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Properties

Label 20.4.891...264.1
Degree $20$
Signature $[4, 8]$
Discriminant $8.918\times 10^{23}$
Root discriminant \(15.76\)
Ramified primes $2,67,641$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^9.A_{10}$ (as 20T1100)

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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^20 - x^19 - 10*x^18 - 2*x^17 + 50*x^16 + 55*x^15 - 108*x^14 - 242*x^13 + 21*x^12 + 435*x^11 + 295*x^10 - 283*x^9 - 434*x^8 - 28*x^7 + 216*x^6 + 74*x^5 - 94*x^4 - 112*x^3 - 52*x^2 - 13*x - 1)
 
Copy content gp:K = bnfinit(y^20 - y^19 - 10*y^18 - 2*y^17 + 50*y^16 + 55*y^15 - 108*y^14 - 242*y^13 + 21*y^12 + 435*y^11 + 295*y^10 - 283*y^9 - 434*y^8 - 28*y^7 + 216*y^6 + 74*y^5 - 94*y^4 - 112*y^3 - 52*y^2 - 13*y - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^19 - 10*x^18 - 2*x^17 + 50*x^16 + 55*x^15 - 108*x^14 - 242*x^13 + 21*x^12 + 435*x^11 + 295*x^10 - 283*x^9 - 434*x^8 - 28*x^7 + 216*x^6 + 74*x^5 - 94*x^4 - 112*x^3 - 52*x^2 - 13*x - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - x^19 - 10*x^18 - 2*x^17 + 50*x^16 + 55*x^15 - 108*x^14 - 242*x^13 + 21*x^12 + 435*x^11 + 295*x^10 - 283*x^9 - 434*x^8 - 28*x^7 + 216*x^6 + 74*x^5 - 94*x^4 - 112*x^3 - 52*x^2 - 13*x - 1)
 

\( x^{20} - x^{19} - 10 x^{18} - 2 x^{17} + 50 x^{16} + 55 x^{15} - 108 x^{14} - 242 x^{13} + 21 x^{12} + \cdots - 1 \) 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:  $20$
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:  $[4, 8]$
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:   \(891807767841222552715264\) \(\medspace = 2^{18}\cdot 67^{4}\cdot 641^{4}\) 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:  \(15.76\)
Copy content comment:Root discriminant
 
