↑

Properties

Label 15.15.116...009.1
Degree $15$
Signature $[15, 0]$
Discriminant $1.162\times 10^{28}$
Root discriminant \(74.31\)
Ramified primes $7,283$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $A_6$ (as 15T20)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 38*x^13 + 196*x^12 + 476*x^11 - 2646*x^10 - 2219*x^9 + 15116*x^8 + 2757*x^7 - 37522*x^6 + 6328*x^5 + 33201*x^4 - 16184*x^3 - 2037*x^2 + 2058*x - 245)
 
gp: K = bnfinit(y^15 - 5*y^14 - 38*y^13 + 196*y^12 + 476*y^11 - 2646*y^10 - 2219*y^9 + 15116*y^8 + 2757*y^7 - 37522*y^6 + 6328*y^5 + 33201*y^4 - 16184*y^3 - 2037*y^2 + 2058*y - 245, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 5*x^14 - 38*x^13 + 196*x^12 + 476*x^11 - 2646*x^10 - 2219*x^9 + 15116*x^8 + 2757*x^7 - 37522*x^6 + 6328*x^5 + 33201*x^4 - 16184*x^3 - 2037*x^2 + 2058*x - 245);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 5*x^14 - 38*x^13 + 196*x^12 + 476*x^11 - 2646*x^10 - 2219*x^9 + 15116*x^8 + 2757*x^7 - 37522*x^6 + 6328*x^5 + 33201*x^4 - 16184*x^3 - 2037*x^2 + 2058*x - 245)
 

\( x^{15} - 5 x^{14} - 38 x^{13} + 196 x^{12} + 476 x^{11} - 2646 x^{10} - 2219 x^{9} + 15116 x^{8} + \cdots - 245 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $15$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[15, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(11621759510846642583679213009\) \(\medspace = 7^{10}\cdot 283^{8}\) 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:  \(74.31\)
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:  $7^{3/4}283^{2/3}\approx 185.50166855022897$
Ramified primes:   \(7\), \(283\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7}a^{9}-\frac{3}{7}a^{8}-\frac{2}{7}a^{6}+\frac{1}{7}a^{5}+\frac{3}{7}a^{4}$, $\frac{1}{7}a^{10}-\frac{2}{7}a^{8}-\frac{2}{7}a^{7}+\frac{2}{7}a^{6}-\frac{1}{7}a^{5}+\frac{2}{7}a^{4}$, $\frac{1}{119}a^{11}+\frac{6}{119}a^{10}-\frac{6}{119}a^{9}+\frac{54}{119}a^{8}-\frac{59}{119}a^{7}-\frac{9}{119}a^{6}+\frac{13}{119}a^{5}-\frac{1}{17}a^{4}-\frac{1}{17}a^{3}+\frac{6}{17}a^{2}+\frac{5}{17}a+\frac{5}{17}$, $\frac{1}{119}a^{12}-\frac{8}{119}a^{10}+\frac{5}{119}a^{9}+\frac{6}{17}a^{8}+\frac{39}{119}a^{7}-\frac{52}{119}a^{6}+\frac{2}{7}a^{5}-\frac{33}{119}a^{4}-\frac{5}{17}a^{3}+\frac{3}{17}a^{2}-\frac{8}{17}a+\frac{4}{17}$, $\frac{1}{119}a^{13}+\frac{2}{119}a^{10}-\frac{6}{119}a^{9}-\frac{22}{119}a^{8}+\frac{54}{119}a^{7}-\frac{3}{17}a^{6}+\frac{3}{119}a^{5}+\frac{45}{119}a^{4}-\frac{5}{17}a^{3}+\frac{6}{17}a^{2}-\frac{7}{17}a+\frac{6}{17}$, $\frac{1}{12\!\cdots\!45}a^{14}-\frac{2218816459322}{12\!\cdots\!45}a^{13}-\frac{177356103447}{184576521450335}a^{12}+\frac{315151008177}{184576521450335}a^{11}+\frac{52816190011}{1551063205465}a^{10}+\frac{1852946366367}{26368074492905}a^{9}+\frac{2189814917013}{36915304290067}a^{8}+\frac{309902078496286}{12\!\cdots\!45}a^{7}+\frac{62482364217810}{258407130030469}a^{6}-\frac{23885772550576}{184576521450335}a^{5}+\frac{77018898749431}{184576521450335}a^{4}+\frac{80804241405446}{184576521450335}a^{3}+\frac{1945164765426}{184576521450335}a^{2}+\frac{10095914468986}{26368074492905}a+\frac{71866310433}{5273614898581}$ 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

Trivial group, which has order $1$ (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:  $14$
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:   $\frac{8860188755869}{12\!\cdots\!45}a^{14}-\frac{44801333673708}{12\!\cdots\!45}a^{13}-\frac{6848935925869}{26368074492905}a^{12}+\frac{250432623765673}{184576521450335}a^{11}+\frac{601332255272851}{184576521450335}a^{10}-\frac{481145010089407}{26368074492905}a^{9}-\frac{82833459236243}{5273614898581}a^{8}+\frac{13\!\cdots\!24}{12\!\cdots\!45}a^{7}+\frac{68\!\cdots\!12}{258407130030469}a^{6}-\frac{46\!\cdots\!79}{184576521450335}a^{5}+\frac{244104972795162}{26368074492905}a^{4}+\frac{40\!\cdots\!34}{184576521450335}a^{3}-\frac{12\!\cdots\!66}{184576521450335}a^{2}-\frac{416137457996986}{26368074492905}a+\frac{23990060546419}{5273614898581}$, $\frac{2993452325042}{12\!\cdots\!45}a^{14}-\frac{10690701124359}{12\!\cdots\!45}a^{13}-\frac{18499313324874}{184576521450335}a^{12}+\frac{59412498196124}{184576521450335}a^{11}+\frac{283350955889418}{184576521450335}a^{10}-\frac{801637433145557}{184576521450335}a^{9}-\frac{370253181137451}{36915304290067}a^{8}+\frac{32\!\cdots\!57}{12\!\cdots\!45}a^{7}+\frac{68\!\cdots\!68}{258407130030469}a^{6}-\frac{745717306153221}{10857442438255}a^{5}-\frac{23\!\cdots\!98}{184576521450335}a^{4}+\frac{13\!\cdots\!57}{184576521450335}a^{3}-\frac{48\!\cdots\!83}{184576521450335}a^{2}-\frac{146276577522743}{26368074492905}a+\frac{8993317488332}{5273614898581}$, $\frac{7106259171114}{12\!\cdots\!45}a^{14}-\frac{32428844118893}{12\!\cdots\!45}a^{13}-\frac{40136352965253}{184576521450335}a^{12}+\frac{179350291292293}{184576521450335}a^{11}+\frac{544989424645006}{184576521450335}a^{10}-\frac{23\!\cdots\!44}{184576521450335}a^{9}-\frac{621956332543474}{36915304290067}a^{8}+\frac{92\!\cdots\!69}{12\!\cdots\!45}a^{7}+\frac{11\!\cdots\!31}{258407130030469}a^{6}-\frac{44\!\cdots\!07}{26368074492905}a^{5}-\frac{80\!\cdots\!91}{184576521450335}a^{4}+\frac{26\!\cdots\!44}{184576521450335}a^{3}-\frac{797341227955071}{184576521450335}a^{2}-\frac{443600087813446}{26368074492905}a+\frac{6800877506766}{5273614898581}$, $\frac{9310387079132}{12\!\cdots\!45}a^{14}-\frac{30001066890619}{12\!\cdots\!45}a^{13}-\frac{63062842209314}{184576521450335}a^{12}+\frac{179048540027554}{184576521450335}a^{11}+\frac{10\!\cdots\!23}{184576521450335}a^{10}-\frac{26\!\cdots\!32}{184576521450335}a^{9}-\frac{17\!\cdots\!55}{36915304290067}a^{8}+\frac{12\!\cdots\!72}{12\!\cdots\!45}a^{7}+\frac{42\!\cdots\!05}{258407130030469}a^{6}-\frac{77\!\cdots\!61}{26368074492905}a^{5}-\frac{39\!\cdots\!88}{184576521450335}a^{4}+\frac{58\!\cdots\!42}{184576521450335}a^{3}+\frac{887554898834197}{184576521450335}a^{2}-\frac{11\!\cdots\!88}{26368074492905}a+\frac{29394677864580}{5273614898581}$, $\frac{5574124374379}{12\!\cdots\!45}a^{14}-\frac{22997212568023}{12\!\cdots\!45}a^{13}-\frac{32886270205148}{184576521450335}a^{12}+\frac{126695810987988}{184576521450335}a^{11}+\frac{479386503089636}{184576521450335}a^{10}-\frac{16\!