↑

Properties

Label 18.0.621...704.1
Degree $18$
Signature $[0, 9]$
Discriminant $-6.218\times 10^{22}$
Root discriminant \(18.46\)
Ramified primes $2,19$
Class number $1$
Class group trivial
Galois group $S_3 \times C_3$ (as 18T3)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 11*x^16 + 29*x^15 + 13*x^14 - 181*x^13 + 260*x^12 + 223*x^11 - 970*x^10 + 567*x^9 + 1007*x^8 - 1329*x^7 + 46*x^6 + 692*x^5 - 526*x^4 + 5*x^3 + 348*x^2 - 173*x + 83)
 
gp: K = bnfinit(y^18 - y^17 - 11*y^16 + 29*y^15 + 13*y^14 - 181*y^13 + 260*y^12 + 223*y^11 - 970*y^10 + 567*y^9 + 1007*y^8 - 1329*y^7 + 46*y^6 + 692*y^5 - 526*y^4 + 5*y^3 + 348*y^2 - 173*y + 83, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - x^17 - 11*x^16 + 29*x^15 + 13*x^14 - 181*x^13 + 260*x^12 + 223*x^11 - 970*x^10 + 567*x^9 + 1007*x^8 - 1329*x^7 + 46*x^6 + 692*x^5 - 526*x^4 + 5*x^3 + 348*x^2 - 173*x + 83);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - x^17 - 11*x^16 + 29*x^15 + 13*x^14 - 181*x^13 + 260*x^12 + 223*x^11 - 970*x^10 + 567*x^9 + 1007*x^8 - 1329*x^7 + 46*x^6 + 692*x^5 - 526*x^4 + 5*x^3 + 348*x^2 - 173*x + 83)
 

\( x^{18} - x^{17} - 11 x^{16} + 29 x^{15} + 13 x^{14} - 181 x^{13} + 260 x^{12} + 223 x^{11} - 970 x^{10} + \cdots + 83 \) 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:   \(-62181896314367173832704\) \(\medspace = -\,2^{12}\cdot 19^{15}\) 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:  \(18.46\)
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:  $2^{2/3}19^{5/6}\approx 18.463526946433323$
Ramified primes:   \(2\), \(19\) 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:  \(\Q(\sqrt{-19}) \)
$\card{ \Gal(K/\Q) }$:  $18$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{197}a^{15}-\frac{31}{197}a^{14}-\frac{44}{197}a^{13}-\frac{66}{197}a^{12}-\frac{89}{197}a^{11}-\frac{4}{197}a^{10}-\frac{86}{197}a^{9}-\frac{13}{197}a^{8}+\frac{66}{197}a^{7}+\frac{72}{197}a^{6}-\frac{22}{197}a^{5}+\frac{14}{197}a^{4}-\frac{50}{197}a^{3}-\frac{89}{197}a^{2}+\frac{41}{197}a-\frac{23}{197}$, $\frac{1}{16351}a^{16}+\frac{22}{16351}a^{15}-\frac{4445}{16351}a^{14}+\frac{1345}{16351}a^{13}+\frac{7445}{16351}a^{12}+\frac{7}{16351}a^{11}+\frac{1081}{16351}a^{10}+\frac{3703}{16351}a^{9}-\frac{5942}{16351}a^{8}-\frac{5492}{16351}a^{7}-\frac{3101}{16351}a^{6}+\frac{7713}{16351}a^{5}+\frac{4435}{16351}a^{4}+\frac{7505}{16351}a^{3}+\frac{5371}{16351}a^{2}+\frac{2347}{16351}a+\frac{85}{197}$, $\frac{1}{60\!\cdots\!21}a^{17}-\frac{3674735660822}{60\!\cdots\!21}a^{16}+\frac{771239323929358}{60\!\cdots\!21}a^{15}+\frac{40\!\cdots\!82}{60\!\cdots\!21}a^{14}-\frac{11\!\cdots\!95}{60\!\cdots\!21}a^{13}+\frac{16\!\cdots\!66}{60\!\cdots\!21}a^{12}+\frac{17\!\cdots\!86}{60\!\cdots\!21}a^{11}-\frac{12\!\cdots\!22}{60\!\cdots\!21}a^{10}+\frac{50\!\cdots\!94}{60\!\cdots\!21}a^{9}-\frac{13\!\cdots\!11}{60\!\cdots\!21}a^{8}-\frac{10\!\cdots\!27}{60\!\cdots\!21}a^{7}+\frac{10\!\cdots\!80}{60\!\cdots\!21}a^{6}-\frac{11\!\cdots\!19}{60\!\cdots\!21}a^{5}-\frac{50\!\cdots\!45}{60\!\cdots\!21}a^{4}-\frac{23\!\cdots\!75}{60\!\cdots\!21}a^{3}-\frac{19\!\cdots\!46}{60\!\cdots\!21}a^{2}+\frac{18\!\cdots\!97}{60\!\cdots\!21}a+\frac{728068576345995}{73\!\cdots\!87}$ Copy content Toggle raw display

