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Properties

Label 18.0.641...375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-6.412\times 10^{28}$
Root discriminant \(39.85\)
Ramified primes $3,5,11$
Class number $18$ (GRH)
Class group [3, 6] (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 60*x^16 - 276*x^15 + 1059*x^14 - 3297*x^13 + 8326*x^12 - 17001*x^11 + 28578*x^10 - 40029*x^9 + 55638*x^8 - 80625*x^7 + 148498*x^6 - 237165*x^5 + 364521*x^4 - 394488*x^3 + 378474*x^2 - 212265*x + 118815)
 
gp: K = bnfinit(y^18 - 9*y^17 + 60*y^16 - 276*y^15 + 1059*y^14 - 3297*y^13 + 8326*y^12 - 17001*y^11 + 28578*y^10 - 40029*y^9 + 55638*y^8 - 80625*y^7 + 148498*y^6 - 237165*y^5 + 364521*y^4 - 394488*y^3 + 378474*y^2 - 212265*y + 118815, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 60*x^16 - 276*x^15 + 1059*x^14 - 3297*x^13 + 8326*x^12 - 17001*x^11 + 28578*x^10 - 40029*x^9 + 55638*x^8 - 80625*x^7 + 148498*x^6 - 237165*x^5 + 364521*x^4 - 394488*x^3 + 378474*x^2 - 212265*x + 118815);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 + 60*x^16 - 276*x^15 + 1059*x^14 - 3297*x^13 + 8326*x^12 - 17001*x^11 + 28578*x^10 - 40029*x^9 + 55638*x^8 - 80625*x^7 + 148498*x^6 - 237165*x^5 + 364521*x^4 - 394488*x^3 + 378474*x^2 - 212265*x + 118815)
 

\( x^{18} - 9 x^{17} + 60 x^{16} - 276 x^{15} + 1059 x^{14} - 3297 x^{13} + 8326 x^{12} - 17001 x^{11} + \cdots + 118815 \) 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:   \(-64119276021132037084693359375\) \(\medspace = -\,3^{21}\cdot 5^{9}\cdot 11^{12}\) 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:  \(39.85\)
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:  $3^{7/6}5^{1/2}11^{2/3}\approx 39.84632346086567$
Ramified primes:   \(3\), \(5\), \(11\) 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:  \(\Q(\sqrt{-15}) \)
$\card{ \Gal(K/\Q) }$:  $18$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{2}$, $\frac{1}{10}a^{10}-\frac{1}{5}a^{6}-\frac{1}{2}a^{3}-\frac{2}{5}a^{2}$, $\frac{1}{10}a^{11}-\frac{1}{5}a^{7}-\frac{1}{2}a^{4}-\frac{2}{5}a^{3}$, $\frac{1}{20}a^{12}-\frac{1}{20}a^{10}+\frac{3}{20}a^{8}-\frac{3}{20}a^{6}+\frac{1}{20}a^{4}-\frac{1}{2}a^{3}+\frac{9}{20}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{100}a^{13}+\frac{1}{100}a^{12}+\frac{1}{100}a^{11}+\frac{1}{100}a^{10}+\frac{13}{100}a^{9}-\frac{7}{100}a^{8}+\frac{13}{100}a^{7}+\frac{23}{100}a^{6}-\frac{49}{100}a^{5}+\frac{31}{100}a^{4}-\frac{29}{100}a^{3}+\frac{11}{100}a^{2}-\frac{1}{20}a+\frac{7}{20}$, $\frac{1}{80100}a^{14}-\frac{7}{80100}a^{13}-\frac{377}{80100}a^{12}+\frac{2353}{80100}a^{11}+\frac{541}{16020}a^{10}+\frac{4789}{80100}a^{9}+\frac{12259}{80100}a^{8}-\frac{11551}{80100}a^{7}-\frac{12083}{80100}a^{6}+\frac{841}{26700}a^{5}+\frac{8251}{26700}a^{4}-\frac{1883}{8900}a^{3}-\frac{5981}{26700}a^{2}-\frac{407}{1068}a-\frac{7}{15}$, $\frac{1}{240300}a^{15}+\frac{5}{3204}a^{13}+\frac{103}{48060}a^{12}+\frac{1319}{80100}a^{11}-\frac{167}{5340}a^{10}-\frac{4781}{48060}a^{9}-\frac{763}{16020}a^{8}-\frac{3479}{80100}a^{7}-\frac{89}{540}a^{6}+\frac{2881}{16020}a^{5}+\frac{4067}{16020}a^{4}+\frac{8143}{80100}a^{3}+\frac{6733}{16020}a^{2}+\frac{1171}{2670}a-\frac{11}{36}$, $\frac{1}{3206927254500}a^{16}-\frac{2}{801731813625}a^{15}-\frac{851857}{534487875750}a^{14}+\frac{17889067}{1603463627250}a^{13}+\frac{38489273503}{1603463627250}a^{12}-\frac{11803639391}{267243937875}a^{11}+\frac{727369319}{320692725450}a^{10}-\frac{303911492459}{1603463627250}a^{9}-\frac{121414886}{10689757515}a^{8}-\frac{83112056138}{801731813625}a^{7}+\frac{213906938273}{1603463627250}a^{6}-\frac{70747803317}{178162625250}a^{5}+\frac{10652325779}{267243937875}a^{4}+\frac{23232585719}{178162625250}a^{3}+\frac{342494942207}{1068975751500}a^{2}-\frac{21620998037}{53448787575}a-\frac{327468887}{1201096350}$, $\frac{1}{77\!\cdots\!00}a^{17}+\frac{4}{25837143913755}a^{16}-\frac{349543793}{38\!\cdots\!50}a^{15}-\frac{233432638}{19\!\cdots\!25}a^{14}-\frac{682366234951}{287079376819500}a^{13}+\frac{9603397504163}{38\!\cdots\!50}a^{12}-\frac{73374961945861}{77\!\cdots\!00}a^{11}-\frac{17410029576763}{12\!\cdots\!50}a^{10}-\frac{22427758542059}{77\!\cdots\!00}a^{9}-\frac{129296524700723}{19\!\cdots\!25}a^{8}+\frac{543272280583}{3827725024260}a^{7}-\frac{198834463862632}{19\!\cdots\!25}a^{6}-\frac{226964898254417}{516742878275100}a^{5}+\frac{71037009419083}{645928597843875}a^{4}-\frac{120189226253089}{645928597843875}a^{3}+\frac{138007843270201}{430619065229250}a^{2}-\frac{56193004897}{11483175072780}a+\frac{529966416232}{1451524938975}$ 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:  $2$, $5$