\cdots\!04}{184576521450335}a^{9}-\frac{614664604365972}{36915304290067}a^{8}+\frac{64\!\cdots\!09}{12\!\cdots\!45}a^{7}+\frac{13\!\cdots\!47}{258407130030469}a^{6}-\frac{30\!\cdots\!22}{26368074492905}a^{5}-\frac{707235745423288}{10857442438255}a^{4}+\frac{18\!\cdots\!69}{184576521450335}a^{3}+\frac{26\!\cdots\!14}{184576521450335}a^{2}-\frac{270898978216241}{26368074492905}a-\frac{4001286803686}{5273614898581}$, $\frac{30577845558557}{12\!\cdots\!45}a^{14}-\frac{151700900965029}{12\!\cdots\!45}a^{13}-\frac{168670029391699}{184576521450335}a^{12}+\frac{854835597203094}{184576521450335}a^{11}+\frac{21\!\cdots\!48}{184576521450335}a^{10}-\frac{11\!\cdots\!27}{184576521450335}a^{9}-\frac{22\!\cdots\!67}{36915304290067}a^{8}+\frac{47\!\cdots\!67}{12\!\cdots\!45}a^{7}+\frac{31\!\cdots\!68}{258407130030469}a^{6}-\frac{17\!\cdots\!97}{184576521450335}a^{5}-\frac{135059348840334}{26368074492905}a^{4}+\frac{15\!\cdots\!37}{184576521450335}a^{3}-\frac{27\!\cdots\!84}{10857442438255}a^{2}-\frac{25\!\cdots\!38}{26368074492905}a+\frac{170259675539466}{5273614898581}$, $\frac{3580487441767}{12\!\cdots\!45}a^{14}-\frac{20841691358969}{12\!\cdots\!45}a^{13}-\frac{991496643097}{10857442438255}a^{12}+\frac{112474438928839}{184576521450335}a^{11}+\frac{150485469040163}{184576521450335}a^{10}-\frac{14\!\cdots\!72}{184576521450335}a^{9}-\frac{13725403770976}{36915304290067}a^{8}+\frac{49\!\cdots\!52}{12\!\cdots\!45}a^{7}-\frac{40\!\cdots\!72}{258407130030469}a^{6}-\frac{14\!\cdots\!72}{184576521450335}a^{5}+\frac{77\!\cdots\!32}{184576521450335}a^{4}+\frac{472117082876946}{10857442438255}a^{3}-\frac{56\!\cdots\!18}{184576521450335}a^{2}+\frac{4410201668896}{1551063205465}a+\frac{2620337448686}{5273614898581}$, $\frac{56316452235018}{12\!\cdots\!45}a^{14}-\frac{262360049102686}{12\!\cdots\!45}a^{13}-\frac{45355094926638}{26368074492905}a^{12}+\frac{14\!\cdots\!86}{184576521450335}a^{11}+\frac{42\!\cdots\!52}{184576521450335}a^{10}-\frac{19\!\cdots\!48}{184576521450335}a^{9}-\frac{48\!\cdots\!22}{36915304290067}a^{8}+\frac{77\!\cdots\!48}{12\!\cdots\!45}a^{7}+\frac{83\!\cdots\!54}{258407130030469}a^{6}-\frac{27\!\cdots\!18}{184576521450335}a^{5}-\frac{45\!\cdots\!87}{184576521450335}a^{4}+\frac{24\!\cdots\!08}{184576521450335}a^{3}-\frac{39\!\cdots\!57}{184576521450335}a^{2}-\frac{45\!\cdots\!07}{26368074492905}a+\frac{163020768098499}{5273614898581}$, $\frac{17941197943087}{12\!\cdots\!45}a^{14}-\frac{81275147298239}{12\!\cdots\!45}a^{13}-\frac{102475314469154}{184576521450335}a^{12}+\frac{454468031817409}{184576521450335}a^{11}+\frac{14\!\cdots\!18}{184576521450335}a^{10}-\frac{876026014107836}{26368074492905}a^{9}-\frac{16\!\cdots\!78}{36915304290067}a^{8}+\frac{24\!\cdots\!02}{12\!\cdots\!45}a^{7}+\frac{27\!\cdots\!83}{258407130030469}a^{6}-\frac{88\!\cdots\!77}{184576521450335}a^{5}-\frac{11\!\cdots\!73}{184576521450335}a^{4}+\frac{47\!\cdots\!26}{10857442438255}a^{3}-\frac{19\!\cdots\!73}{184576521450335}a^{2}-\frac{14\!\cdots\!58}{26368074492905}a+\frac{84867875668676}{5273614898581}$, $\frac{32451081600076}{12\!\cdots\!45}a^{14}-\frac{157050813168227}{12\!