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Properties

Label 18.0.500...864.1
Degree $18$
Signature $[0, 9]$
Discriminant $-5.008\times 10^{29}$
Root discriminant \(44.67\)
Ramified primes $2,7,37,151$
Class number $171$ (GRH)
Class group [171] (GRH)
Galois group $C_6\times S_4$ (as 18T61)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 13*x^16 - 14*x^15 + 102*x^14 - 85*x^13 + 597*x^12 - 117*x^11 + 2104*x^10 - 147*x^9 + 5814*x^8 - 1244*x^7 + 10992*x^6 - 4320*x^5 + 5856*x^4 - 2240*x^3 + 2176*x^2 - 512*x + 512)
 
gp: K = bnfinit(y^18 - y^17 + 13*y^16 - 14*y^15 + 102*y^14 - 85*y^13 + 597*y^12 - 117*y^11 + 2104*y^10 - 147*y^9 + 5814*y^8 - 1244*y^7 + 10992*y^6 - 4320*y^5 + 5856*y^4 - 2240*y^3 + 2176*y^2 - 512*y + 512, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - x^17 + 13*x^16 - 14*x^15 + 102*x^14 - 85*x^13 + 597*x^12 - 117*x^11 + 2104*x^10 - 147*x^9 + 5814*x^8 - 1244*x^7 + 10992*x^6 - 4320*x^5 + 5856*x^4 - 2240*x^3 + 2176*x^2 - 512*x + 512);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - x^17 + 13*x^16 - 14*x^15 + 102*x^14 - 85*x^13 + 597*x^12 - 117*x^11 + 2104*x^10 - 147*x^9 + 5814*x^8 - 1244*x^7 + 10992*x^6 - 4320*x^5 + 5856*x^4 - 2240*x^3 + 2176*x^2 - 512*x + 512)
 

\( x^{18} - x^{17} + 13 x^{16} - 14 x^{15} + 102 x^{14} - 85 x^{13} + 597 x^{12} - 117 x^{11} + 2104 x^{10} + \cdots + 512 \) 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:   \(-500815330257541456373423140864\) \(\medspace = -\,2^{12}\cdot 7^{12}\cdot 37^{6}\cdot 151^{3}\) 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:  \(44.67\)
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:  $2^{2/3}7^{2/3}37^{1/2}151^{1/2}\approx 434.1848772346385$
Ramified primes:   \(2\), \(7\), \(37\), \(151\) 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:  \(\Q(\sqrt{-151}) \)
$\card{ \Aut(K/\Q) }$:  $6$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{256}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{10}+\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{11}+\frac{1}{8}a^{10}-\frac{1}{4}a^{9}+\frac{1}{4}a^{8}+\frac{3}{8}a^{7}-\frac{3}{8}a^{6}-\frac{1}{8}a^{5}-\frac{1}{2}a^{4}+\frac{1}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{16}a^{13}-\frac{1}{16}a^{12}+\frac{1}{16}a^{11}-\frac{1}{8}a^{10}-\frac{3}{8}a^{9}+\frac{3}{16}a^{8}-\frac{3}{16}a^{7}+\frac{7}{16}a^{6}-\frac{1}{4}a^{5}-\frac{7}{16}a^{4}+\frac{3}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{14}-\frac{1}{32}a^{13}+\frac{1}{32}a^{12}-\frac{1}{16}a^{11}-\frac{3}{16}a^{10}+\frac{3}{32}a^{9}-\frac{3}{32}a^{8}+\frac{7}{32}a^{7}-\frac{1}{8}a^{6}-\frac{7}{32}a^{5}+\frac{3}{16}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{64}a^{15}-\frac{1}{64}a^{14}+\frac{1}{64}a^{13}-\frac{1}{32}a^{12}-\frac{3}{32}a^{11}+\frac{3}{64}a^{10}+\frac{29}{64}a^{9}+\frac{7}{64}a^{8}+\frac{7}{16}a^{7}-\frac{7}{64}a^{6}-\frac{13}{32}a^{5}-\frac{1}{8}a^{4}+\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{287550362752}a^{16}-\frac{1216518055}{287550362752}a^{15}+\frac{2146558727}{287550362752}a^{14}-\frac{951852075}{35943795344}a^{13}-\frac{8504579589}{143775181376}a^{12}+\frac{8587263223}{287550362752}a^{11}+\frac{34678346283}{287550362752}a^{10}-\frac{142371614759}{287550362752}a^{9}-\frac{20737833727}{143775181376}a^{8}-\frac{96064469151}{287550362752}a^{7}-\frac{4137684421}{8985948836}a^{6}-\frac{12506434987}{71887590688}a^{5}-\frac{10729229607}{35943795344}a^{4}+\frac{8235279129}{17971897672}a^{3}-\frac{137071648}{2246487209}a^{2}-\frac{766727092}{2246487209}a+\frac{710883531}{2246487209}$, $\frac{1}{20\!\cdots\!68}a^{17}+\frac{7637}{20\!\cdots\!68}a^{16}+\frac{174994411383535}{20\!\cdots\!68}a^{15}+\frac{80411312543755}{12\!\cdots\!48}a^{14}+\frac{18\!\cdots\!21}{10\!\cdots\!84}a^{13}-\frac{98\!\cdots\!41}{20\!\cdots\!68}a^{12}+\frac{23\!\cdots\!47}{20\!\cdots\!68}a^{11}-\frac{97\!\cdots\!51}{20\!\cdots\!68}a^{10}+\frac{25\!\cdots\!17}{10\!\cdots\!84}a^{9}+\frac{49\!\cdots\!85}{20\!\cdots\!68}a^{8}+\frac{15\!\cdots\!51}{51\!\cdots\!92}a^{7}-\frac{211745483471897}{12\!\cdots\!48}a^{6}+\frac{41\!\cdots\!93}{12\!\cdots\!48}a^{5}+\frac{73622921312185}{64\!\cdots\!24}a^{4}+\frac{245040351308309}{494388933059048}a^{3}+\frac{4622042421921}{16\!\cdots\!06}a^{2}-\frac{181438645946718}{803382016220953}a-\frac{276803381138847}{803382016220953}$ 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

