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Properties

Label 18.16.174...000.1
Degree $18$
Signature $[16, 1]$
Discriminant $-1.746\times 10^{32}$
Root discriminant \(61.83\)
Ramified primes $2,5,139,197$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $A_4^3.(C_2^2\times S_4)$ (as 18T835)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^18 - 26*x^16 + 163*x^14 + 565*x^12 - 8221*x^10 + 27766*x^8 - 38974*x^6 + 21878*x^4 - 4485*x^2 + 245)
 
Copy content gp:K = bnfinit(y^18 - 26*y^16 + 163*y^14 + 565*y^12 - 8221*y^10 + 27766*y^8 - 38974*y^6 + 21878*y^4 - 4485*y^2 + 245, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 26*x^16 + 163*x^14 + 565*x^12 - 8221*x^10 + 27766*x^8 - 38974*x^6 + 21878*x^4 - 4485*x^2 + 245);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 26*x^16 + 163*x^14 + 565*x^12 - 8221*x^10 + 27766*x^8 - 38974*x^6 + 21878*x^4 - 4485*x^2 + 245)
 

\( x^{18} - 26 x^{16} + 163 x^{14} + 565 x^{12} - 8221 x^{10} + 27766 x^{8} - 38974 x^{6} + 21878 x^{4} + \cdots + 245 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $18$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[16, 1]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(-174560728208376204026312000000000\) \(\medspace = -\,2^{12}\cdot 5^{9}\cdot 139^{4}\cdot 197^{6}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(61.83\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{63/32}5^{3/4}139^{2/3}197^{1/2}\approx 4929.353384853861$
Ramified primes:   \(2\), \(5\), \(139\), \(197\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-5}) \)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{748}a^{14}+\frac{45}{187}a^{12}-\frac{89}{374}a^{10}-\frac{239}{748}a^{8}+\frac{139}{748}a^{6}+\frac{83}{748}a^{4}-\frac{111}{748}a^{2}+\frac{261}{748}$, $\frac{1}{1496}a^{15}-\frac{1}{1496}a^{14}+\frac{45}{374}a^{13}-\frac{45}{374}a^{12}-\frac{89}{748}a^{11}+\frac{89}{748}a^{10}+\frac{509}{1496}a^{9}-\frac{509}{1496}a^{8}+\frac{139}{1496}a^{7}-\frac{139}{1496}a^{6}-\frac{665}{1496}a^{5}+\frac{665}{1496}a^{4}+\frac{637}{1496}a^{3}-\frac{637}{1496}a^{2}-\frac{487}{1496}a+\frac{487}{1496}$, $\frac{1}{64837432880}a^{16}+\frac{11834807}{64837432880}a^{14}+\frac{15549449677}{32418716440}a^{12}+\frac{19806633967}{64837432880}a^{10}+\frac{3100750101}{6483743288}a^{8}+\frac{56485787}{114150410}a^{6}-\frac{1421131033}{2947156040}a^{4}-\frac{389232886}{810467911}a^{2}-\frac{5903514737}{12967486576}$, $\frac{1}{907724060320}a^{17}-\frac{1}{129674865760}a^{16}+\frac{5795051}{53395532960}a^{15}+\frac{74846253}{129674865760}a^{14}-\frac{6718673113}{41260184560}a^{13}+\frac{1451209539}{3813966640}a^{12}+\frac{198889703927}{907724060320}a^{11}+\frac{29601570233}{129674865760}a^{10}-\frac{18422157097}{90772406032}a^{9}+\frac{1311315853}{12967486576}a^{8}+\frac{155396459}{3196211480}a^{7}-\frac{70546689}{456601640}a^{6}+\frac{52802255507}{453862030160}a^{5}+\frac{19229705353}{64837432880}a^{4}-\frac{18247369647}{45386203016}a^{3}-\frac{2166019983}{6483743288}a^{2}+\frac{24556209747}{181544812064}a+\frac{10428266069}{25934973152}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{2}$, which