\cdots\!45}a^{13}-\frac{178280236942452}{184576521450335}a^{12}+\frac{870523994545512}{184576521450335}a^{11}+\frac{23\!\cdots\!99}{184576521450335}a^{10}-\frac{11\!\cdots\!76}{184576521450335}a^{9}-\frac{340892574616260}{5273614898581}a^{8}+\frac{44\!\cdots\!31}{12\!\cdots\!45}a^{7}+\frac{35\!\cdots\!50}{258407130030469}a^{6}-\frac{21\!\cdots\!78}{26368074492905}a^{5}-\frac{13\!\cdots\!34}{184576521450335}a^{4}+\frac{12\!\cdots\!91}{184576521450335}a^{3}-\frac{23\!\cdots\!59}{184576521450335}a^{2}-\frac{22\!\cdots\!04}{26368074492905}a+\frac{4976928621887}{310212641093}$, $\frac{31539747232986}{12\!\cdots\!45}a^{14}-\frac{136366292020622}{12\!\cdots\!45}a^{13}-\frac{186670663546982}{184576521450335}a^{12}+\frac{770319682745722}{184576521450335}a^{11}+\frac{27\!\cdots\!24}{184576521450335}a^{10}-\frac{10\!\cdots\!76}{184576521450335}a^{9}-\frac{35\!\cdots\!04}{36915304290067}a^{8}+\frac{43\!\cdots\!96}{12\!\cdots\!45}a^{7}+\frac{43\!\cdots\!58}{15200419413557}a^{6}-\frac{15\!\cdots\!86}{184576521450335}a^{5}-\frac{85\!\cdots\!87}{26368074492905}a^{4}+\frac{14\!\cdots\!56}{184576521450335}a^{3}-\frac{10\!\cdots\!59}{184576521450335}a^{2}-\frac{23\!\cdots\!39}{26368074492905}a+\frac{74485465745621}{5273614898581}$, $\frac{1173993847185}{258407130030469}a^{14}-\frac{7238718787151}{258407130030469}a^{13}-\frac{326946490583}{2171488487651}a^{12}+\frac{40787239839035}{36915304290067}a^{11}+\frac{48926510228476}{36915304290067}a^{10}-\frac{79352686543993}{5273614898581}a^{9}+\frac{22836921123962}{36915304290067}a^{8}+\frac{22\!\cdots\!60}{258407130030469}a^{7}-\frac{11\!\cdots\!31}{258407130030469}a^{6}-\frac{82\!\cdots\!01}{36915304290067}a^{5}+\frac{55\!\cdots\!28}{36915304290067}a^{4}+\frac{79\!\cdots\!03}{36915304290067}a^{3}-\frac{60\!\cdots\!52}{36915304290067}a^{2}-\frac{8291514362864}{310212641093}a+\frac{100941394271309}{5273614898581}$, $\frac{617457802712}{76002097067785}a^{14}-\frac{46947970999348}{12\!\cdots\!45}a^{13}-\frac{61403084852468}{184576521450335}a^{12}+\frac{266148443864453}{184576521450335}a^{11}+\frac{52001305324848}{10857442438255}a^{10}-\frac{524230396548812}{26368074492905}a^{9}-\frac{11\!\cdots\!24}{36915304290067}a^{8}+\frac{89\!\cdots\!27}{76002097067785}a^{7}+\frac{22\!\cdots\!40}{258407130030469}a^{6}-\frac{56\!\cdots\!54}{184576521450335}a^{5}-\frac{17\!\cdots\!81}{184576521450335}a^{4}+\frac{52\!\cdots\!44}{184576521450335}a^{3}-\frac{44\!\cdots\!26}{184576521450335}a^{2}-\frac{52534342483988}{1551063205465}a+\frac{28351664072897}{5273614898581}$, $\frac{2930282039421}{184576521450335}a^{14}-\frac{7259032451877}{184576521450335}a^{13}-\frac{131219702129219}{184576521450335}a^{12}+\frac{245976462916299}{184576521450335}a^{11}+\frac{122937636906209}{10857442438255}a^{10}-\frac{150991487737331}{10857442438255}a^{9}-\frac{407525861780515}{5273614898581}a^{8}+\frac{91\!\cdots\!21}{184576521450335}a^{7}+\frac{81\!\cdots\!62}{36915304290067}a^{6}-\frac{96\!\cdots\!47}{184576521450335}a^{5}-\frac{35\!\cdots\!68}{184576521450335}a^{4}+\frac{12\!\cdots\!41}{26368074492905}a^{3}+\frac{730350804972626}{26368074492905}a^{2}-\frac{140126674088748}{26368074492905}a+\frac{74129477103}{5273614898581}$ 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:  \( 4903571851.42 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{15}\cdot(2\pi)^{0}\cdot 4903571851.42 \cdot 1}{2\cdot\sqrt{11621759510846642583679213009}}\cr\approx \mathstrut & 0.745240657747 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 38*x^13 + 196*x^12 + 476*x^11 - 2646*x^10 - 2219*x^9 + 15116*x^8 + 2757*x^7 - 37522*x^6 + 6328*x^5 + 33201*x^4 - 16184*x^3 - 2037*x^2 + 2058*x - 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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^15 - 5*x^14 - 38*x^13 + 196*x^12 + 476*x^11 - 2646*x^10 - 2219*x^9 + 15116*x^8 + 2757*x^7 - 37522*x^6 + 6328*x^5 + 33201*x^4 - 16184*x^3 - 2037*x^2 + 2058*x - 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^15 - 5*x^14 - 38*x^13 + 196*x^12 + 476*x^11 - 2646*x^10 - 2219*x^9 + 15116*x^8 + 2757*x^7 - 37522*x^6 + 6328*x^5 + 33201*x^4 - 16184*x^3 - 2037*x^2 + 2058*x - 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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 5*x^14 - 38*x^13 + 196*x^12 + 476*x^11 - 2646*x^10 - 2219*x^9 + 15116*x^8 + 2757*x^7 - 37522*x^6 + 6328*x^5 + 33201*x^4 - 16184*x^3 - 2037*x^2 + 2058*x - 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

$A_6$ (as 15T20):

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 non-solvable group of order 360
The 7 conjugacy class representatives for $A_6$
Character table for $A_6$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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 6 siblings: 6.6.15400609258321.1, 6.6.192293689.1
Degree 10 sibling: 10.10.60437550349593857881.1
Degree 15 sibling: deg 15
Degree 20 sibling: deg 20
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 sibling: data not computed
Degree 45 sibling: data not computed
Minimal sibling: 6.6.192293689.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.5.0.1}{5} }^{3}$ ${\href{/padicField/3.4.0.1}{4} }^{3}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.4.0.1}{4} }^{3}{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ R ${\href{/padicField/11.5.0.1}{5} }^{3}$ ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.3.0.1}{3} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ ${\href{/padicField/19.5.0.1}{5} }^{3}$ ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.3.0.1}{3} }^{5}$ ${\href{/padicField/41.5.0.1}{5} }^{3}$ ${\href{/padicField/43.3.0.1}{3} }^{5}$ ${\href{/padicField/47.5.0.1}{5} }^{3}$ ${\href{/padicField/53.2.0.1}{2} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ ${\href{/padicField/59.5.0.1}{5} }^{3}$

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
\(7\) Copy content Toggle raw display $\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.3.1$x^{4} + 7$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.8.6.2$x^{8} + 24 x^{7} + 228 x^{6} + 1080 x^{5} + 2660 x^{4} + 3408 x^{3} + 3312 x^{2} + 5184 x + 6304$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
\(283\) Copy content Toggle raw display Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$3$$1$$2$
Deg $9$$3$$3$$6$