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:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{39/16}67^{2/3}641^{2/3}\approx 6643.279161271313$
Ramified primes:   \(2\), \(67\), \(641\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{9192351081637}a^{19}-\frac{1212495565468}{9192351081637}a^{18}+\frac{17566487188}{9192351081637}a^{17}+\frac{2789527878365}{9192351081637}a^{16}-\frac{3074776038300}{9192351081637}a^{15}-\frac{4059252428289}{9192351081637}a^{14}+\frac{4042374792463}{9192351081637}a^{13}-\frac{1803031553298}{9192351081637}a^{12}+\frac{2640860813274}{9192351081637}a^{11}-\frac{483924161061}{9192351081637}a^{10}-\frac{2539638571840}{9192351081637}a^{9}+\frac{712275933603}{9192351081637}a^{8}-\frac{3534662334271}{9192351081637}a^{7}+\frac{1106549932694}{9192351081637}a^{6}+\frac{899550148663}{9192351081637}a^{5}-\frac{3729839013266}{9192351081637}a^{4}+\frac{46638221673}{9192351081637}a^{3}+\frac{4443682422933}{9192351081637}a^{2}+\frac{2156493812624}{9192351081637}a-\frac{2670111446221}{9192351081637}$ 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:  Trivial group, which has order $1$ (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:  $11$
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:   $\frac{3101127938640}{9192351081637}a^{19}-\frac{3798750626860}{9192351081637}a^{18}-\frac{30881522152443}{9192351081637}a^{17}+\frac{1555580084801}{9192351081637}a^{16}+\frac{162388568221886}{9192351081637}a^{15}+\frac{133443406227919}{9192351081637}a^{14}-\frac{405529750713305}{9192351081637}a^{13}-\frac{690024321654374}{9192351081637}a^{12}+\frac{326980308393445}{9192351081637}a^{11}+\frac{14\cdots 77}{9192351081637}a^{10}+\frac{485472473054700}{9192351081637}a^{9}-\frac{13\cdots 01}{9192351081637}a^{8}-\frac{11\cdots 67}{9192351081637}a^{7}+\frac{522377484037397}{9192351081637}a^{6}+\frac{778909331666757}{9192351081637}a^{5}-\frac{112346738340942}{9192351081637}a^{4}-\frac{434473307437001}{9192351081637}a^{3}-\frac{199367746813814}{9192351081637}a^{2}-\frac{28496818390460}{9192351081637}a+\frac{6335154115592}{9192351081637}$, $\frac{8172568800432}{9192351081637}a^{19}-\frac{12356814942275}{9192351081637}a^{18}-\frac{77079245171433}{9192351081637}a^{17}+\frac{26082228922711}{9192351081637}a^{16}+\frac{410337758263790}{9192351081637}a^{15}+\frac{230295971926381}{9192351081637}a^{14}-\frac{10\cdots 50}{9192351081637}a^{13}-\frac{14\cdots 57}{9192351081637}a^{12}+\frac{11\cdots 16}{9192351081637}a^{11}+\frac{32\cdots 66}{9192351081637}a^{10}+\frac{469953060082206}{9192351081637}a^{9}-\frac{31\cdots 27}{9192351081637}a^{8}-\frac{18\cdots 76}{9192351081637}a^{7}+\frac{13\cdots 16}{9192351081637}a^{6}+\frac{12\cdots 86}{9192351081637}a^{5}-\frac{406238355890530}{9192351081637}a^{4}-\frac{731117633007181}{9192351081637}a^{3}-\frac{397571762963929}{9192351081637}a^{2}-\frac{99075962985807}{9192351081637}a-\frac{14178163565728}{9192351081637}$, $\frac{8415152302772}{9192351081637}a^{19}-\frac{11977562521054}{9192351081637}a^{18}-\frac{79738299444952}{9192351081637}a^{17}+\frac{17645249681424}{9192351081637}a^{16}+\frac{420298219479772}{9192351081637}a^{15}+\frac{284505761933773}{9192351081637}a^{14}-\frac{10\cdots 77}{9192351081637}a^{13}-\frac{16\cdots 00}{9192351081637}a^{12}+\frac{955020925834445}{9192351081637}a^{11}+\frac{34\cdots 59}{9192351081637}a^{10}+\frac{953114236725234}{9192351081637}a^{9}-\frac{31\cdots 88}{9192351081637}a^{8}-\frac{24\cdots 11}{9192351081637}a^{7}+\frac{10\cdots 92}{9192351081637}a^{6}+\frac{15\cdots 09}{9192351081637}a^{5}-\frac{181422575427055}{9192351081637}a^{4}-\frac{851063835095536}{9192351081637}a^{3}-\frac{524687212430300}{9192351081637}a^{2}-\frac{137471172560397}{9192351081637}a-\frac{21276284328136}{9192351081637}$, $\frac{242583502340}{9192351081637}a^{19}+\frac{379252421221}{9192351081637}a^{18}-\frac{2659054273519}{9192351081637}a^{17}-\frac{8436979241287}{9192351081637}a^{16}+\frac{9960461215982}{9192351081637}a^{15}+\frac{54209790007392}{9192351081637}a^{14}+\frac{16524576554473}{9192351081637}a^{13}-\frac{164520911203843}{9192351081637}a^{12}-\frac{187977723873171}{9192351081637}a^{11}+\frac{199607093473493}{9192351081637}a^{10}+\frac{483161176643028}{9192351081637}a^{9}+\frac{26955511697039}{9192351081637}a^{8}-\frac{534173385327835}{9192351081637}a^{7}-\frac{272923723248624}{9192351081637}a^{6}+\frac{284922965059923}{9192351081637}a^{5}+\frac{224815780463475}{9192351081637}a^{4}-\frac{119946202088355}{9192351081637}a^{3}-\frac{127115449466371}{9192351081637}a^{2}-\frac{38395209574590}{9192351081637}a-\frac{7098120762408}{9192351081637}$, $\frac{3163789408582}{9192351081637}a^{19}-\frac{1561743918057}{9192351081637}a^{18}-\frac{35445155656976}{9192351081637}a^{17}-\frac{18137783736714}{9192351081637}a^{16}+\frac{173386912663401}{9192351081637}a^{15}+\frac{241914266373276}{9192351081637}a^{14}-\frac{354495423331463}{9192351081637}a^{13}-\frac{970723427026784}{9192351081637}a^{12}-\frac{48047018604624}{9192351081637}a^{11}+\frac{17\cdots 67}{9192351081637}a^{10}+\frac{13\cdots 11}{9192351081637}a^{9}-\frac{11\cdots 33}{9192351081637}a^{8}-\frac{18\cdots 95}{9192351081637}a^{7}-\frac{67252179558121}{9192351081637}a^{6}+\frac{10\cdots 87}{9192351081637}a^{5}+\frac{260156364769323}{9192351081637}a^{4}-\frac{461007125201701}{9192351081637}a^{3}-\frac{432741292756252}{9192351081637}a^{2}-\frac{161039560406377}{9192351081637}a-\frac{24740115415726}{9192351081637}$, $\frac{3340590282521}{9192351081637}a^{19}-\frac{2403980176712}{9192351081637}a^{18}-\frac{36403542781117}{9192351081637}a^{17}-\frac{12590027394233}{9192351081637}a^{16}+\frac{183514523904742}{9192351081637}a^{15}+\frac{222079242213276}{9192351081637}a^{14}-\frac{408252216429959}{9192351081637}a^{13}-\frac{955063709981194}{9192351081637}a^{12}+\frac{108438377336067}{9192351081637}a^{11}+\frac{17\cdots 63}{9192351081637}a^{10}+\frac{10\cdots 30}{9192351081637}a^{9}-\frac{14\cdots 50}{9192351081637}a^{8}-\frac{17\cdots 21}{9192351081637}a^{7}+\frac{256920986409980}{9192351081637}a^{6}+\frac{10\cdots 28}{9192351081637}a^{5}+\frac{86238185414764}{9192351081637}a^{4}-\frac{533608318444866}{9192351081637}a^{3}-\frac{348642162477392}{9192351081637}a^{2}-\frac{92137073605199}{9192351081637}a-\frac{12791397547761}{9192351081637}$, $\frac{6120487479278}{9192351081637}a^{19}-\frac{6911888429219}{9192351081637}a^{18}-\frac{62194475285453}{9192351081637}a^{17}-\frac{285906172389}{9192351081637}a^{16}+\frac{321279394434757}{9192351081637}a^{15}+\frac{281758694359530}{9192351081637}a^{14}-\frac{782732435778219}{9192351081637}a^{13}-\frac{13\cdots 74}{9192351081637}a^{12}+\frac{550696983033575}{9192351081637}a^{11}+\frac{28\cdots 84}{9192351081637}a^{10}+\frac{11\cdots 19}{9192351081637}a^{9}-\frac{24\cdots 43}{9192351081637}a^{8}-\frac{22\cdots 91}{9192351081637}a^{7}+\frac{735305076786430}{9192351081637}a^{6}+\frac{14\cdots 60}{9192351081637}a^{5}-\frac{53587518623942}{9192351081637}a^{4}-\frac{727725556252124}{9192351081637}a^{3}-\frac{487934086134387}{9192351081637}a^{2}-\frac{135810164955432}{9192351081637}a-\frac{14584846821698}{9192351081637}$, $\frac{955857745303}{9192351081637}a^{19}-\frac{1609939541642}{9192351081637}a^{18}-\frac{9324717344162}{9192351081637}a^{17}+\frac{5390055377436}{9192351081637}a^{16}+\frac{53011125039633}{9192351081637}a^{15}+\frac{17610992842989}{9192351081637}a^{14}-\frac{161899237235861}{9192351081637}a^{13}-\frac{168867191410261}{9192351081637}a^{12}+\frac{244576124170954}{9192351081637}a^{11}+\frac{476939750861563}{9192351081637}a^{10}-\frac{92018178601787}{9192351081637}a^{9}-\frac{655684562989076}{9192351081637}a^{8}-\frac{221168346805243}{9192351081637}a^{7}+\frac{486911745004953}{9192351081637}a^{6}+\frac{319974595455615}{9192351081637}a^{5}-\frac{210368707366031}{9192351081637}a^{4}-\frac{234955484923272}{9192351081637}a^{3}+\frac{14057057969889}{9192351081637}a^{2}+\frac{89289140939436}{9192351081637}a+\frac{29644547542363}{9192351081637}$, $\frac{369049810445}{9192351081637}a^{19}+\frac{497485834869}{9192351081637}a^{18}-\frac{4769696336105}{9192351081637}a^{17}-\frac{8959273737873}{9192351081637}a^{16}+\frac{18258036204870}{9192351081637}a^{15}+\frac{62718087982918}{9192351081637}a^{14}-\frac{798981543225}{9192351081637}a^{13}-\frac{186641338777134}{9192351081637}a^{12}-\frac{183368909213823}{9192351081637}a^{11}+\frac{208237645884619}{9192351081637}a^{10}+\frac{478496008571655}{9192351081637}a^{9}+\frac{100482834431199}{9192351081637}a^{8}-\frac{452266076367454}{9192351081637}a^{7}-\frac{369829636316997}{9192351081637}a^{6}+\frac{119059898578907}{9192351081637}a^{5}+\frac{256237314913683}{9192351081637}a^{4}-\frac{976562180453}{9192351081637}a^{3}-\frac{160881668817240}{9192351081637}a^{2}-\frac{97941556283201}{9192351081637}a-\frac{22355883389045}{9192351081637}$, $\frac{9634338123162}{9192351081637}a^{19}-\frac{11030324817491}{9192351081637}a^{18}-\frac{96448382676528}{9192351081637}a^{17}-\frac{1600205890607}{9192351081637}a^{16}+\frac{495414605387650}{9192351081637}a^{15}+\frac{444644599624691}{9192351081637}a^{14}-\frac{11\cdots 14}{9192351081637}a^{13}-\frac{21\cdots 71}{9192351081637}a^{12}+\frac{735065456453483}{9192351081637}a^{11}+\frac{42\cdots 41}{9192351081637}a^{10}+\frac{19\cdots 12}{9192351081637}a^{9}-\frac{34\cdots 43}{9192351081637}a^{8}-\frac{35\cdots 36}{9192351081637}a^{7}+\frac{773669125584015}{9192351081637}a^{6}+\frac{20\cdots 68}{9192351081637}a^{5}+\frac{114172770444988}{9192351081637}a^{4}-\frac{10\cdots 12}{9192351081637}a^{3}-\frac{830203561551803}{9192351081637}a^{2}-\frac{279568363561781}{9192351081637}a-\frac{42859340319263}{9192351081637}$, $\frac{6238818027071}{9192351081637}a^{19}-\frac{9413720712739}{9192351081637}a^{18}-\frac{57232010892197}{9192351081637}a^{17}+\frac{15817316660438}{9192351081637}a^{16}+\frac{301236483283643}{9192351081637}a^{15}+\frac{192089699842138}{9192351081637}a^{14}-\frac{756224395117269}{9192351081637}a^{13}-\frac{11\cdots 53}{9192351081637}a^{12}+\frac{662525778178332}{9192351081637}a^{11}+\frac{23\cdots 50}{9192351081637}a^{10}+\frac{689993101271563}{9192351081637}a^{9}-\frac{20\cdots 44}{9192351081637}a^{8}-\frac{16\cdots 56}{9192351081637}a^{7}+\frac{594302833354535}{9192351081637}a^{6}+\frac{972321481860837}{9192351081637}a^{5}-\frac{37605668713008}{9192351081637}a^{4}-\frac{536853724004321}{9192351081637}a^{3}-\frac{411287702869551}{9192351081637}a^{2}-\frac{145879477578830}{9192351081637}a-\frac{32330849902588}{9192351081637}$ 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:  \( 7497.55761136 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 7497.55761136 \cdot 1}{2\cdot\sqrt{891807767841222552715264}}\cr\approx \mathstrut & 0.154281239738 \end{aligned}\] (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^20 - x^19 - 10*x^18 - 2*x^17 + 50*x^16 + 55*x^15 - 108*x^14 - 242*x^13 + 21*x^12 + 435*x^11 + 295*x^10 - 283*x^9 - 434*x^8 - 28*x^7 + 216*x^6 + 74*x^5 - 94*x^4 - 112*x^3 - 52*x^2 - 13*x - 1) 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^20 - x^19 - 10*x^18 - 2*x^17 + 50*x^16 + 55*x^15 - 108*x^14 - 242*x^13 + 21*x^12 + 435*x^11 + 295*x^10 - 283*x^9 - 434*x^8 - 28*x^7 + 216*x^6 + 74*x^5 - 94*x^4 - 112*x^3 - 52*x^2 - 13*x - 1, 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^20 - x^19 - 10*x^18 - 2*x^17 + 50*x^16 + 55*x^15 - 108*x^14 - 242*x^13 + 21*x^12 + 435*x^11 + 295*x^10 - 283*x^9 - 434*x^8 - 28*x^7 + 216*x^6 + 74*x^5 - 94*x^4 - 112*x^3 - 52*x^2 - 13*x - 1); 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 = polynomial_ring(QQ); K, a = number_field(x^20 - x^19 - 10*x^18 - 2*x^17 + 50*x^16 + 55*x^15 - 108*x^14 - 242*x^13 + 21*x^12 + 435*x^11 + 295*x^10 - 283*x^9 - 434*x^8 - 28*x^7 + 216*x^6 + 74*x^5 - 94*x^4 - 112*x^3 - 52*x^2 - 13*x - 1); 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

$C_2^9.A_{10}$ (as 20T1100):

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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A non-solvable group of order 928972800
The 139 conjugacy class representatives for $C_2^9.A_{10}$
Character table for $C_2^9.A_{10}$

Intermediate fields

10.2.1844444809.1

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(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-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 $16{,}\,{\href{/padicField/3.4.0.1}{4} }$ ${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.6.0.1}{6} }$ ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.8.0.1}{8} }$ ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.6.0.1}{6} }$ ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ $18{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ ${\href{/padicField/23.5.0.1}{5} }^{4}$ ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ $16{,}\,{\href{/padicField/41.4.0.1}{4} }$ ${\href{/padicField/43.7.0.1}{7} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ $18{,}\,{\href{/padicField/59.2.0.1}{2} }$

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 2.8.1.0a1.1$x^{8} + x^{4} + x^{3} + x^{2} + 1$$1$$8$$0$$C_8$$$[\ ]^{8}$$
2.6.2.18a1.12$x^{12} + 2 x^{10} + 2 x^{9} + 5 x^{8} + 8 x^{7} + 7 x^{6} + 10 x^{5} + 8 x^{4} + 14 x^{3} + 9 x^{2} + 6 x + 3$$2$$6$$18$12T105$$[2, 2, 2, 2, 3]^{6}$$
\(67\) Copy content Toggle raw display $\Q_{67}$$x + 65$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{67}$$x + 65$$1$$1$$0$Trivial$$[\ ]$$
67.1.3.2a1.3$x^{3} + 268$$3$$1$$2$$C_3$$$[\ ]_{3}$$
67.3.1.0a1.1$x^{3} + 6 x + 65$$1$$3$$0$$C_3$$$[\ ]^{3}$$
67.3.1.0a1.1$x^{3} + 6 x + 65$$1$$3$$0$$C_3$$$[\ ]^{3}$$
67.1.3.2a1.3$x^{3} + 268$$3$$1$$2$$C_3$$$[\ ]_{3}$$
67.3.1.0a1.1$x^{3} + 6 x + 65$$1$$3$$0$$C_3$$$[\ ]^{3}$$
67.3.1.0a1.1$x^{3} + 6 x + 65$$1$$3$$0$$C_3$$$[\ ]^{3}$$
\(641\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
Deg $6$$3$$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)