Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 6*x^19 + 8*x^18 + 15*x^17 - 28*x^16 - 39*x^15 + 74*x^14 + 112*x^13 - 127*x^12 - 227*x^11 + 89*x^10 + 306*x^9 + x^8 - 244*x^7 - 55*x^6 + 122*x^5 + 40*x^4 - 41*x^3 - 6*x^2 + 8*x - 1)
gp: K = bnfinit(y^21 - y^20 - 6*y^19 + 8*y^18 + 15*y^17 - 28*y^16 - 39*y^15 + 74*y^14 + 112*y^13 - 127*y^12 - 227*y^11 + 89*y^10 + 306*y^9 + y^8 - 244*y^7 - 55*y^6 + 122*y^5 + 40*y^4 - 41*y^3 - 6*y^2 + 8*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - x^20 - 6*x^19 + 8*x^18 + 15*x^17 - 28*x^16 - 39*x^15 + 74*x^14 + 112*x^13 - 127*x^12 - 227*x^11 + 89*x^10 + 306*x^9 + x^8 - 244*x^7 - 55*x^6 + 122*x^5 + 40*x^4 - 41*x^3 - 6*x^2 + 8*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - x^20 - 6*x^19 + 8*x^18 + 15*x^17 - 28*x^16 - 39*x^15 + 74*x^14 + 112*x^13 - 127*x^12 - 227*x^11 + 89*x^10 + 306*x^9 + x^8 - 244*x^7 - 55*x^6 + 122*x^5 + 40*x^4 - 41*x^3 - 6*x^2 + 8*x - 1)
\( x^{21} - x^{20} - 6 x^{19} + 8 x^{18} + 15 x^{17} - 28 x^{16} - 39 x^{15} + 74 x^{14} + 112 x^{13} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[5, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(16253945603436050603615041\)
\(\medspace = 13^{8}\cdot 109^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.87\) | 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: | $13^{1/2}109^{1/2}\approx 37.64306044943742$ | ||
| Ramified primes: |
\(13\), \(109\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{13}a^{17}+\frac{3}{13}a^{16}-\frac{1}{13}a^{15}+\frac{2}{13}a^{14}-\frac{3}{13}a^{13}-\frac{5}{13}a^{12}-\frac{1}{13}a^{11}-\frac{2}{13}a^{10}+\frac{6}{13}a^{8}+\frac{5}{13}a^{7}-\frac{5}{13}a^{5}+\frac{4}{13}a^{4}-\frac{6}{13}a^{3}+\frac{6}{13}a^{2}-\frac{1}{13}a+\frac{1}{13}$, $\frac{1}{13}a^{18}+\frac{3}{13}a^{16}+\frac{5}{13}a^{15}+\frac{4}{13}a^{14}+\frac{4}{13}a^{13}+\frac{1}{13}a^{12}+\frac{1}{13}a^{11}+\frac{6}{13}a^{10}+\frac{6}{13}a^{9}-\frac{2}{13}a^{7}-\frac{5}{13}a^{6}+\frac{6}{13}a^{5}-\frac{5}{13}a^{4}-\frac{2}{13}a^{3}-\frac{6}{13}a^{2}+\frac{4}{13}a-\frac{3}{13}$, $\frac{1}{13}a^{19}-\frac{4}{13}a^{16}-\frac{6}{13}a^{15}-\frac{2}{13}a^{14}-\frac{3}{13}a^{13}+\frac{3}{13}a^{12}-\frac{4}{13}a^{11}-\frac{1}{13}a^{10}+\frac{6}{13}a^{8}+\frac{6}{13}a^{7}+\frac{6}{13}a^{6}-\frac{3}{13}a^{5}-\frac{1}{13}a^{4}-\frac{1}{13}a^{3}-\frac{1}{13}a^{2}-\frac{3}{13}$, $\frac{1}{54746195699}a^{20}-\frac{1527135170}{54746195699}a^{19}-\frac{1923090249}{54746195699}a^{18}+\frac{600428357}{54746195699}a^{17}+\frac{3509160196}{54746195699}a^{16}-\frac{1994377962}{4211245823}a^{15}+\frac{1969036218}{54746195699}a^{14}-\frac{16609214244}{54746195699}a^{13}-\frac{4648215350}{54746195699}a^{12}+\frac{20830433798}{54746195699}a^{11}-\frac{26314124665}{54746195699}a^{10}-\frac{23928899390}{54746195699}a^{9}+\frac{21131991303}{54746195699}a^{8}-\frac{20467156920}{54746195699}a^{7}-\frac{26186883091}{54746195699}a^{6}+\frac{4504342602}{54746195699}a^{5}-\frac{1048801418}{54746195699}a^{4}-\frac{1285799224}{54746195699}a^{3}+\frac{19159610627}{54746195699}a^{2}-\frac{9682737660}{54746195699}a+\frac{7063160961}{54746195699}$