Class group and class number

$C_{3}\times C_{6}$, which has order $18$ (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:   $\frac{10601979859}{430619065229250}a^{17}-\frac{329048802374}{19\!\cdots\!25}a^{16}+\frac{7810748068831}{77\!\cdots\!00}a^{15}-\frac{9677982694649}{25\!\cdots\!00}a^{14}+\frac{23748453235808}{19\!\cdots\!25}a^{13}-\frac{231199455452869}{77\!\cdots\!00}a^{12}+\frac{6381043559066}{129185719568775}a^{11}-\frac{352958349242641}{77\!\cdots\!00}a^{10}-\frac{52060899467}{10917103062150}a^{9}+\frac{228936871318927}{25\!\cdots\!00}a^{8}+\frac{188470623823181}{38\!\cdots\!50}a^{7}-\frac{14\!\cdots\!57}{77\!\cdots\!00}a^{6}+\frac{152761099432826}{215309532614625}a^{5}-\frac{147425035266187}{25\!\cdots\!00}a^{4}-\frac{252062134785811}{430619065229250}a^{3}+\frac{238525462199363}{103348575655020}a^{2}-\frac{11\!\cdots\!47}{516742878275100}a+\frac{30754816}{69500835}$, $\frac{12585944437}{430619065229250}a^{17}-\frac{213961055429}{861238130458500}a^{16}+\frac{24657524337}{15948854267750}a^{15}-\frac{1141415249581}{172247626091700}a^{14}+\frac{20223384776921}{861238130458500}a^{13}-\frac{964583033102}{14353968840975}a^{12}+\frac{127438397230519}{861238130458500}a^{11}-\frac{1506459524581}{6065057256750}a^{10}+\frac{31665067877719}{95693125606500}a^{9}-\frac{79181599798417}{215309532614625}a^{8}+\frac{515038332934213}{861238130458500}a^{7}-\frac{165037348593311}{143539688409750}a^{6}+\frac{72147912652817}{31897708535500}a^{5}-\frac{223715099250334}{71769844204875}a^{4}+\frac{38864554838909}{11483175072780}a^{3}-\frac{701547250659749}{287079376819500}a^{2}+\frac{65529399803527}{57415875363900}a-\frac{160071171109}{215040731700}$, $\frac{6856610629}{95693125606500}a^{17}-\frac{149982861253}{172247626091700}a^{16}+\frac{5054667511939}{861238130458500}a^{15}-\frac{2106042573466}{71769844204875}a^{14}+\frac{96867133394483}{861238130458500}a^{13}-\frac{77856276728866}{215309532614625}a^{12}+\frac{264635252286893}{287079376819500}a^{11}-\frac{398527826134792}{215309532614625}a^{10}+\frac{25\!\cdots\!41}{861238130458500}a^{9}-\frac{276904850649091}{71769844204875}a^{8}+\frac{846279085708439}{172247626091700}a^{7}-\frac{40825559545451}{4838416463250}a^{6}+\frac{3638276398955}{255181668284}a^{5}-\frac{38\!\cdots\!29}{143539688409750}a^{4}+\frac{805835791276702}{23923281401625}a^{3}-\frac{124586965290881}{3225610975500}a^{2}+\frac{138114117712939}{5741587536390}a-\frac{5564361632021}{322561097550}$, $\frac{26287198621}{77\!\cdots\!00}a^{17}-\frac{352197380179}{77\!\cdots\!00}a^{16}+\frac{2484114928231}{77\!\cdots\!00}a^{15}-\frac{3130029272536}{19\!\cdots\!25}a^{14}+\frac{24164052767501}{38\!\cdots\!50}a^{13}-\frac{37668681089737}{19\!\cdots\!25}a^{12}+\frac{92799194918213}{19\!\cdots\!25}a^{11}-\frac{163571974865252}{19\!\cdots\!25}a^{10}+\frac{363524659003099}{38\!\cdots\!50}a^{9}-\frac{79988294018371}{38\!\cdots\!50}a^{8}-\frac{520974205217831}{38\!\cdots\!50}a^{7}+\frac{604440624549329}{38\!\cdots\!50}a^{6}+\frac{140935499856002}{645928597843875}a^{5}-\frac{603558153727933}{645928597843875}a^{4}+\frac{43\!\cdots\!93}{25\!\cdots\!00}a^{3}-\frac{32\!\cdots\!77}{25\!\cdots\!00}a^{2}+\frac{611104572864077}{516742878275100}a+\frac{1839387023317}{2903049877950}$, $\frac{26287198621}{77\!\cdots\!00}a^{17}-\frac{47342498189}{38\!\cdots\!50}a^{16}+\frac{424015857823}{77\!\cdots\!00}a^{15}-\frac{353216274121}{77\!\cdots\!00}a^{14}-\frac{788466445019}{77\!