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

Trivial group, which has order $1$

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:   $\frac{28\!\cdots\!70}{60\!\cdots\!21}a^{17}+\frac{11\!\cdots\!95}{60\!\cdots\!21}a^{16}-\frac{31\!\cdots\!14}{60\!\cdots\!21}a^{15}+\frac{36\!\cdots\!22}{60\!\cdots\!21}a^{14}+\frac{11\!\cdots\!13}{60\!\cdots\!21}a^{13}-\frac{36\!\cdots\!86}{60\!\cdots\!21}a^{12}+\frac{56\!\cdots\!11}{60\!\cdots\!21}a^{11}+\frac{10\!\cdots\!15}{60\!\cdots\!21}a^{10}-\frac{94\!\cdots\!83}{60\!\cdots\!21}a^{9}-\frac{17\!\cdots\!22}{60\!\cdots\!21}a^{8}+\frac{23\!\cdots\!86}{60\!\cdots\!21}a^{7}+\frac{21\!\cdots\!65}{60\!\cdots\!21}a^{6}-\frac{28\!\cdots\!34}{60\!\cdots\!21}a^{5}-\frac{12\!\cdots\!82}{60\!\cdots\!21}a^{4}+\frac{13\!\cdots\!39}{60\!\cdots\!21}a^{3}+\frac{46\!\cdots\!66}{60\!\cdots\!21}a^{2}+\frac{27\!\cdots\!02}{60\!\cdots\!21}a+\frac{12\!\cdots\!42}{73\!\cdots\!87}$, $\frac{28\!\cdots\!70}{60\!\cdots\!21}a^{17}+\frac{11\!\cdots\!95}{60\!\cdots\!21}a^{16}-\frac{31\!\cdots\!14}{60\!\cdots\!21}a^{15}+\frac{36\!\cdots\!22}{60\!\cdots\!21}a^{14}+\frac{11\!\cdots\!13}{60\!\cdots\!21}a^{13}-\frac{36\!\cdots\!86}{60\!\cdots\!21}a^{12}+\frac{56\!\cdots\!11}{60\!\cdots\!21}a^{11}+\frac{10\!\cdots\!15}{60\!\cdots\!21}a^{10}-\frac{94\!\cdots\!83}{60\!\cdots\!21}a^{9}-\frac{17\!\cdots\!22}{60\!\cdots\!21}a^{8}+\frac{23\!\cdots\!86}{60\!\cdots\!21}a^{7}+\frac{21\!\cdots\!65}{60\!\cdots\!21}a^{6}-\frac{28\!\cdots\!34}{60\!\cdots\!21}a^{5}-\frac{12\!\cdots\!82}{60\!\cdots\!21}a^{4}+\frac{13\!\cdots\!39}{60\!\cdots\!21}a^{3}+\frac{46\!\cdots\!66}{60\!\cdots\!21}a^{2}+\frac{27\!\cdots\!02}{60\!\cdots\!21}a+\frac{85\!\cdots\!29}{73\!\cdots\!87}$, $\frac{24\!\cdots\!50}{60\!\cdots\!21}a^{17}+\frac{942541891682375}{60\!\cdots\!21}a^{16}-\frac{24\!\cdots\!55}{60\!\cdots\!21}a^{15}+\frac{30\!\cdots\!27}{60\!\cdots\!21}a^{14}+\frac{68\!\cdots\!11}{60\!\cdots\!21}a^{13}-\frac{26\!\cdots\!32}{60\!\cdots\!21}a^{12}+\frac{13\!\cdots\!82}{60\!\cdots\!21}a^{11}+\frac{56\!\cdots\!34}{60\!\cdots\!21}a^{10}-\frac{63\!\cdots\!96}{60\!\cdots\!21}a^{9}-\frac{63\!\cdots\!78}{60\!\cdots\!21}a^{8}+\frac{73\!\cdots\!98}{60\!\cdots\!21}a^{7}+\frac{12\!\cdots\!78}{60\!\cdots\!21}a^{6}-\frac{65\!\cdots\!53}{60\!\cdots\!21}a^{5}-\frac{57\!\cdots\!45}{60\!\cdots\!21}a^{4}+\frac{37\!\cdots\!59}{60\!\cdots\!21}a^{3}-\frac{89\!\cdots\!82}{60\!\cdots\!21}a^{2}+\frac{26\!\cdots\!94}{60\!\cdots\!21}a-\frac{64\!\cdots\!46}{73\!\cdots\!87}$, $\frac{24\!\cdots\!50}{60\!\cdots\!21}a^{17}+\frac{942541891682375}{60\!\cdots\!21}a^{16}-\frac{24\!\cdots\!55}{60\!\cdots\!21}a^{15}+\frac{30\!\cdots\!27}{60\!\cdots\!21}a^{14}+\frac{68\!\cdots\!11}{60\!\cdots\!21}a^{13}-\frac{26\!\cdots\!32}{60\!\cdots\!21}a^{12}+\frac{13\!\cdots\!82}{60\!\cdots\!21}a^{11}+\frac{56\!\cdots\!34}{60\!\cdots\!21}a^{10}-\frac{63\!\cdots\!96}{60\!\cdots\!21}a^{9}-\frac{63\!\cdots\!78}{60\!\cdots\!21}a^{8}+\frac{73\!\cdots\!98}{60\!\cdots\!21}a^{7}+\frac{12\!\cdots\!78}{60\!\cdots\!21}a^{6}-\frac{65\!\cdots\!53}{60\!\cdots\!21}a^{5}-\frac{57\!\cdots\!45}{60\!\cdots\!21}a^{4}+\frac{37\!\cdots\!59}{60\!\cdots\!21}a^{3}-\frac{89\!\cdots\!82}{60\!\cdots\!21}a^{2}+\frac{26\!\cdots\!94}{60\!\cdots\!21}a+\frac{873157970759041}{73\!\cdots\!87}$, $\frac{65\!\cdots\!38}{60\!\cdots\!21}a^{17}-\frac{30\!\cdots\!02}{60\!\cdots\!21}a^{16}-\frac{75\!\cdots\!79}{60\!\cdots\!21}a^{15}+\frac{15\!\cdots\!70}{60\!\cdots\!21}a^{14}+\frac{18\!\cdots\!22}{60\!\cdots\!21}a^{13}-\frac{11\!\cdots\!88}{60\!\cdots\!21}a^{12}+\frac{10\!\cdots\!06}{60\!\cdots\!21}a^{11}+\frac{22\!\cdots\!45}{60\!\cdots\!21}a^{10}-\frac{53\!\cdots\!38}{60\!\cdots\!21}a^{9}+\frac{28\!\cdots\!07}{60\!\cdots\!21}a^{8}+\frac{79\!\cdots\!82}{60\!\cdots\!21}a^{7}-\frac{43\!\cdots\!81}{60\!\cdots\!21}a^{6}-\frac{40\!\cdots\!79}{60\!\cdots\!21}a^{5}+\frac{28\!\cdots\!10}{60\!\cdots\!21}a^{4}+\frac{71\!\cdots\!68}{60\!\cdots\!21}a^{3}-\frac{51\!\cdots\!54}{60\!\cdots\!21}a^{2}+\frac{57\!\cdots\!98}{60\!\cdots\!21}a+\frac{52\!\cdots\!45}{73\!\cdots\!87}$, $\frac{59\!\cdots\!23}{60\!\cdots\!21}a^{17}-\frac{27\!\cdots\!13}{60\!\cdots\!21}a^{16}-\frac{69\!\cdots\!12}{60\!\cdots\!21}a^{15}+\frac{13\!\cdots\!48}{60\!\cdots\!21}a^{14}+\frac{18\!\cdots\!83}{60\!\cdots\!21}a^{13}-\frac{10\!\cdots\!22}{60\!\cdots\!21}a^{12}+\frac{91\!\cdots\!96}{60\!\cdots\!21}a^{11}+\frac{23\!\cdots\!93}{60\!\cdots\!21}a^{10}-\frac{50\!\cdots\!41}{60\!\cdots\!21}a^{9}-\frac{29\!\cdots\!40}{60\!\cdots\!21}a^{8}+\frac{84\!\cdots\!35}{60\!\cdots\!21}a^{7}-\frac{39\!\cdots\!15}{60\!