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: | $12$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{1436288338}{2881378721}a^{20}-\frac{2583734640}{2881378721}a^{19}-\frac{6344711817}{2881378721}a^{18}+\frac{16486790530}{2881378721}a^{17}+\frac{6833186837}{2881378721}a^{16}-\frac{3445660380}{221644517}a^{15}-\frac{15334744072}{2881378721}a^{14}+\frac{114300253386}{2881378721}a^{13}+\frac{56513315863}{2881378721}a^{12}-\frac{16619896746}{221644517}a^{11}-\frac{116645400979}{2881378721}a^{10}+\frac{208605650551}{2881378721}a^{9}+\frac{198871887986}{2881378721}a^{8}-\frac{176463246462}{2881378721}a^{7}-\frac{128349622166}{2881378721}a^{6}+\frac{84468496206}{2881378721}a^{5}+\frac{71076842401}{2881378721}a^{4}-\frac{50872289791}{2881378721}a^{3}-\frac{17789353872}{2881378721}a^{2}+\frac{26662614564}{2881378721}a-\frac{6003513695}{2881378721}$, $\frac{5219280887}{4211245823}a^{20}-\frac{6602455450}{4211245823}a^{19}-\frac{363983969274}{54746195699}a^{18}+\frac{608767002225}{54746195699}a^{17}+\frac{756061133002}{54746195699}a^{16}-\frac{1890965491045}{54746195699}a^{15}-\frac{1981314090030}{54746195699}a^{14}+\frac{4929145314391}{54746195699}a^{13}+\frac{5900259409289}{54746195699}a^{12}-\frac{8610414450607}{54746195699}a^{11}-\frac{11885026846565}{54746195699}a^{10}+\frac{6324495800878}{54746195699}a^{9}+\frac{16600745087151}{54746195699}a^{8}-\frac{1703885640712}{54746195699}a^{7}-\frac{12364545917462}{54746195699}a^{6}-\frac{2273235945348}{54746195699}a^{5}+\frac{6019035589082}{54746195699}a^{4}+\frac{1551941837825}{54746195699}a^{3}-\frac{1643900889206}{54746195699}a^{2}-\frac{70877906727}{54746195699}a+\frac{128021192378}{54746195699}$, $\frac{24516199875}{54746195699}a^{20}-\frac{30288707539}{54746195699}a^{19}-\frac{123770576550}{54746195699}a^{18}+\frac{209770263988}{54746195699}a^{17}+\frac{227370721113}{54746195699}a^{16}-\frac{619033304621}{54746195699}a^{15}-\frac{603778632979}{54746195699}a^{14}+\frac{1555708049419}{54746195699}a^{13}+\frac{1837471385783}{54746195699}a^{12}-\frac{2537367028030}{54746195699}a^{11}-\frac{3387035807081}{54746195699}a^{10}+\frac{1409252808541}{54746195699}a^{9}+\frac{4202851479689}{54746195699}a^{8}-\frac{261381308089}{54746195699}a^{7}-\frac{2391682169958}{54746195699}a^{6}-\frac{376647364746}{54746195699}a^{5}+\frac{819813718141}{54746195699}a^{4}+\frac{22810656255}{54746195699}a^{3}-\frac{211410583860}{54746195699}a^{2}+\frac{234410218550}{54746195699}a+\frac{6912801615}{54746195699}$, $\frac{1731707363}{4211245823}a^{20}-\frac{55919220441}{54746195699}a^{19}-\frac{5857437001}{4211245823}a^{18}+\frac{338118371998}{54746195699}a^{17}-\frac{50773445258}{54746195699}a^{16}-\frac{846017082554}{54746195699}a^{15}+\frac{228489499466}{54746195699}a^{14}+\frac{2141746874325}{54746195699}a^{13}-\frac{330519206562}{54746195699}a^{12}-\frac{4483932940493}{54746195699}a^{11}+\frac{378572098136}{54746195699}a^{10}+\frac{428782684613}{4211245823}a^{9}+\frac{1333609094819}{54746195699}a^{8}-\frac{6083302859325}{54746195699}a^{7}-\frac{1265288923168}{54746195699}a^{6}+\frac{3484151894962}{54746195699}a^{5}+\frac{1167928373426}{54746195699}a^{4}-\frac{1673624686843}{54746195699}a^{3}-\frac{219146642854}{54746195699}a^{2}+\frac{624287605431}{54746195699}a-\frac{215546707133}{54746195699}$, $\frac{55083243131}{54746195699}a^{20}-\frac{115