\cdots\!00}a^{13}+\frac{9801762120953}{77\!\cdots\!00}a^{12}-\frac{9376242233969}{15\!\cdots\!00}a^{11}+\frac{75820207188307}{77\!\cdots\!00}a^{10}-\frac{3245225096507}{15\!\cdots\!00}a^{9}-\frac{295893167199407}{77\!\cdots\!00}a^{8}+\frac{927384215316071}{77\!\cdots\!00}a^{7}-\frac{648095079736091}{77\!\cdots\!00}a^{6}+\frac{69387810612371}{25\!\cdots\!00}a^{5}+\frac{6417189203119}{36390343540500}a^{4}+\frac{84648857055211}{12\!\cdots\!50}a^{3}-\frac{294444132409747}{516742878275100}a^{2}+\frac{317592841767127}{258371439137550}a-\frac{453570375653}{290304987795}$, $\frac{62766714161}{38\!\cdots\!50}a^{17}-\frac{376393353047}{15\!\cdots\!00}a^{16}+\frac{10809595713523}{77\!\cdots\!00}a^{15}-\frac{60807519508219}{77\!\cdots\!00}a^{14}+\frac{115333733804303}{38\!\cdots\!50}a^{13}-\frac{215562409509562}{19\!\cdots\!25}a^{12}+\frac{598482963416597}{19\!\cdots\!25}a^{11}-\frac{14\!\cdots\!04}{19\!\cdots\!25}a^{10}+\frac{31\!\cdots\!13}{19\!\cdots\!25}a^{9}-\frac{11\!\cdots\!77}{38\!\cdots\!50}a^{8}+\frac{29\!\cdots\!67}{775114317412650}a^{7}-\frac{94\!\cdots\!29}{19\!\cdots\!25}a^{6}+\frac{11\!\cdots\!73}{258371439137550}a^{5}-\frac{19\!\cdots\!13}{12\!\cdots\!50}a^{4}+\frac{12\!\cdots\!47}{645928597843875}a^{3}-\frac{11\!\cdots\!93}{25\!\cdots\!00}a^{2}+\frac{20\!\cdots\!67}{103348575655020}a-\frac{180905627112559}{5806099755900}$, $\frac{609885011233}{77\!\cdots\!00}a^{17}-\frac{4934172288997}{77\!\cdots\!00}a^{16}+\frac{14881609034159}{38\!\cdots\!50}a^{15}-\frac{30612175417603}{19\!\cdots\!25}a^{14}+\frac{105153578243674}{19\!\cdots\!25}a^{13}-\frac{11\!\cdots\!39}{77\!\cdots\!00}a^{12}+\frac{12\!\cdots\!03}{38\!\cdots\!50}a^{11}-\frac{36\!\cdots\!59}{77\!\cdots\!00}a^{10}+\frac{23\!\cdots\!77}{38\!\cdots\!50}a^{9}-\frac{52\!\cdots\!31}{77\!\cdots\!00}a^{8}+\frac{49\!\cdots\!57}{38\!\cdots\!50}a^{7}-\frac{18\!\cdots\!91}{77\!\cdots\!00}a^{6}+\frac{29\!\cdots\!46}{645928597843875}a^{5}-\frac{14\!\cdots\!51}{25\!\cdots\!00}a^{4}+\frac{14\!\cdots\!29}{25\!\cdots\!00}a^{3}-\frac{20\!\cdots\!99}{645928597843875}a^{2}+\frac{547385614188163}{258371439137550}a-\frac{3245752176943}{5806099755900}$, $\frac{30367566113437}{77\!\cdots\!00}a^{17}-\frac{55037141462257}{15\!\cdots\!00}a^{16}+\frac{148102599394969}{645928597843875}a^{15}-\frac{20\!\cdots\!86}{19\!\cdots\!25}a^{14}+\frac{74\!\cdots\!39}{19\!\cdots\!25}a^{13}-\frac{75\!\cdots\!69}{645928597843875}a^{12}+\frac{54\!\cdots\!17}{19\!\cdots\!25}a^{11}-\frac{20\!\cdots\!73}{38\!\cdots\!50}a^{10}+\frac{54\!\cdots\!11}{645928597843875}a^{9}-\frac{41\!\cdots\!47}{38\!\cdots\!50}a^{8}+\frac{56\!\cdots\!96}{387557158706325}a^{7}-\frac{99\!\cdots\!67}{430619065229250}a^{6}+\frac{11\!\cdots\!29}{258371439137550}a^{5}-\frac{31\!\cdots\!81}{430619065229250}a^{4}+\frac{24\!\cdots\!93}{25\!\cdots\!00}a^{3}-\frac{27\!\cdots\!03}{25\!\cdots\!00}a^{2}+\frac{31\!\cdots\!26}{5167428782751}a-\frac{148365578092873}{322561097550}$ 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:  \( 13204003.8996 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 13204003.8996 \cdot 18}{2\cdot\sqrt{64119276021132037084693359375}}\cr\approx \mathstrut & 7.16263241086 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 60*x^16 - 276*x^15 + 1059*x^14 - 3297*x^13 + 8326*x^12 - 17001*x^11 + 28578*x^10 - 40029*x^9 + 55638*x^8 - 80625*x^7 + 148498*x^6 - 237165*x^5 + 364521*x^4 - 394488*x^3 + 378474*x^2 - 212265*x + 118815)
 
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 - 9*x^17 + 60*x^16 - 276*x^15 + 1059*x^14 - 3297*x^13 + 8326*x^12 - 17001*x^11 + 28578*x^10 - 40029*x^9 + 55638*x^8 - 80625*x^7 + 148498*x^6 - 237165*x^5 + 364521*x^4 - 394488*x^3 + 378474*x^2 - 212265*x + 118815, 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 - 9*x^17 + 60*x^16 - 276*x^15 + 1059*x^14 - 3297*x^13 + 8326*x^12 - 17001*x^11 + 28578*x^10 - 40029*x^9 + 55638*x^8 - 80625*x^7 + 148498*x^6 - 237165*x^5 + 364521*x^4 - 394488*x^3 + 378474*x^2 - 212265*x + 118815);
 
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 - 9*x^17 + 60*x^16 - 276*x^15 + 1059*x^14 - 3297*x^13 + 8326*x^12 - 17001*x^11 + 28578*x^10 - 40029*x^9 + 55638*x^8 - 80625*x^7 + 148498*x^6 - 237165*x^5 + 364521*x^4 - 394488*x^3 + 378474*x^2 - 212265*x + 118815);
 
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

$C_3:S_3$ (as 18T4):

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 $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-15}) \), 3.1.1815.1 x3, 3.1.16335.2 x3, 3.1.135.1 x3, 3.1.16335.1 x3, 6.0.49413375.1, 6.0.4002483375.3, 6.0.273375.1, 6.0.4002483375.2, 9.1.65380565930625.1 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: 9.1.65380565930625.1
Minimal sibling: 9.1.65380565930625.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.3.0.1}{3} }^{6}$ R R ${\href{/padicField/7.2.0.1}{2} }^{9}$ R ${\href{/padicField/13.2.0.1}{2} }^{9}$ ${\href{/padicField/17.3.0.1}{3} }^{6}$ ${\href{/padicField/19.3.0.1}{3} }^{6}$ ${\href{/padicField/23.3.0.1}{3} }^{6}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.2.0.1}{2} }^{9}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.2.0.1}{2} }^{9}$ ${\href{/padicField/47.3.0.1}{3} }^{6}$ ${\href{/padicField/53.3.0.1}{3} }^{6}$ ${\href{/padicField/59.2.0.1}{2} }^{9}$

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
\(3\) Copy content Toggle raw display 3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
\(5\) Copy content Toggle raw display 5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
\(11\) 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}$