has order $2$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $16$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{2370705}{294715604}a^{16}-\frac{29516747}{147357802}a^{14}+\frac{161908271}{147357802}a^{12}+\frac{1642689929}{294715604}a^{10}-\frac{17460590647}{294715604}a^{8}+\frac{683974283}{4150924}a^{6}-\frac{52580723537}{294715604}a^{4}+\frac{18537444263}{294715604}a^{2}-\frac{586014743}{147357802}$, $\frac{49251401}{12967486576}a^{16}-\frac{1368559213}{12967486576}a^{14}+\frac{4961789485}{6483743288}a^{12}+\frac{22495639615}{12967486576}a^{10}-\frac{245445389333}{6483743288}a^{8}+\frac{6173043131}{45660164}a^{6}-\frac{1073901941677}{6483743288}a^{4}+\frac{12923606427}{294715604}a^{2}-\frac{32637473321}{12967486576}$, $\frac{300766333}{64837432880}a^{16}-\frac{7999399809}{64837432880}a^{14}+\frac{26498298881}{32418716440}a^{12}+\frac{156631950331}{64837432880}a^{10}-\frac{15521979893}{381396664}a^{8}+\frac{32519862827}{228300820}a^{6}-\frac{6390877632609}{32418716440}a^{4}+\frac{313358366077}{3241871644}a^{2}-\frac{158294448505}{12967486576}$, $\frac{420301991}{64837432880}a^{16}-\frac{9997945403}{64837432880}a^{14}+\frac{23661634587}{32418716440}a^{12}+\frac{320133134817}{64837432880}a^{10}-\frac{264713143099}{6483743288}a^{8}+\frac{23119305439}{228300820}a^{6}-\frac{348377355653}{2947156040}a^{4}+\frac{242054815577}{3241871644}a^{2}-\frac{170195738835}{12967486576}$, $\frac{325533377}{113465507540}a^{17}-\frac{80157751}{6483743288}a^{16}-\frac{2327542444}{28366376885}a^{15}+\frac{1887971135}{6483743288}a^{14}+\frac{18026135642}{28366376885}a^{13}-\frac{4269315121}{3241871644}a^{12}+\frac{114356048819}{113465507540}a^{11}-\frac{64066829421}{6483743288}a^{10}-\frac{687580631915}{22693101508}a^{9}+\frac{22445818113}{294715604}a^{8}+\frac{190046340117}{1598105740}a^{7}-\frac{1931781560}{11415041}a^{6}-\frac{19101980273017}{113465507540}a^{5}+\frac{477963626223}{3241871644}a^{4}+\frac{1550993316105}{22693101508}a^{3}-\frac{39684308892}{810467911}a^{2}+\frac{131555213}{1031504614}a+\frac{19749637799}{6483743288}$, $\frac{325533377}{113465507540}a^{17}+\frac{80157751}{6483743288}a^{16}-\frac{2327542444}{28366376885}a^{15}-\frac{1887971135}{6483743288}a^{14}+\frac{18026135642}{28366376885}a^{13}+\frac{4269315121}{3241871644}a^{12}+\frac{114356048819}{113465507540}a^{11}+\frac{64066829421}{6483743288}a^{10}-\frac{687580631915}{22693101508}a^{9}-\frac{22445818113}{294715604}a^{8}+\frac{190046340117}{1598105740}a^{7}+\frac{1931781560}{11415041}a^{6}-\frac{19101980273017}{113465507540}a^{5}-\frac{477963626223}{3241871644}a^{4}+\frac{1550993316105}{22693101508}a^{3}+\frac{39684308892}{810467911}a^{2}+\frac{131555213}{1031504614}a-\frac{19749637799}{6483743288}$, $\frac{46993663}{907724060320}a^{17}-\frac{133764239}{25934973152}a^{16}+\frac{1208536321}{907724060320}a^{15}+\frac{277287957}{2357724832}a^{14}-\frac{25262192869}{453862030160}a^{13}-\frac{5966786427}{12967486576}a^{12}+\frac{304553323761}{907724060320}a^{11}-\frac{622358339}