$C_{171}$, which has order $171$ (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:   $\frac{25840562979}{10\!\cdots\!84}a^{17}-\frac{5461226003059}{10\!\cdots\!84}a^{16}+\frac{827692316187}{10\!\cdots\!84}a^{15}-\frac{2708667610187}{64\!\cdots\!24}a^{14}+\frac{5214667370691}{51\!\cdots\!92}a^{13}-\frac{254744296635461}{10\!\cdots\!84}a^{12}-\frac{88379998914689}{10\!\cdots\!84}a^{11}-\frac{12\!\cdots\!51}{10\!\cdots\!84}a^{10}-\frac{10\!\cdots\!63}{51\!\cdots\!92}a^{9}-\frac{25\!\cdots\!03}{10\!\cdots\!84}a^{8}-\frac{39383632937745}{12\!\cdots\!48}a^{7}-\frac{18\!\cdots\!03}{25\!\cdots\!96}a^{6}+\frac{21389693134535}{803382016220953}a^{5}-\frac{515326183723427}{12\!\cdots\!48}a^{4}+\frac{3779721535173}{247194466529524}a^{3}-\frac{24940145388497}{16\!\cdots\!06}a^{2}+\frac{3149510570169}{803382016220953}a-\frac{18\!\cdots\!27}{803382016220953}$, $\frac{55331379581}{20\!\cdots\!68}a^{17}-\frac{5065806506779}{20\!\cdots\!68}a^{16}+\frac{1257154646687}{20\!\cdots\!68}a^{15}-\frac{10063148891083}{51\!\cdots\!92}a^{14}+\frac{6335953721761}{10\!\cdots\!84}a^{13}-\frac{232663097392717}{20\!\cdots\!68}a^{12}-\frac{66622374795209}{20\!\cdots\!68}a^{11}-\frac{11\!\cdots\!11}{20\!\cdots\!68}a^{10}-\frac{897756466830033}{10\!\cdots\!84}a^{9}-\frac{22\!\cdots\!03}{20\!\cdots\!68}a^{8}-\frac{2984458763885}{32\!\cdots\!12}a^{7}-\frac{823647024842979}{25\!\cdots\!96}a^{6}+\frac{78193773748215}{64\!\cdots\!24}a^{5}-\frac{232201154692651}{12\!\cdots\!48}a^{4}+\frac{1714839071659}{247194466529524}a^{3}-\frac{11212138332661}{16\!\cdots\!06}a^{2}+\frac{1426562060637}{803382016220953}a-\frac{14\!\cdots\!09}{803382016220953}$, $\frac{59979479}{287550362752}a^{17}+\frac{608381043}{287550362752}a^{16}-\frac{1310211835}{287550362752}a^{15}+\frac{3883295411}{143775181376}a^{14}-\frac{6695345937}{143775181376}a^{13}+\frac{58356965535}{287550362752}a^{12}-\frac{78096865447}{287550362752}a^{11}+\frac{314035592443}{287550362752}a^{10}-\frac{41139443103}{71887590688}a^{9}+\frac{1050482169813}{287550362752}a^{8}-\frac{187275555653}{143775181376}a^{7}+\frac{179417058423}{17971897672}a^{6}-\frac{307511114929}{71887590688}a^{5}+\frac{163879246869}{8985948836}a^{4}-\frac{17850509627}{2246487209}a^{3}+\frac{10323135063}{2246487209}a^{2}-\frac{7493787577}{4492974418}a+\frac{5624286958}{2246487209}$, $\frac{348150628946461}{20\!