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Properties

Label 18.18.409...064.1
Degree $18$
Signature $[18, 0]$
Discriminant $4.097\times 10^{34}$
Root discriminant \(83.74\)
Ramified primes $2,3,7,29$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $A_4^3:(C_2\times A_4)$ (as 18T701)

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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 - 54*x^16 + 1203*x^14 - 14349*x^12 + 99573*x^10 - 409398*x^8 + 972425*x^6 - 1238373*x^4 + 722274*x^2 - 142129)
 
Copy content gp:K = bnfinit(y^18 - 54*y^16 + 1203*y^14 - 14349*y^12 + 99573*y^10 - 409398*y^8 + 972425*y^6 - 1238373*y^4 + 722274*y^2 - 142129, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 54*x^16 + 1203*x^14 - 14349*x^12 + 99573*x^10 - 409398*x^8 + 972425*x^6 - 1238373*x^4 + 722274*x^2 - 142129);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 54*x^16 + 1203*x^14 - 14349*x^12 + 99573*x^10 - 409398*x^8 + 972425*x^6 - 1238373*x^4 + 722274*x^2 - 142129)
 

\( x^{18} - 54 x^{16} + 1203 x^{14} - 14349 x^{12} + 99573 x^{10} - 409398 x^{8} + 972425 x^{6} + \cdots - 142129 \) 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:  $[18, 0]$
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:   \(40971612301023960236709043943768064\) \(\medspace = 2^{18}\cdot 3^{18}\cdot 7^{14}\cdot 29^{6}\) 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:  \(83.74\)
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:  not computed
Ramified primes:   \(2\), \(3\), \(7\), \(29\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7}a^{10}+\frac{1}{7}a^{8}+\frac{2}{7}a^{6}-\frac{1}{7}a^{4}+\frac{1}{7}a^{2}+\frac{3}{7}$, $\frac{1}{7}a^{11}+\frac{1}{7}a^{9}+\frac{2}{7}a^{7}-\frac{1}{7}a^{5}+\frac{1}{7}a^{3}+\frac{3}{7}a$, $\frac{1}{7}a^{12}+\frac{1}{7}a^{8}-\frac{3}{7}a^{6}+\frac{2}{7}a^{4}+\frac{2}{7}a^{2}-\frac{3}{7}$, $\frac{1}{7}a^{13}+\frac{1}{7}a^{9}-\frac{3}{7}a^{7}+\frac{2}{7}a^{5}+\frac{2}{7}a^{3}-\frac{3}{7}a$, $\frac{1}{49}a^{14}-\frac{2}{49}a^{12}+\frac{3}{49}a^{10}+\frac{4}{49}a^{8}+\frac{12}{49}a^{6}+\frac{10}{49}a^{4}-\frac{19}{49}a^{2}+\frac{12}{49}$, $\frac{1}{637}a^{15}-\frac{9}{637}a^{13}+\frac{31}{637}a^{11}-\frac{318}{637}a^{9}-\frac{303}{637}a^{7}-\frac{10}{49}a^{5}+\frac{142}{637}a^{3}+\frac{313}{637}a$, $\frac{1}{25521375237449}a^{16}-\frac{257375523765}{25521375237449}a^{14}-\frac{707060902543}{25521375237449}a^{12}-\frac{428832799674}{25521375237449}a^{10}+\frac{6958956030321}{25521375237449}a^{8}-\frac{845146040667}{1963182710573}a^{6}+\frac{2863113830303}{25521375237449}a^{4}-\frac{10994195392478}{25521375237449}a^{2}-\frac{23744692918}{67695955537}$, $\frac{1}{25521375237449}a^{17}-\frac{16985804103}{25521375237449}a^{15}+\frac{775342368706}{25521375237449}a^{13}-\frac{268572986566}{25521375237449}a^{11}+\frac{3433240141945}{25521375237449}a^{9}-\frac{7260857873910}{25521375237449}a^{7}+\frac{11717468504520}{25521375237449}a^{5}-\frac{183094649071}{1963182710573}a^{3}+\frac{39190448536}{125721060283}a$ 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:  $C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  $17$
