!28}a^{3}+\frac{10\!\cdots\!97}{15\!\cdots\!84}a^{2}-\frac{78\!\cdots\!57}{15\!\cdots\!84}a-\frac{61\!\cdots\!97}{52\!\cdots\!28}$, $\frac{13\!\cdots\!87}{79\!\cdots\!92}a^{17}-\frac{13\!\cdots\!17}{79\!\cdots\!92}a^{16}+\frac{33\!\cdots\!91}{52\!\cdots\!28}a^{15}-\frac{43\!\cdots\!71}{15\!\cdots\!84}a^{14}+\frac{15\!\cdots\!57}{15\!\cdots\!84}a^{13}-\frac{15\!\cdots\!63}{15\!\cdots\!84}a^{12}+\frac{31\!\cdots\!93}{15\!\cdots\!84}a^{11}-\frac{30\!\cdots\!01}{52\!\cdots\!28}a^{10}+\frac{20\!\cdots\!43}{15\!\cdots\!84}a^{9}-\frac{23\!\cdots\!67}{15\!\cdots\!84}a^{8}-\frac{82\!\cdots\!49}{79\!\cdots\!92}a^{7}+\frac{22\!\cdots\!55}{26\!\cdots\!64}a^{6}-\frac{20\!\cdots\!73}{26\!\cdots\!64}a^{5}-\frac{29\!\cdots\!77}{99\!\cdots\!74}a^{4}+\frac{90\!\cdots\!35}{15\!\cdots\!84}a^{3}+\frac{64\!\cdots\!13}{15\!\cdots\!84}a^{2}+\frac{46\!\cdots\!23}{52\!\cdots\!28}a-\frac{23\!\cdots\!89}{15\!\cdots\!84}$, $\frac{40\!\cdots\!93}{15\!\cdots\!84}a^{17}-\frac{72\!\cdots\!13}{15\!\cdots\!84}a^{16}-\frac{60\!\cdots\!77}{99\!\cdots\!74}a^{15}-\frac{16\!\cdots\!29}{39\!\cdots\!96}a^{14}+\frac{27\!\cdots\!45}{13\!\cdots\!32}a^{13}-\frac{23\!\cdots\!71}{19\!\cdots\!48}a^{12}-\frac{47\!\cdots\!64}{49\!\cdots\!37}a^{11}-\frac{11\!\cdots\!45}{79\!\cdots\!92}a^{10}+\frac{49\!\cdots\!85}{15\!\cdots\!84}a^{9}+\frac{18\!\cdots\!39}{52\!\cdots\!28}a^{8}-\frac{14\!\cdots\!59}{26\!\cdots\!64}a^{7}+\frac{18\!\cdots\!73}{39\!\cdots\!96}a^{6}-\frac{60\!\cdots\!65}{15\!\cdots\!84}a^{5}+\frac{31\!\cdots\!21}{15\!\cdots\!84}a^{4}+\frac{12\!\cdots\!09}{39\!\cdots\!96}a^{3}-\frac{21\!\cdots\!39}{39\!\cdots\!96}a^{2}+\frac{52\!\cdots\!59}{52\!\cdots\!28}a-\frac{77\!\cdots\!23}{15\!\cdots\!84}$, $\frac{21\!\cdots\!01}{19\!\cdots\!48}a^{17}+\frac{29\!\cdots\!93}{52\!\cdots\!28}a^{16}-\frac{78\!\cdots\!91}{39\!\cdots\!96}a^{15}-\frac{12\!\cdots\!15}{66\!\cdots\!16}a^{14}+\frac{33\!\cdots\!67}{66\!\cdots\!16}a^{13}+\frac{33\!\cdots\!28}{49\!\cdots\!37}a^{12}+\frac{56\!\cdots\!07}{39\!\cdots\!96}a^{11}-\frac{98\!\cdots\!21}{19\!\cdots\!48}a^{10}+\frac{31\!\cdots\!55}{79\!\cdots\!92}a^{9}+\frac{23\!\cdots\!49}{15\!\cdots\!84}a^{8}-\frac{15\!\cdots\!19}{19\!\cdots\!48}a^{7}-\frac{14\!\cdots\!41}{26\!\cdots\!64}a^{6}+\frac{15\!\cdots\!17}{26\!\cdots\!64}a^{5}+\frac{26\!\cdots\!31}{15\!\cdots\!84}a^{4}+\frac{24\!\cdots\!37}{79\!\cdots\!92}a^{3}+\frac{26\!\cdots\!17}{79\!\cdots\!92}a^{2}-\frac{18\!\cdots\!59}{79\!\cdots\!92}a-\frac{18\!\cdots\!99}{15\!\cdots\!84}$, $\frac{56\!\cdots\!65}{16\!\cdots\!79}a^{17}-\frac{93\!\cdots\!39}{19\!\cdots\!48}a^{16}-\frac{21\!\cdots\!11}{79\!\cdots\!92}a^{15}-\frac{49\!\cdots\!43}{79\!\cdots\!92}a^{14}+\frac{12\!\cdots\!07}{79\!\cdots\!92}a^{13}+\frac{28\!\cdots\!33}{79\!\cdots\!92}a^{12}-\frac{27\!\cdots\!87}{26\!\cdots\!64}a^{11}-\frac{12\!\cdots\!53}{79\!\cdots\!92}a^{10}+\frac{26\!\cdots\!17}{79\!\cdots\!92}a^{9}+\frac{26\!\cdots\!45}{26\!\cdots\!64}a^{8}-\frac{68\!\cdots\!23}{39\!\cdots\!96}a^{7}-\frac{10\!\cdots\!33}{16\!\cdots\!79}a^{6}+\frac{13\!\cdots\!69}{13\!\cdots\!32}a^{5}-\frac{29\!\cdots\!29}{66\!\cdots\!16}a^{4}+\frac{39\!\cdots\!07}{79\!\cdots\!92}a^{3}-\frac{14\!\cdots\!63}{26\!\cdots\!64}a^{2}-\frac{82\!\cdots\!31}{79\!\cdots\!92}a-\frac{72\!\cdots\!75}{79\!\cdots\!92}$, $\frac{32\!\cdots\!05}{39\!\cdots\!96}a^{17}-\frac{34\!\cdots\!11}{15\!\cdots\!84}a^{16}-\frac{69\!\cdots\!45}{79\!\cdots\!92}a^{15}-\frac{15\!\cdots\!47}{79\!\cdots\!92}a^{14}+\frac{32\!\cdots\!03}{79\!\cdots\!92}a^{13}-\frac{96\!\cdots\!75}{26\!\cdots\!64}a^{12}+\frac{96\!\cdots\!61}{79\!\cdots\!92}a^{11}-\frac{42\!\cdots\!39}{79\!\cdots\!92}a^{10}+\frac{67\!\cdots\!37}{13\!\cdots\!32}a^{9}-\frac{44\!\cdots\!59}{15\!\cdots\!84}a^{8}+\frac{12\!\cdots\!41}{13\!\cdots\!32}a^{7}-\frac{26\!\cdots\!23}{26\!\cdots\!64}a^{6}-\frac{25\!\cdots\!07}{26\!\cdots\!64}a^{5}+\frac{19\!\cdots\!41}{15\!\cdots\!84}a^{4}-\frac{20\!\cdots\!27}{66\!\cdots\!16}a^{3}+\frac{19\!\cdots\!05}{39\!\cdots\!96}a^{2}-\frac{57\!\cdots\!97}{19\!\cdots\!48}a-\frac{74\!\cdots\!21}{52\!\cdots\!28}$ Copy content Toggle raw display
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:  \( 29368.4164899 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 29368.4164899 \cdot 5}{2\cdot\sqrt{15490206293606403634785167}}\cr\approx \mathstrut & 0.284715265757 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 13*x^15 + 83*x^14 - 101*x^13 + 27*x^12 - 399*x^11 + 1430*x^10 - 743*x^9 - 2237*x^8 + 4368*x^7 - 3469*x^6 + 3460*x^5 + 1340*x^4 - 4333*x^3 + 6206*x^2 - 3917*x + 13411)
 