522831107}{54746195699}a^{19}-\frac{219863666910}{54746195699}a^{18}+\frac{722947003429}{54746195699}a^{17}+\frac{7687681311}{4211245823}a^{16}-\frac{1908235156843}{54746195699}a^{15}-\frac{80425044585}{54746195699}a^{14}+\frac{4853942810724}{54746195699}a^{13}+\frac{809608560937}{54746195699}a^{12}-\frac{9682376479017}{54746195699}a^{11}-\frac{2031902460181}{54746195699}a^{10}+\frac{10984630090244}{54746195699}a^{9}+\frac{5708002736422}{54746195699}a^{8}-\frac{10948227335795}{54746195699}a^{7}-\frac{4366483363148}{54746195699}a^{6}+\frac{5898425248330}{54746195699}a^{5}+\frac{3094112123551}{54746195699}a^{4}-\frac{3076919862116}{54746195699}a^{3}-\frac{734757904974}{54746195699}a^{2}+\frac{1152952154977}{54746195699}a-\frac{18992671337}{4211245823}$, $\frac{791292600}{2881378721}a^{20}-\frac{2068131972}{2881378721}a^{19}-\frac{1830566871}{2881378721}a^{18}+\frac{11425002934}{2881378721}a^{17}-\frac{5844765794}{2881378721}a^{16}-\frac{24678616746}{2881378721}a^{15}+\frac{14979096540}{2881378721}a^{14}+\frac{62922285659}{2881378721}a^{13}-\frac{32146029151}{2881378721}a^{12}-\frac{129196805457}{2881378721}a^{11}+\frac{5459009164}{221644517}a^{10}+\frac{156438220534}{2881378721}a^{9}-\frac{41926428053}{2881378721}a^{8}-\frac{227916351773}{2881378721}a^{7}+\frac{59432094994}{2881378721}a^{6}+\frac{161037514988}{2881378721}a^{5}-\frac{2902760430}{2881378721}a^{4}-\frac{106212786621}{2881378721}a^{3}+\frac{5130579081}{2881378721}a^{2}+\frac{40629581847}{2881378721}a-\frac{12969069962}{2881378721}$, $\frac{20967304845}{54746195699}a^{20}-\frac{23761340004}{54746195699}a^{19}-\frac{8585265139}{4211245823}a^{18}+\frac{157450354808}{54746195699}a^{17}+\frac{248239222645}{54746195699}a^{16}-\frac{460427065536}{54746195699}a^{15}-\frac{746442206522}{54746195699}a^{14}+\frac{1238095428573}{54746195699}a^{13}+\frac{2229262218896}{54746195699}a^{12}-\frac{1972385246053}{54746195699}a^{11}-\frac{4383838280064}{54746195699}a^{10}+\frac{433142433214}{54746195699}a^{9}+\frac{5890592843318}{54746195699}a^{8}+\frac{1415125335517}{54746195699}a^{7}-\frac{3996771866645}{54746195699}a^{6}-\frac{2156026257414}{54746195699}a^{5}+\frac{1684164423444}{54746195699}a^{4}+\frac{1129714851299}{54746195699}a^{3}-\frac{504142936391}{54746195699}a^{2}-\frac{58591948970}{54746195699}a+\frac{78596660910}{54746195699}$, $\frac{6003513695}{2881378721}a^{20}-\frac{7439802033}{2881378721}a^{19}-\frac{33437347530}{2881378721}a^{18}+\frac{54372821377}{2881378721}a^{17}+\frac{73565914895}{2881378721}a^{16}-\frac{174931570297}{2881378721}a^{15}-\frac{14564880705}{221644517}a^{14}+\frac{459594757502}{2881378721}a^{13}+\frac{558093280454}{2881378721}a^{12}-\frac{62996888856}{221644517}a^{11}-\frac{1146738951067}{2881378721}a^{10}+\frac{650958119834}{2881378721}a^{9}+\frac{1628469540119}{2881378721}a^{8}-\frac{192868374291}{2881378721}a^{7}-\frac{99107238086}{221644517}a^{6}-\frac{201843631059}{2881378721}a^{5}+\frac{647960174584}{2881378721}a^{4}+\frac{169063705399}{2881378721}a^{3}-\frac{195271771704}{2881378721}a^{2}-\frac{18231728298}{2881378721}a+\frac{21365494996}{2881378721}$, $\frac{17515436720}{54746195699}a^{20}-\frac{5607738444}{54746195699}a^{19}-\frac{122907827240}{54746195699}a^{18}+