\cdots\!21}a^{6}-\frac{55\!\cdots\!75}{60\!\cdots\!21}a^{5}+\frac{38\!\cdots\!38}{60\!\cdots\!21}a^{4}+\frac{79\!\cdots\!27}{60\!\cdots\!21}a^{3}-\frac{14\!\cdots\!77}{60\!\cdots\!21}a^{2}+\frac{15\!\cdots\!35}{60\!\cdots\!21}a+\frac{896012181920771}{73\!\cdots\!87}$, $\frac{59\!\cdots\!23}{60\!\cdots\!21}a^{17}-\frac{27\!\cdots\!13}{60\!\cdots\!21}a^{16}-\frac{69\!\cdots\!12}{60\!\cdots\!21}a^{15}+\frac{13\!\cdots\!48}{60\!\cdots\!21}a^{14}+\frac{18\!\cdots\!83}{60\!\cdots\!21}a^{13}-\frac{10\!\cdots\!22}{60\!\cdots\!21}a^{12}+\frac{91\!\cdots\!96}{60\!\cdots\!21}a^{11}+\frac{23\!\cdots\!93}{60\!\cdots\!21}a^{10}-\frac{50\!\cdots\!41}{60\!\cdots\!21}a^{9}-\frac{29\!\cdots\!40}{60\!\cdots\!21}a^{8}+\frac{84\!\cdots\!35}{60\!\cdots\!21}a^{7}-\frac{39\!\cdots\!15}{60\!\cdots\!21}a^{6}-\frac{55\!\cdots\!75}{60\!\cdots\!21}a^{5}+\frac{38\!\cdots\!38}{60\!\cdots\!21}a^{4}+\frac{79\!\cdots\!27}{60\!\cdots\!21}a^{3}-\frac{14\!\cdots\!77}{60\!\cdots\!21}a^{2}+\frac{15\!\cdots\!35}{60\!\cdots\!21}a+\frac{82\!\cdots\!58}{73\!\cdots\!87}$, $\frac{43818350318250}{30\!\cdots\!93}a^{17}-\frac{20\!\cdots\!59}{60\!\cdots\!21}a^{16}-\frac{10\!\cdots\!05}{60\!\cdots\!21}a^{15}+\frac{17\!\cdots\!01}{60\!\cdots\!21}a^{14}+\frac{29\!\cdots\!17}{60\!\cdots\!21}a^{13}-\frac{13\!\cdots\!97}{60\!\cdots\!21}a^{12}+\frac{92\!\cdots\!48}{60\!\cdots\!21}a^{11}+\frac{33\!\cdots\!10}{60\!\cdots\!21}a^{10}-\frac{57\!\cdots\!66}{60\!\cdots\!21}a^{9}-\frac{25\!\cdots\!99}{60\!\cdots\!21}a^{8}+\frac{10\!\cdots\!60}{60\!\cdots\!21}a^{7}-\frac{74\!\cdots\!36}{60\!\cdots\!21}a^{6}-\frac{97\!\cdots\!25}{60\!\cdots\!21}a^{5}+\frac{26\!\cdots\!13}{60\!\cdots\!21}a^{4}+\frac{29\!\cdots\!76}{60\!\cdots\!21}a^{3}-\frac{23\!\cdots\!21}{60\!\cdots\!21}a^{2}+\frac{13\!\cdots\!92}{60\!\cdots\!21}a+\frac{13\!\cdots\!25}{73\!\cdots\!87}$ 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:  \( 6329.31955748 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 6329.31955748 \cdot 1}{2\cdot\sqrt{62181896314367173832704}}\cr\approx \mathstrut & 0.193692971350 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 11*x^16 + 29*x^15 + 13*x^14 - 181*x^13 + 260*x^12 + 223*x^11 - 970*x^10 + 567*x^9 + 1007*x^8 - 1329*x^7 + 46*x^6 + 692*x^5 - 526*x^4 + 5*x^3 + 348*x^2 - 173*x + 83)
 