{138689696}a^{10}+\frac{152163232519}{90772406032}a^{9}+\frac{368716740105}{12967486576}a^{8}-\frac{6433196816}{399526435}a^{7}-\frac{200376759}{4150924}a^{6}+\frac{17224860774831}{453862030160}a^{5}+\frac{292101868581}{12967486576}a^{4}-\frac{684894850109}{22693101508}a^{3}-\frac{3258293727}{810467911}a^{2}+\frac{836766638425}{181544812064}a+\frac{27442029259}{25934973152}$, $\frac{742941217}{32418716440}a^{17}-\frac{105422533}{5894312080}a^{16}-\frac{12067031}{21670265}a^{15}+\frac{26250329749}{64837432880}a^{14}+\frac{45792736399}{16209358220}a^{13}-\frac{49399747231}{32418716440}a^{12}+\frac{49809739399}{2947156040}a^{11}-\frac{1031463952421}{64837432880}a^{10}-\frac{1012546134981}{6483743288}a^{9}+\frac{77935802940}{810467911}a^{8}+\frac{16704128327}{41509240}a^{7}-\frac{67172700109}{456601640}a^{6}-\frac{13621546591047}{32418716440}a^{5}+\frac{627080603257}{16209358220}a^{4}+\frac{1080722297905}{6483743288}a^{3}+\frac{55690477299}{6483743288}a^{2}-\frac{32354203099}{1620935822}a-\frac{38823448155}{12967486576}$, $\frac{97229823}{32418716440}a^{16}-\frac{232084059}{2947156040}a^{14}+\frac{8125400411}{16209358220}a^{12}+\frac{5140054671}{2947156040}a^{10}-\frac{41823442303}{1620935822}a^{8}+\frac{1723147989}{20754620}a^{6}-\frac{383528552266}{4052339555}a^{4}+\frac{67567036803}{3241871644}a^{2}+\frac{1883496663}{6483743288}$, $\frac{17662173733}{907724060320}a^{17}+\frac{240695289}{25934973152}a^{16}-\frac{441294768829}{907724060320}a^{15}-\frac{341356833}{1525586656}a^{14}+\frac{1228972972081}{453862030160}a^{13}+\frac{14281726805}{12967486576}a^{12}+\frac{11852627192971}{907724060320}a^{11}+\frac{183005005143}{25934973152}a^{10}-\frac{13057435855703}{90772406032}a^{9}-\frac{801555705231}{12967486576}a^{8}+\frac{670051929061}{1598105740}a^{7}+\frac{6876991625}{45660164}a^{6}-\frac{229426739074119}{453862030160}a^{5}-\frac{1890107681043}{12967486576}a^{4}+\frac{2682923128313}{11346550754}a^{3}+\frac{93325368035}{1620935822}a^{2}-\frac{5941325925669}{181544812064}a-\frac{178083266653}{25934973152}$, $\frac{2162530299}{453862030160}a^{17}+\frac{258848111}{32418716440}a^{16}-\frac{52647380417}{453862030160}a^{15}-\frac{1654640387}{8104679110}a^{14}+\frac{7788097739}{13348883240}a^{13}+\frac{1792422667}{1473578020}a^{12}+\frac{1676728695553}{453862030160}a^{11}+\frac{161305358147}{32418716440}a^{10}-\frac{189193542432}{5673275377}a^{9}-\frac{24160848871}{381396664}a^{8}+\frac{245115176297}{3196211480}a^{7}+\frac{89981439871}{456601640}a^{6}-\frac{5095191031921}{113465507540}a^{5}-\frac{7680742551461}{32418716440}a^{4}-\frac{103072548133}{4126018456}a^{3}+\frac{542815924939}{6483743288}a^{2}+\frac{868639219967}{90772406032}a-\frac{5509800004}{810467911}$, $\frac{1515624147}{82520369120}a^{17}-\frac{22500717}{7627933280}a^{16}-\frac{413511040381}{907724060320}a^{15}+\frac{9748060617}{129674865760}a^{14}+\frac{1122455136629}{453862030160}a^{13}-\frac{29079707713}{64837432880}a^{12}+\frac{11587677332999}{907724060320}a^{11}-\frac{228023579483}{129674865760}a^{10}-\frac{12101139050421}{90772406032}a^{9