\frac{79889619632}{54746195699}a^{17}+\frac{379825085600}{54746195699}a^{16}-\frac{375444397123}{54746195699}a^{15}-\frac{1025851822257}{54746195699}a^{14}+\frac{978337400805}{54746195699}a^{13}+\frac{2889401889757}{54746195699}a^{12}-\frac{1274524413026}{54746195699}a^{11}-\frac{5685157709758}{54746195699}a^{10}-\frac{444222744335}{54746195699}a^{9}+\frac{6758133881191}{54746195699}a^{8}+\frac{3161411627281}{54746195699}a^{7}-\frac{4887849022127}{54746195699}a^{6}-\frac{3375500602956}{54746195699}a^{5}+\frac{1751875839615}{54746195699}a^{4}+\frac{1948426865378}{54746195699}a^{3}-\frac{478703572014}{54746195699}a^{2}-\frac{476590657443}{54746195699}a+\frac{142595590679}{54746195699}$, $\frac{16039534671}{54746195699}a^{20}-\frac{21477798333}{54746195699}a^{19}-\frac{69641249434}{54746195699}a^{18}+\frac{132015978597}{54746195699}a^{17}+\frac{83342545162}{54746195699}a^{16}-\frac{332233979049}{54746195699}a^{15}-\frac{236556181347}{54746195699}a^{14}+\frac{792760040646}{54746195699}a^{13}+\frac{59229119504}{4211245823}a^{12}-\frac{1034472976761}{54746195699}a^{11}-\frac{1083959623109}{54746195699}a^{10}-\frac{207897002580}{54746195699}a^{9}+\frac{520383376221}{54746195699}a^{8}+\frac{431165857747}{54746195699}a^{7}+\frac{1716419799463}{54746195699}a^{6}+\frac{110385157709}{54746195699}a^{5}-\frac{1735203725868}{54746195699}a^{4}-\frac{1075373161383}{54746195699}a^{3}+\frac{839477951328}{54746195699}a^{2}+\frac{677331550041}{54746195699}a-\frac{167677306478}{54746195699}$, $\frac{22672027987}{54746195699}a^{20}-\frac{24438493509}{54746195699}a^{19}-\frac{122894069394}{54746195699}a^{18}+\frac{176118022705}{54746195699}a^{17}+\frac{270019225690}{54746195699}a^{16}-\frac{551563690231}{54746195699}a^{15}-\frac{57055572757}{4211245823}a^{14}+\frac{1423869436910}{54746195699}a^{13}+\frac{2176312132510}{54746195699}a^{12}-\frac{2267799676602}{54746195699}a^{11}-\frac{4197919364940}{54746195699}a^{10}+\frac{993792121514}{54746195699}a^{9}+\frac{5387728776110}{54746195699}a^{8}+\frac{652228893624}{54746195699}a^{7}-\frac{3530945691963}{54746195699}a^{6}-\frac{1515536712290}{54746195699}a^{5}+\frac{102047291193}{4211245823}a^{4}+\frac{715258651396}{54746195699}a^{3}-\frac{241917586272}{54746195699}a^{2}+\frac{45525132061}{54746195699}a-\frac{43440530993}{54746195699}$, $\frac{16176974161}{54746195699}a^{20}-\frac{40135508713}{54746195699}a^{19}-\frac{60891978856}{54746195699}a^{18}+\frac{246227477445}{54746195699}a^{17}+\frac{5546812512}{54746195699}a^{16}-\frac{653051785489}{54746195699}a^{15}+\frac{4296988760}{4211245823}a^{14}+\frac{1735346980028}{54746195699}a^{13}-\frac{19203071792}{54746195699}a^{12}-\frac{3675472807060}{54746195699}a^{11}-\frac{336296593197}{54746195699}a^{10}+\frac{4680025062055}{54746195699}a^{9}+\frac{2085447368343}{54746195699}a^{8}-\frac{374355885047}{4211245823}a^{7}-\frac{1905414674801}{54746195699}a^{6}+\frac{2578013353012}{54746195699}a^{5}+\frac{1416082419916}{54746195699}a^{4}-\frac{1377069324963}{54746195699}a^{3}-\frac{320884197618}{54746195699}a^{2}+\frac{541035195732}{54746195699}a-\frac{11472905855}{4211245823}$