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:   $\frac{9887277}{1963182710573}a^{16}-\frac{581494302}{1963182710573}a^{14}+\frac{2104600301}{280454672939}a^{12}-\frac{210432684093}{1963182710573}a^{10}+\frac{1838839360530}{1963182710573}a^{8}-\frac{1398115208780}{280454672939}a^{6}+\frac{29090434483260}{1963182710573}a^{4}-\frac{39064728851553}{1963182710573}a^{2}+\frac{474191072947}{67695955537}$, $\frac{17897937}{280454672939}a^{16}-\frac{6711270081}{1963182710573}a^{14}+\frac{146767812208}{1963182710573}a^{12}-\frac{1687042198647}{1963182710573}a^{10}+\frac{10926360512670}{1963182710573}a^{8}-\frac{39660429061551}{1963182710573}a^{6}+\frac{75287037819888}{1963182710573}a^{4}-\frac{62992316951175}{1963182710573}a^{2}+\frac{553728421830}{67695955537}$, $\frac{9179273333}{25521375237449}a^{16}-\frac{67408820160}{3645910748207}a^{14}+\frac{9817375288155}{25521375237449}a^{12}-\frac{106246989406638}{25521375237449}a^{10}+\frac{639658943988639}{25521375237449}a^{8}-\frac{163476035898190}{1963182710573}a^{6}+\frac{36\cdots 04}{25521375237449}a^{4}-\frac{27\cdots 88}{25521375237449}a^{2}+\frac{1555748856413}{67695955537}$, $\frac{184610030}{3645910748207}a^{16}-\frac{9436781209}{3645910748207}a^{14}+\frac{190163850015}{3645910748207}a^{12}-\frac{273036289818}{520844392601}a^{10}+\frac{9968364551507}{3645910748207}a^{8}-\frac{1930268729159}{280454672939}a^{6}+\frac{21082771372981}{3645910748207}a^{4}+\frac{13638902348325}{3645910748207}a^{2}-\frac{42554092843}{9670850791}$, $\frac{2359864397}{25521375237449}a^{16}-\frac{126664661303}{25521375237449}a^{14}+\frac{2810060041083}{25521375237449}a^{12}-\frac{33448803593767}{25521375237449}a^{10}+\frac{231845040569036}{25521375237449}a^{8}-\frac{72891102798718}{1963182710573}a^{6}+\frac{21\cdots 86}{25521375237449}a^{4}-\frac{25\cdots 54}{25521375237449}a^{2}+\frac{2674560016651}{67695955537}$, $\frac{3856836483}{25521375237449}a^{16}-\frac{205692433169}{25521375237449}a^{14}+\frac{4500768731671}{25521375237449}a^{12}-\frac{52112318624935}{25521375237449}a^{10}+\frac{342227734803333}{25521375237449}a^{8}-\frac{96916372011013}{1963182710573}a^{6}+\frac{23\cdots 81}{25521375237449}a^{4}-\frac{17\cdots 32}{25521375237449}a^{2}+\frac{22537679777}{1381550113}$, $\frac{4086793917}{25521375237449}a^{16}-\frac{187543140195}{25521375237449}a^{14}+\frac{68860949663}{520844392601}a^{12}-\frac{30220934956512}{25521375237449}a^{10}+\frac{141577389648651}{25521375237449}a^{8}-\frac{527295008939}{40064953277}a^{6}+\frac{362555973655696}{25521375237449}a^{4}-\frac{148495244189723}{25521375237449}a^{2}+\frac{12826311724}{67695955537}$, $\frac{13302285288}{25521375237449}a^{16}-\frac{688223856023}{25521375237449}a^{14}+\frac{14451869266215}{25521375237449}a^{12}-\frac{158588167768391}{25521375237449}a^{10}+\frac{139359998029321}{3645910748207}a^{8}-\frac{257687727227472}{1963182710573}a^{6}+\frac{857443498577810}{3645910748207}a^{4}-\frac{47\cdots 60}{25521375237449}a^{2}+\frac{3522824571166}{67695955537}$, $\frac{9670603623}{25521375237449}a^{17}-\frac{11070955492}{25521375237449}a^{16}-\frac{38563184822}{1963182710573}a^{15}+\frac{569118620646}{25521375237449}a^{14}+\frac{10536024676642}{25521375237449}a^{13}-\frac{11833327852523}{25521375237449}a^{12}-\frac{8875437501835}{1963182710573}a^{11}+\frac{18267855581119}{3645910748207}a^{10}+\frac{703604632579130}{25521375237449}a^{9}-\frac{767579857323101}{25521375237449}a^{8}-\frac{23\cdots 66}{25521375237449}a^{7}+\frac{194583430890214}{1963182710573}a^{6}+\frac{40\cdots 11}{25521375237449}a^{5}-\frac{41\cdots 64}{25521375237449}a^{4}-\frac{28\cdots 86}{25521375237449}a^{3}+\frac{57195601897555}{520844392601}a^{2}+\frac{22173559772613}{880047421981}a-\frac{1525543494073}{67695955537}$, $\frac{27288360}{25521375237449}a^{17}+\frac{13584466015}{25521375237449}a^{16}-\frac{7464359366}{25521375237449}a^{15}-\frac{682207650827}{25521375237449}a^{14}+\frac{313838059944}{25521375237449}a^{13}+\frac{13789342069921}{25521375237449}a^{12}-\frac{5558142007827}{25521375237449}a^{11}-\frac{143995312874860}{25521375237449}a^{10}+\frac{49856137096421}{25521375237449}a^{9}+\frac{829878345020908}{25521375237449}a^{8}-\frac{234248667773411}{25521375237449}a^{7}-\frac{201345410467584}{1963182710573}a^{6}+\frac{553229716244340}{25521375237449}a^{5}+\frac{42\cdots 14}{25521375237449}a^{4}-\frac{589458649299081}{25521375237449}a^{3}-\frac{29\cdots 08}{25521375237449}a^{2}+\frac{116571882043}{17960151469}a+\frac{1716567414735}{67695955537}$, $\frac{3295635828}{25521375237449}a^{17}-\frac{1829247542}{25521375237449}a^{16}-\frac{170163630947}{25521375237449}a^{15}+\frac{89091233419}{25521375237449}a^{14}+\frac{3570291479225}{25521375237449}a^{13}-\frac{1731003073851}{25521375237449}a^{12}-\frac{5607515669619}{3645910748207}a^{11}+\frac{17066658936539}{25521375237449}a^{10}+\frac{243223552371153}{25521375237449}a^{9}-\frac{89008999172346}{25521375237449}a^{8}-\frac{849732582646190}{25521375237449}a^{7}+\frac{17331696347275}{1963182710573}a^{6}+\frac{15\cdots 84}{25521375237449}a^{5}-\frac{177106223693524}{25521375237449}a^{4}-\frac{26420420232082}{520844392601}a^{3}-\frac{127513292296083}{25521375237449}a^{2}+\frac{9691444367713}{880047421981}a+\frac{15299990865}{9670850791}$, $\frac{3322924188}{25521375237449}a^{17}-\frac{2490527562}{25521375237449}a^{16}-\frac{177627990313}{25521375237449}a^{15}+\frac{130277508258}{25521375237449}a^{14}+\frac{3884129539169}{25521375237449}a^{13}-\frac{2762812090807}{25521375237449}a^{12}-\frac{44810751695160}{25521375237449}a^{11}+\frac{30486958462966}{25521375237449}a^{10}+\frac{293079689467574}{25521375237449}a^{9}-\frac{186490710103881}{25521375237449}a^{8}-\frac{10\cdots 01}{25521375237449}a^{7}+\frac{47765765178424}{1963182710573}a^{6}+\frac{21\cdots 24}{25521375237449}a^{5}-\frac{10\cdots 59}{25521375237449}a^{4}-\frac{18\cdots 99}{25521375237449}a^{3}+\frac{723426881796746}{25521375237449}a^{2}+\frac{15403466587820}{880047421981}a-\frac{450866157765}{67695955537}$, $\frac{373815783}{3645910748207}a^{17}-\frac{571345289}{3645910748207}a^{16}-\frac{127204037985}{25521375237449}a^{15}+\frac{193227937822}{25521375237449}a^{14}+\frac{2449151201664}{25521375237449}a^{13}-\frac{3657123116940}{25521375237449}a^{12}-\frac{23775772779957}{25521375237449}a^{11}+\frac{34213536942986}{25521375237449}a^{10}+\frac{122954200699374}{25521375237449}a^{9}-\frac{164141784703101}{25521375237449}a^{8}-\frac{333214165050803}{25521375237449}a^{7}+\frac{29304470440713}{1963182710573}a^{6}+\frac{452669700528505}{25521375237449}a^{5}-\frac{368341074873014}{25521375237449}a^{4}-\frac{299217150678564}{25521375237449}a^{3}+\frac{107411277291548}{25521375237449}a^{2}+\frac{2750803039130}{880047421981}a+\frac{12140064726}{67695955537}$, $\frac{1198397895}{25521375237449}a^{17}-\frac{3500654839}{25521375237449}a^{16}-\frac{69910030956}{25521375237449}a^{15}+\frac{163020140051}{25521375237449}a^{14}+\frac{1674202450183}{25521375237449}a^{13}-\frac{2976609136167}{25521375237449}a^{12}-\frac{21254275114236}{25521375237449}a^{11}+\frac{26878949922757}{25521375237449}a^{10}+\frac{22054976715831}{3645910748207}a^{9}-\frac{122896342080345}{25521375237449}a^{8}-\frac{647603074569853}{25521375237449}a^{7}+\frac{18805768767907}{1963182710573}a^{6}+\frac{30853122569089}{520844392601}a^{5}-\frac{29654296773576}{25521375237449}a^{4}-\frac{17\cdots 70}{25521375237449}a^{3}-\frac{454137143596935}{25521375237449}a^{2}+\frac{27653358383782}{880047421981}a+\frac{136000916549}{9670850791}$, $\frac{85846374}{520844392601}a^{17}-\frac{1694956804}{25521375237449}a^{16}-\frac{208993872179}{25521375237449}a^{15}+\frac{54919210886}{25521375237449}a^{14}+\frac{4180305975421}{25521375237449}a^{13}-\frac{53822039533}{3645910748207}a^{12}-\frac{43316029013834}{25521375237449}a^{11}-\frac{5164081011379}{25521375237449}a^{10}+\frac{249449497299348}{25521375237449}a^{9}+\frac{90732314793581}{25521375237449}a^{8}-\frac{794553699111975}{25521375237449}a^{7}-\frac{5513224211273}{280454672939}a^{6}+\frac{12\cdots 93}{25521375237449}a^{5}+\frac{11\cdots 14}{25521375237449}a^{4}-\frac{870644086164806}{25521375237449}a^{3}-\frac{10\cdots 54}{25521375237449}a^{2}+\frac{5777545761003}{880047421981}a+\frac{776579305139}{67695955537}$, $\frac{19372809730}{25521375237449}a^{17}+\frac{538580326}{1963182710573}a^{16}-\frac{995573675956}{25521375237449}a^{15}-\frac{22849000138}{1963182710573}a^{14}+\frac{20798731353901}{25521375237449}a^{13}+\frac{346043492621}{1963182710573}a^{12}-\frac{228099017042023}{25521375237449}a^{11}-\frac{1930870577152}{1963182710573}a^{10}+\frac{14\cdots 86}{25521375237449}a^{9}-\frac{375457725053}{280454672939}a^{8}-\frac{50\cdots 11}{25521375237449}a^{7}+\frac{70731246448528}{1963182710573}a^{6}+\frac{97\cdots 98}{25521375237449}a^{5}-\frac{37107626909516}{280454672939}a^{4}-\frac{89\cdots 16}{25521375237449}a^{3}+\frac{329010082481342}{1963182710573}a^{2}+\frac{82030696351290}{880047421981}a-\frac{3143558033074}{67695955537}$, $\frac{233777318}{1963182710573}a^{17}+\frac{18946814308}{25521375237449}a^{16}-\frac{157347725211}{25521375237449}a^{15}-\frac{137757129059}{3645910748207}a^{14}+\frac{3319514633639}{25521375237449}a^{13}+\frac{19793561395963}{25521375237449}a^{12}-\frac{36896299241937}{25521375237449}a^{11}-\frac{210138493823021}{25521375237449}a^{10}+\frac{233920890445157}{25521375237449}a^{9}+\frac{12\cdots 39}{25521375237449}a^{8}-\frac{860218848200305}{25521375237449}a^{7}-\frac{299071614384730}{1963182710573}a^{6}+\frac{137887823524324}{1963182710573}a^{5}+\frac{60\cdots 21}{25521375237449}a^{4}-\frac{19\cdots 68}{25521375237449}a^{3}-\frac{38\cdots 86}{25521375237449}a^{2}+\frac{21221873285097}{880047421981}a+\frac{2190796975004}{67695955537}$ 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:  \( 458673138470 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 458673138470 \cdot 1}{2\cdot\sqrt{40971612301023960236709043943768064}}\cr\approx \mathstrut & 0.297010433795579 \end{aligned}\] (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 - 54*x^16 + 1203*x^14 - 14349*x^12 + 99573*x^10 - 409398*x^8 + 972425*x^6 - 1238373*x^4 + 722274*x^2 - 142129) 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 - 54*x^16 + 1203*x^14 - 14349*x^12 + 99573*x^10 - 409398*x^8 + 972425*x^6 - 1238373*x^4 + 722274*x^2 - 142129, 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 - 54*x^16 + 1203*x^14 - 14349*x^12 + 99573*x^10 - 409398*x^8 + 972425*x^6 - 1238373*x^4 + 722274*x^2 - 142129); 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 - 54*x^16 + 1203*x^14 - 14349*x^12 + 99573*x^10 - 409398*x^8 + 972425*x^6 - 1238373*x^4 + 722274*x^2 - 142129); 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

$A_4^3:(C_2\times A_4)$ (as 18T701):

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 41472
The 72 conjugacy class representatives for $A_4^3:(C_2\times A_4)$
Character table for $A_4^3:(C_2\times A_4)$

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.13632439166829.1

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

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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 R ${\href{/padicField/5.9.0.1}{9} }^{2}$ R ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.6.0.1}{6} }$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.9.0.1}{9} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ R ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.6.0.1}{6} }$ ${\href{/padicField/37.9.0.1}{9} }^{2}$ ${\href{/padicField/41.3.0.1}{3} }^{6}$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.3.0.1}{3} }^{6}$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ ${\href{/padicField/59.9.0.1}{9} }^{2}$

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

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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 2.3.2.6a2.2$x^{6} + 4 x^{4} + 2 x^{3} + 3 x^{2} + 4 x + 7$$2$$3$$6$$A_4\times C_2$$$[2, 2, 2]^{3}$$
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}$$
\(3\) Copy content Toggle raw display 3.3.3.9a8.1$x^{9} + 6 x^{7} + 3 x^{6} + 15 x^{5} + 12 x^{4} + 20 x^{3} + 15 x^{2} + 12 x + 7$$3$$3$$9$$C_3^2 : S_3 $$$[\frac{3}{2}, \frac{3}{2}]_{2}^{3}$$
3.3.3.9a8.1$x^{9} + 6 x^{7} + 3 x^{6} + 15 x^{5} + 12 x^{4} + 20 x^{3} + 15 x^{2} + 12 x + 7$$3$$3$$9$$C_3^2 : S_3 $$$[\frac{3}{2}, \frac{3}{2}]_{2}^{3}$$
\(7\) Copy content Toggle raw display 7.1.3.2a1.1$x^{3} + 7$$3$$1$$2$$C_3$$$[\ ]_{3}$$
7.1.3.2a1.1$x^{3} + 7$$3$$1$$2$$C_3$$$[\ ]_{3}$$
7.2.6.10a1.2$x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 192996 x^{6} + 266328 x^{5} + 234495 x^{4} + 131220 x^{3} + 45198 x^{2} + 8748 x + 736$$6$$2$$10$$C_6\times C_2$$$[\ ]_{6}^{2}$$
\(29\) Copy content Toggle raw display 29.1.2.1a1.1$x^{2} + 29$$2$$1$$1$$C_2$$$[\ ]_{2}$$
29.1.2.1a1.2$x^{2} + 58$$2$$1$$1$$C_2$$$[\ ]_{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.3.4a1.2$x^{6} + 72 x^{5} + 1734 x^{4} + 14112 x^{3} + 3468 x^{2} + 288 x + 37$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$

Spectrum of ring of integers

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