\cdots\!68}a^{17}+\frac{352001072692589}{20\!\cdots\!68}a^{16}-\frac{45\!\cdots\!73}{20\!\cdots\!68}a^{15}+\frac{24\!\cdots\!33}{10\!\cdots\!84}a^{14}-\frac{18\!\cdots\!29}{10\!\cdots\!84}a^{13}+\frac{30\!\cdots\!81}{20\!\cdots\!68}a^{12}-\frac{21\!\cdots\!57}{20\!\cdots\!68}a^{11}+\frac{45\!\cdots\!53}{20\!\cdots\!68}a^{10}-\frac{18\!\cdots\!31}{51\!\cdots\!92}a^{9}+\frac{55\!\cdots\!23}{20\!\cdots\!68}a^{8}-\frac{10\!\cdots\!55}{10\!\cdots\!84}a^{7}+\frac{27\!\cdots\!89}{12\!\cdots\!48}a^{6}-\frac{24\!\cdots\!25}{12\!\cdots\!48}a^{5}+\frac{96\!\cdots\!25}{12\!\cdots\!48}a^{4}-\frac{12\!\cdots\!43}{123597233264762}a^{3}+\frac{31\!\cdots\!49}{803382016220953}a^{2}-\frac{12\!\cdots\!87}{803382016220953}a-\frac{734166606897153}{803382016220953}$, $\frac{87038569800021}{51\!\cdots\!92}a^{17}+\frac{183994665602743}{10\!\cdots\!84}a^{16}-\frac{22\!\cdots\!17}{10\!\cdots\!84}a^{15}+\frac{25\!\cdots\!91}{10\!\cdots\!84}a^{14}-\frac{563289018395471}{32\!\cdots\!12}a^{13}+\frac{38\!\cdots\!15}{25\!\cdots\!96}a^{12}-\frac{10\!\cdots\!35}{10\!\cdots\!84}a^{11}+\frac{24\!\cdots\!33}{10\!\cdots\!84}a^{10}-\frac{37\!\cdots\!03}{10\!\cdots\!84}a^{9}+\frac{78\!\cdots\!29}{25\!\cdots\!96}a^{8}-\frac{10\!\cdots\!75}{10\!\cdots\!84}a^{7}+\frac{14\!\cdots\!15}{64\!\cdots\!24}a^{6}-\frac{24\!\cdots\!15}{12\!\cdots\!48}a^{5}+\frac{96\!\cdots\!03}{12\!\cdots\!48}a^{4}-\frac{12\!\cdots\!59}{123597233264762}a^{3}+\frac{31\!\cdots\!28}{803382016220953}a^{2}-\frac{12\!\cdots\!93}{803382016220953}a+\frac{17\!\cdots\!30}{803382016220953}$, $\frac{10883051295375}{64\!\cdots\!24}a^{17}-\frac{173072213596625}{10\!\cdots\!84}a^{16}+\frac{22\!\cdots\!43}{10\!\cdots\!84}a^{15}-\frac{24\!\cdots\!07}{10\!\cdots\!84}a^{14}+\frac{45\!\cdots\!77}{25\!\cdots\!96}a^{13}-\frac{75\!\cdots\!69}{51\!\cdots\!92}a^{12}+\frac{10\!\cdots\!13}{10\!\cdots\!84}a^{11}-\frac{21\!\cdots\!31}{10\!\cdots\!84}a^{10}+\frac{37\!\cdots\!55}{10\!\cdots\!84}a^{9}-\frac{13\!\cdots\!55}{51\!\cdots\!92}a^{8}+\frac{10\!\cdots\!95}{10\!\cdots\!84}a^{7}-\frac{26\!\cdots\!27}{12\!\cdots\!48}a^{6}+\frac{24\!\cdots\!95}{12\!\cdots\!48}a^{5}-\frac{95\!\cdots\!49}{12\!