}+\frac{297288366185}{12967486576}a^{8}+\frac{1186956198793}{3196211480}a^{7}-\frac{34718263441}{456601640}a^{6}-\frac{188898653706881}{453862030160}a^{5}+\frac{612698769007}{5894312080}a^{4}+\frac{7718508830687}{45386203016}a^{3}-\frac{293018885191}{6483743288}a^{2}-\frac{3041014856181}{181544812064}a+\frac{87963927905}{25934973152}$, $\frac{309308913}{20630092280}a^{17}-\frac{80157751}{6483743288}a^{16}-\frac{84557844269}{226931015080}a^{15}+\frac{1887971135}{6483743288}a^{14}+\frac{231823254581}{113465507540}a^{13}-\frac{4269315121}{3241871644}a^{12}+\frac{2315105908461}{226931015080}a^{11}-\frac{64066829421}{6483743288}a^{10}-\frac{619018423160}{5673275377}a^{9}+\frac{22445818113}{294715604}a^{8}+\frac{502134086059}{1598105740}a^{7}-\frac{1931781560}{11415041}a^{6}-\frac{21176124461757}{56732753770}a^{5}+\frac{477963626223}{3241871644}a^{4}+\frac{3741369866523}{22693101508}a^{3}-\frac{39684308892}{810467911}a^{2}-\frac{881940938377}{45386203016}a+\frac{19749637799}{6483743288}$, $\frac{16738470041}{907724060320}a^{17}-\frac{567768081}{129674865760}a^{16}-\frac{424444617053}{907724060320}a^{15}+\frac{15306049853}{129674865760}a^{14}+\frac{1234412948397}{453862030160}a^{13}-\frac{52207319077}{64837432880}a^{12}+\frac{10741835066447}{907724060320}a^{11}-\frac{284298115047}{129674865760}a^{10}-\frac{12950482801577}{90772406032}a^{9}+\frac{518492070117}{12967486576}a^{8}+\frac{1383623094699}{3196211480}a^{7}-\frac{64406044009}{456601640}a^{6}-\frac{236820543214333}{453862030160}a^{5}+\frac{12225260934593}{64837432880}a^{4}+\frac{9509197505433}{45386203016}a^{3}-\frac{515766650687}{6483743288}a^{2}-\frac{2578671606517}{181544812064}a+\frac{131249785397}{25934973152}$, $\frac{16738470041}{907724060320}a^{17}+\frac{567768081}{129674865760}a^{16}-\frac{424444617053}{907724060320}a^{15}-\frac{15306049853}{129674865760}a^{14}+\frac{1234412948397}{453862030160}a^{13}+\frac{52207319077}{64837432880}a^{12}+\frac{10741835066447}{907724060320}a^{11}+\frac{284298115047}{129674865760}a^{10}-\frac{12950482801577}{90772406032}a^{9}-\frac{518492070117}{12967486576}a^{8}+\frac{1383623094699}{3196211480}a^{7}+\frac{64406044009}{456601640}a^{6}-\frac{236820543214333}{453862030160}a^{5}-\frac{12225260934593}{64837432880}a^{4}+\frac{9509197505433}{45386203016}a^{3}+\frac{515766650687}{6483743288}a^{2}-\frac{2578671606517}{181544812064}a-\frac{131249785397}{25934973152}$, $\frac{3825656071}{453862030160}a^{17}+\frac{1918342589}{64837432880}a^{16}-\frac{110463476083}{453862030160}a^{15}-\frac{2734563521}{3813966640}a^{14}+\frac{436883415247}{226931015080}a^{13}+\frac{116749676933}{32418716440}a^{12}+\frac{1197672025657}{453862030160}a^{11}+\frac{1428956026803}{64837432880}a^{10}-\frac{4115964892899}{45386203016}a^{9}-\frac{1297674770049}{6483743288}a^{8}+\frac{586513478389}{1598105740}a^{7}+\frac{115797211511}{228300820}a^{6}-\frac{122388775131743}{226931015080}a^{5}-\frac{16605144750797}{32418716440}a^{4}+\frac{5543270439955}{22693101508}a^{3}+\frac{594193800969}{3241871644}a^{2}-\frac{1580206513955}{90772406032}a-\frac{162774695521}{12967486576}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 