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 - x^17 - 11*x^16 + 29*x^15 + 13*x^14 - 181*x^13 + 260*x^12 + 223*x^11 - 970*x^10 + 567*x^9 + 1007*x^8 - 1329*x^7 + 46*x^6 + 692*x^5 - 526*x^4 + 5*x^3 + 348*x^2 - 173*x + 83, 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 - x^17 - 11*x^16 + 29*x^15 + 13*x^14 - 181*x^13 + 260*x^12 + 223*x^11 - 970*x^10 + 567*x^9 + 1007*x^8 - 1329*x^7 + 46*x^6 + 692*x^5 - 526*x^4 + 5*x^3 + 348*x^2 - 173*x + 83);
 
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 - x^17 - 11*x^16 + 29*x^15 + 13*x^14 - 181*x^13 + 260*x^12 + 223*x^11 - 970*x^10 + 567*x^9 + 1007*x^8 - 1329*x^7 + 46*x^6 + 692*x^5 - 526*x^4 + 5*x^3 + 348*x^2 - 173*x + 83);
 
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

\(\Q(\sqrt{-19}) \), 3.1.76.1 x3, 3.3.361.1, 6.0.109744.2, 6.0.2476099.1, 6.0.39617584.1 x2, 9.3.57207791296.1 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: 6.0.39617584.1
Degree 9 sibling: 9.3.57207791296.1
Minimal sibling: 6.0.39617584.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{3}$ ${\href{/padicField/5.3.0.1}{3} }^{6}$ ${\href{/padicField/7.3.0.1}{3} }^{6}$ ${\href{/padicField/11.3.0.1}{3} }^{6}$ ${\href{/padicField/13.6.0.1}{6} }^{3}$ ${\href{/padicField/17.3.0.1}{3} }^{6}$ R ${\href{/padicField/23.3.0.1}{3} }^{6}$ ${\href{/padicField/29.6.0.1}{6} }^{3}$ ${\href{/padicField/31.2.0.1}{2} }^{9}$ ${\href{/padicField/37.2.0.1}{2} }^{9}$ ${\href{/padicField/41.6.0.1}{6} }^{3}$ ${\href{/padicField/43.3.0.1}{3} }^{6}$ ${\href{/padicField/47.3.0.1}{3} }^{6}$ ${\href{/padicField/53.6.0.1}{6} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }^{3}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 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}$
\(19\) Copy content Toggle raw display 19.6.5.5$x^{6} + 19$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.5$x^{6} + 19$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.5$x^{6} + 19$$6$$1$$5$$C_6$$[\ ]_{6}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.19.2t1.a.a$1$ $ 19 $ \(\Q(\sqrt{-19}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.19.3t1.a.a$1$ $ 19 $ 3.3.361.1 $C_3$ (as 3T1) $0$ $1$
* 1.19.6t1.a.a$1$ $ 19 $ 6.0.2476099.1 $C_6$ (as 6T1) $0$ $-1$
* 1.19.6t1.a.b$1$ $ 19 $ 6.0.2476099.1 $C_6$ (as 6T1) $0$ $-1$
* 1.19.3t1.a.b$1$ $ 19 $ 3.3.361.1 $C_3$ (as 3T1) $0$ $1$
*2 2.76.3t2.a.a$2$ $ 2^{2} \cdot 19 $ 3.1.76.1 $S_3$ (as 3T2) $1$ $0$
*2 2.1444.6t5.a.a$2$ $ 2^{2} \cdot 19^{2}$ 18.0.62181896314367173832704.1 $S_3 \times C_3$ (as 18T3) $0$ $0$
*2 2.1444.6t5.a.b$2$ $ 2^{2} \cdot 19^{2}$ 18.0.62181896314367173832704.1 $S_3 \times C_3$ (as 18T3) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.