\cdots\!48}a^{4}+\frac{627357510129993}{61798616632381}a^{3}-\frac{31\!\cdots\!31}{803382016220953}a^{2}+\frac{12\!\cdots\!55}{803382016220953}a+\frac{10\!\cdots\!71}{803382016220953}$, $\frac{348261291705623}{20\!\cdots\!68}a^{17}-\frac{362132685706147}{20\!\cdots\!68}a^{16}+\frac{45\!\cdots\!47}{20\!\cdots\!68}a^{15}-\frac{25\!\cdots\!65}{10\!\cdots\!84}a^{14}+\frac{18\!\cdots\!51}{10\!\cdots\!84}a^{13}-\frac{30\!\cdots\!15}{20\!\cdots\!68}a^{12}+\frac{21\!\cdots\!39}{20\!\cdots\!68}a^{11}-\frac{47\!\cdots\!75}{20\!\cdots\!68}a^{10}+\frac{93\!\cdots\!99}{25\!\cdots\!96}a^{9}-\frac{60\!\cdots\!29}{20\!\cdots\!68}a^{8}+\frac{10\!\cdots\!15}{10\!\cdots\!84}a^{7}-\frac{70\!\cdots\!17}{32\!\cdots\!12}a^{6}+\frac{24\!\cdots\!85}{12\!\cdots\!48}a^{5}-\frac{96\!\cdots\!27}{12\!\cdots\!48}a^{4}+\frac{628731150281701}{61798616632381}a^{3}-\frac{31\!\cdots\!10}{803382016220953}a^{2}+\frac{12\!\cdots\!61}{803382016220953}a-\frac{13\!\cdots\!12}{803382016220953}$, $\frac{348098947820503}{20\!\cdots\!68}a^{17}-\frac{362923524698707}{20\!\cdots\!68}a^{16}+\frac{45\!\cdots\!47}{20\!\cdots\!68}a^{15}-\frac{25\!\cdots\!25}{10\!\cdots\!84}a^{14}+\frac{18\!\cdots\!11}{10\!\cdots\!84}a^{13}-\frac{30\!\cdots\!03}{20\!\cdots\!68}a^{12}+\frac{21\!\cdots\!79}{20\!\cdots\!68}a^{11}-\frac{47\!\cdots\!55}{20\!\cdots\!68}a^{10}+\frac{29\!\cdots\!37}{803382016220953}a^{9}-\frac{60\!\cdots\!29}{20\!\cdots\!68}a^{8}+\frac{10\!\cdots\!95}{10\!\cdots\!84}a^{7}-\frac{56\!\cdots\!81}{25\!\cdots\!96}a^{6}+\frac{24\!\cdots\!85}{12\!\cdots\!48}a^{5}-\frac{60\!\cdots\!47}{803382016220953}a^{4}+\frac{25\!\cdots\!59}{247194466529524}a^{3}-\frac{62\!\cdots\!95}{16\!\cdots\!06}a^{2}+\frac{12\!\cdots\!56}{803382016220953}a-\frac{270609004882421}{803382016220953}$ 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:  \( 146431.64361735608 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 146431.64361735608 \cdot 171}{2\cdot\sqrt{500815330257541456373423140864}}\cr\approx \mathstrut & 0.270010752922176 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 13*x^16 - 14*x^15 + 102*x^14 - 85*x^13 + 597*x^12 - 117*x^11 + 2104*x^10 - 147*x^9 + 5814*x^8 - 1244*x^7 + 10992*x^6 - 4320*x^5 + 5856*x^4 - 2240*x^3 + 2176*x^2 - 512*x + 512)
 
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 + 13*x^16 - 14*x^15 + 102*x^14 - 85*x^13 + 597*x^12 - 117*x^11 + 2104*x^10 - 147*x^9 + 5814*x^8 - 1244*x^7 + 10992*x^6 - 4320*x^5 + 5856*x^4 - 2240*x^3 + 2176*x^2 - 512*x + 512, 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 + 13*x^16 - 14*x^15 + 102*x^14 - 85*x^13 + 597*x^12 - 117*x^11 + 2104*x^10 - 147*x^9 + 5814*x^8 - 1244*x^7 + 10992*x^6 - 4320*x^5 + 5856*x^4 - 2240*x^3 + 2176*x^2 - 512*x + 512);
 
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 + 13*x^16 - 14*x^15 + 102*x^14 - 85*x^13 + 597*x^12 - 117*x^11 + 2104*x^10 - 147*x^9 + 5814*x^8 - 1244*x^7 + 10992*x^6 - 4320*x^5 + 5856*x^4 - 2240*x^3 + 2176*x^2 - 512*x + 512);
 
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

$C_6\times S_4$ (as 18T61):

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 144
The 30 conjugacy class representatives for $C_6\times S_4$
Character table for $C_6\times S_4$

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.148.1, 6.0.3307504.1, 9.9.381393587008.1

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 18 sibling: data not computed
Degree 24 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{3}$ ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ R ${\href{/padicField/11.3.0.1}{3} }^{6}$ ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{6}$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.6.0.1}{6} }$ ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.6.0.1}{6} }$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ R ${\href{/padicField/41.6.0.1}{6} }^{3}$ ${\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ ${\href{/padicField/47.3.0.1}{3} }^{6}$ ${\href{/padicField/53.6.0.1}{6} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$

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
\(2\) Copy content Toggle raw display 2.9.6.1$x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
\(7\) Copy content Toggle raw display 7.18.12.1$x^{18} + 3 x^{16} + 57 x^{15} + 15 x^{14} + 90 x^{13} + 424 x^{12} - 921 x^{11} - 3090 x^{10} - 6496 x^{9} - 10560 x^{8} + 6912 x^{7} + 28033 x^{6} + 33237 x^{5} + 188463 x^{4} - 139476 x^{3} + 351552 x^{2} - 514905 x + 582014$$3$$6$$12$$C_6 \times C_3$$[\ ]_{3}^{6}$
\(37\) Copy content Toggle raw display 37.3.0.1$x^{3} + 6 x + 35$$1$$3$$0$$C_3$$[\ ]^{3}$
37.3.0.1$x^{3} + 6 x + 35$$1$$3$$0$$C_3$$[\ ]^{3}$
37.12.6.1$x^{12} + 5550 x^{11} + 12834597 x^{10} + 15830089320 x^{9} + 10983311908793 x^{8} + 4064880121459160 x^{7} + 627211360763929855 x^{6} + 150784964321118570 x^{5} + 157305214180175641 x^{4} + 21952692460017249320 x^{3} + 5259154456776668411 x^{2} + 19105722471444557170 x + 2009397505586138476$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
\(151\) Copy content Toggle raw display 151.6.3.2$x^{6} + 455 x^{4} + 290 x^{3} + 68404 x^{2} - 131080 x + 3418525$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
151.6.0.1$x^{6} + 125 x^{3} + 18 x^{2} + 15 x + 6$$1$$6$$0$$C_6$$[\ ]^{6}$
151.6.0.1$x^{6} + 125 x^{3} + 18 x^{2} + 15 x + 6$$1$$6$$0$$C_6$$[\ ]^{6}$