26124025175.4 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{1}\cdot 26124025175.4 \cdot 1}{2\cdot\sqrt{174560728208376204026312000000000}}\cr\approx \mathstrut & 0.407095802482 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^18 - 26*x^16 + 163*x^14 + 565*x^12 - 8221*x^10 + 27766*x^8 - 38974*x^6 + 21878*x^4 - 4485*x^2 + 245) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^18 - 26*x^16 + 163*x^14 + 565*x^12 - 8221*x^10 + 27766*x^8 - 38974*x^6 + 21878*x^4 - 4485*x^2 + 245, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 26*x^16 + 163*x^14 + 565*x^12 - 8221*x^10 + 27766*x^8 - 38974*x^6 + 21878*x^4 - 4485*x^2 + 245); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 26*x^16 + 163*x^14 + 565*x^12 - 8221*x^10 + 27766*x^8 - 38974*x^6 + 21878*x^4 - 4485*x^2 + 245); 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

$A_4^3.(C_2^2\times S_4)$ (as 18T835):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 165888
The 168 conjugacy class representatives for $A_4^3.(C_2^2\times S_4)$
Character table for $A_4^3.(C_2^2\times S_4)$

Intermediate fields

3.3.985.1, 9.9.92322657333125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 18 siblings: 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 ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ R ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{7}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ $18$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{3}$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.9.0.1}{9} }^{2}$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.2.0.1}{2} }^{9}$ $18$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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 2.6.1.0a1.1$x^{6} + x^{4} + x^{3} + x + 1$$1$$6$$0$$C_6$$$[\ ]^{6}$$
2.6.2.12a8.1$x^{12} + 2 x^{11} + 2 x^{10} + 4 x^{9} + 5 x^{8} + 4 x^{7} + 7 x^{6} + 6 x^{5} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 3$$2$$6$$12$12T134$$[2, 2, 2, 2, 2, 2]^{6}$$
\(5\) Copy content Toggle raw display 5.2.1.0a1.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
5.1.4.3a1.3$x^{4} + 15$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(139\) Copy content Toggle raw display 139.1.3.2a1.1$x^{3} + 139$$3$$1$$2$$C_3$$$[\ ]_{3}$$
139.1.3.2a1.1$x^{3} + 139$$3$$1$$2$$C_3$$$[\ ]_{3}$$
139.12.1.0a1.1$x^{12} + 120 x^{7} + 75 x^{6} + 41 x^{5} + 77 x^{4} + 106 x^{3} + 8 x^{2} + 10 x + 2$$1$$12$$0$$C_{12}$$$[\ ]^{12}$$
\(197\) Copy content Toggle raw display 197.2.1.0a1.1$x^{2} + 192 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
197.2.1.0a1.1$x^{2} + 192 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
197.2.1.0a1.1$x^{2} + 192 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
197.2.2.2a1.2$x^{4} + 384 x^{3} + 36868 x^{2} + 768 x + 201$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
197.4.2.4a1.2$x^{8} + 32 x^{6} + 248 x^{5} + 260 x^{4} + 3968 x^{3} + 15440 x^{2} + 496 x + 201$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)