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 - 2*x^17 - 13*x^15 + 83*x^14 - 101*x^13 + 27*x^12 - 399*x^11 + 1430*x^10 - 743*x^9 - 2237*x^8 + 4368*x^7 - 3469*x^6 + 3460*x^5 + 1340*x^4 - 4333*x^3 + 6206*x^2 - 3917*x + 13411, 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 - 2*x^17 - 13*x^15 + 83*x^14 - 101*x^13 + 27*x^12 - 399*x^11 + 1430*x^10 - 743*x^9 - 2237*x^8 + 4368*x^7 - 3469*x^6 + 3460*x^5 + 1340*x^4 - 4333*x^3 + 6206*x^2 - 3917*x + 13411);
 
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 - 2*x^17 - 13*x^15 + 83*x^14 - 101*x^13 + 27*x^12 - 399*x^11 + 1430*x^10 - 743*x^9 - 2237*x^8 + 4368*x^7 - 3469*x^6 + 3460*x^5 + 1340*x^4 - 4333*x^3 + 6206*x^2 - 3917*x + 13411);
 
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

$C_3\times S_3$ (as 18T3):

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 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-47}) \), 3.1.2303.1 x3, \(\Q(\zeta_{7})^+\), 6.0.249279023.1, 6.0.5087327.1 x2, 6.0.249279023.2, 9.3.12214672127.1 x3

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 6 sibling: 6.0.5087327.1
Degree 9 sibling: 9.3.12214672127.1
Minimal sibling: 6.0.5087327.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.3.0.1}{3} }^{6}$ ${\href{/padicField/3.3.0.1}{3} }^{6}$ ${\href{/padicField/5.6.0.1}{6} }^{3}$ R ${\href{/padicField/11.6.0.1}{6} }^{3}$ ${\href{/padicField/13.2.0.1}{2} }^{9}$ ${\href{/padicField/17.3.0.1}{3} }^{6}$ ${\href{/padicField/19.6.0.1}{6} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.6.0.1}{6} }^{3}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.2.0.1}{2} }^{9}$ R ${\href{/padicField/53.3.0.1}{3} }^{6}$ ${\href{/padicField/59.3.0.1}{3} }^{6}$

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
\(7\) Copy content Toggle raw display 7.9.6.1$x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
\(47\) Copy content Toggle raw display 47.6.3.2$x^{6} + 147 x^{4} + 84 x^{3} + 6636 x^{2} - 11592 x + 92756$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
47.6.3.2$x^{6} + 147 x^{4} + 84 x^{3} + 6636 x^{2} - 11592 x + 92756$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
47.6.3.2$x^{6} + 147 x^{4} + 84 x^{3} + 6636 x^{2} - 11592 x + 92756$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$