| 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: | \( 17240.5464964 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{8}\cdot 17240.5464964 \cdot 1}{2\cdot\sqrt{16253945603436050603615041}}\cr\approx \mathstrut & 0.166199822890 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 6*x^19 + 8*x^18 + 15*x^17 - 28*x^16 - 39*x^15 + 74*x^14 + 112*x^13 - 127*x^12 - 227*x^11 + 89*x^10 + 306*x^9 + x^8 - 244*x^7 - 55*x^6 + 122*x^5 + 40*x^4 - 41*x^3 - 6*x^2 + 8*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))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^21 - x^20 - 6*x^19 + 8*x^18 + 15*x^17 - 28*x^16 - 39*x^15 + 74*x^14 + 112*x^13 - 127*x^12 - 227*x^11 + 89*x^10 + 306*x^9 + x^8 - 244*x^7 - 55*x^6 + 122*x^5 + 40*x^4 - 41*x^3 - 6*x^2 + 8*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)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^21 - x^20 - 6*x^19 + 8*x^18 + 15*x^17 - 28*x^16 - 39*x^15 + 74*x^14 + 112*x^13 - 127*x^12 - 227*x^11 + 89*x^10 + 306*x^9 + x^8 - 244*x^7 - 55*x^6 + 122*x^5 + 40*x^4 - 41*x^3 - 6*x^2 + 8*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)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - x^20 - 6*x^19 + 8*x^18 + 15*x^17 - 28*x^16 - 39*x^15 + 74*x^14 + 112*x^13 - 127*x^12 - 227*x^11 + 89*x^10 + 306*x^9 + x^8 - 244*x^7 - 55*x^6 + 122*x^5 + 40*x^4 - 41*x^3 - 6*x^2 + 8*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
$\GL(3,2)$ (as 21T14):
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 168 |
| The 6 conjugacy class representatives for $\PSL(2,7)$ |
| Character table for $\PSL(2,7)$ |
Intermediate fields
| 7.3.2007889.1, 7.3.2007889.2 |
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 7 siblings: | 7.3.2007889.1, 7.3.2007889.2 |
| Degree 8 sibling: | 8.0.4031618236321.1 |
| Degree 14 siblings: | deg 14, deg 14 |
| Degree 24 sibling: | data not computed |
| Degree 28 sibling: | data not computed |
| Degree 42 siblings: | data not computed |
| Minimal sibling: | 7.3.2007889.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.7.0.1}{7} }^{3}$ | ${\href{/padicField/3.7.0.1}{7} }^{3}$ | ${\href{/padicField/5.7.0.1}{7} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{7}$ | ${\href{/padicField/11.3.0.1}{3} }^{7}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{7}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.7.0.1}{7} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{7}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.3.0.1}{3} }^{7}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.7.0.1}{7} }^{3}$ | ${\href{/padicField/47.7.0.1}{7} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
|
\(109\)
| $\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 109.2.1.2 | $x^{2} + 218$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 109.2.1.1 | $x^{2} + 109$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 109.2.0.1 | $x^{2} + 108 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 109.4.2.1 | $x^{4} + 14604 x^{3} + 54096386 x^{2} + 5674982964 x + 401153281$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 109.4.2.1 | $x^{4} + 14604 x^{3} + 54096386 x^{2} + 5674982964 x + 401153281$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 109.4.2.1 | $x^{4} + 14604 x^{3} + 54096386 x^{2} + 5674982964 x + 401153281$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |