Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - x^18 + 14*x^17 - 21*x^16 + 20*x^15 - 6*x^14 - 91*x^13 + 224*x^12 - 130*x^11 - 184*x^10 + 330*x^9 - 224*x^8 + 85*x^7 + 27*x^6 - 99*x^5 + 81*x^4 - 13*x^3 - 16*x^2 + 8*x - 1)
gp: K = bnfinit(y^20 - 3*y^19 - y^18 + 14*y^17 - 21*y^16 + 20*y^15 - 6*y^14 - 91*y^13 + 224*y^12 - 130*y^11 - 184*y^10 + 330*y^9 - 224*y^8 + 85*y^7 + 27*y^6 - 99*y^5 + 81*y^4 - 13*y^3 - 16*y^2 + 8*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 3*x^19 - x^18 + 14*x^17 - 21*x^16 + 20*x^15 - 6*x^14 - 91*x^13 + 224*x^12 - 130*x^11 - 184*x^10 + 330*x^9 - 224*x^8 + 85*x^7 + 27*x^6 - 99*x^5 + 81*x^4 - 13*x^3 - 16*x^2 + 8*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 3*x^19 - x^18 + 14*x^17 - 21*x^16 + 20*x^15 - 6*x^14 - 91*x^13 + 224*x^12 - 130*x^11 - 184*x^10 + 330*x^9 - 224*x^8 + 85*x^7 + 27*x^6 - 99*x^5 + 81*x^4 - 13*x^3 - 16*x^2 + 8*x - 1)
\( x^{20} - 3 x^{19} - x^{18} + 14 x^{17} - 21 x^{16} + 20 x^{15} - 6 x^{14} - 91 x^{13} + 224 x^{12} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[8, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(11149939773041205947265625\)
\(\medspace = 5^{10}\cdot 97^{2}\cdot 3319^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.88\) | 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: | $5^{1/2}97^{1/2}3319^{1/2}\approx 1268.7454433415712$ | ||
| Ramified primes: |
\(5\), \(97\), \(3319\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{57}a^{17}+\frac{1}{57}a^{16}+\frac{22}{57}a^{15}+\frac{6}{19}a^{14}+\frac{5}{57}a^{13}-\frac{1}{19}a^{12}+\frac{28}{57}a^{11}-\frac{1}{3}a^{10}+\frac{20}{57}a^{9}+\frac{11}{57}a^{8}+\frac{4}{19}a^{7}-\frac{10}{57}a^{6}+\frac{3}{19}a^{5}-\frac{13}{57}a^{4}-\frac{2}{57}a^{3}-\frac{9}{19}a^{2}+\frac{20}{57}a+\frac{7}{57}$, $\frac{1}{57}a^{18}+\frac{2}{57}a^{16}-\frac{4}{57}a^{15}+\frac{25}{57}a^{14}+\frac{11}{57}a^{13}-\frac{26}{57}a^{12}+\frac{10}{57}a^{11}+\frac{20}{57}a^{10}-\frac{28}{57}a^{9}-\frac{6}{19}a^{8}+\frac{16}{57}a^{7}-\frac{1}{3}a^{6}-\frac{22}{57}a^{5}+\frac{11}{57}a^{4}-\frac{2}{19}a^{3}+\frac{3}{19}a^{2}+\frac{25}{57}a-\frac{26}{57}$, $\frac{1}{9805336707}a^{19}-\frac{80420005}{9805336707}a^{18}-\frac{14581244}{9805336707}a^{17}+\frac{1562594411}{9805336707}a^{16}-\frac{460940291}{9805336707}a^{15}-\frac{3938310281}{9805336707}a^{14}-\frac{380621266}{3268445569}a^{13}-\frac{669215217}{3268445569}a^{12}-\frac{1133177172}{3268445569}a^{11}+\frac{3553441687}{9805336707}a^{10}+\frac{1981420484}{9805336707}a^{9}+\frac{70410017}{516070353}a^{8}-\frac{1865314331}{9805336707}a^{7}-\frac{1281949136}{9805336707}a^{6}-\frac{459536672}{3268445569}a^{5}+\frac{2443166669}{9805336707}a^{4}+\frac{3910008680}{9805336707}a^{3}+\frac{3602786455}{9805336707}a^{2}-\frac{3743484014}{9805336707}a+\frac{4010606353}{9805336707}$
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: | $13$ | 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: |
$\frac{1868225110}{172023451}a^{19}-\frac{4652286946}{172023451}a^{18}-\frac{4236148280}{172023451}a^{17}+\frac{23995445440}{172023451}a^{16}-\frac{27040667823}{172023451}a^{15}+\frac{23626653274}{172023451}a^{14}+\frac{919509232}{172023451}a^{13}-\frac{169722767995}{172023451}a^{12}+\frac{332193448276}{172023451}a^{11}-\frac{73998791644}{172023451}a^{10}-\frac{381622025928}{172023451}a^{9}+\frac{424203362988}{172023451}a^{8}-\frac{204730630561}{172023451}a^{7}+\frac{53813487349}{172023451}a^{6}+\frac{80669072150}{172023451}a^{5}-\frac{146243058099}{172023451}a^{4}+\frac{78547628994}{172023451}a^{3}+\frac{15090909657}{172023451}a^{2}-\frac{22670901702}{172023451}a+\frac{3891917619}{172023451}$, $\frac{32660771635}{9805336707}a^{19}-\frac{76074163031}{9805336707}a^{18}-\frac{84921400208}{9805336707}a^{17}+\frac{402754803770}{9805336707}a^{16}-\frac{412009445491}{9805336707}a^{15}+\frac{363104522918}{9805336707}a^{14}+\frac{60209263820}{9805336707}a^{13}-\frac{2943414993748}{9805336707}a^{12}+\frac{5335010545187}{9805336707}a^{11}-\frac{558726297494}{9805336707}a^{10}-\frac{6554927852143}{9805336707}a^{9}+\frac{6371292446254}{9805336707}a^{8}-\frac{2809664531155}{9805336707}a^{7}+\frac{228414887351}{3268445569}a^{6}+\frac{1422332937043}{9805336707}a^{5}-\frac{756595407284}{3268445569}a^{4}+\frac{352053937815}{3268445569}a^{3}+\frac{355871325032}{9805336707}a^{2}-\frac{16506959338}{516070353}a+\frac{37023936965}{9805336707}$, $\frac{2343552097}{9805336707}a^{19}-\frac{11597950103}{9805336707}a^{18}+\frac{2528549286}{3268445569}a^{17}+\frac{46797399055}{9805336707}a^{16}-\frac{104231051633}{9805336707}a^{15}+\frac{93419108777}{9805336707}a^{14}-\frac{17376161607}{3268445569}a^{13}-\frac{228921013213}{9805336707}a^{12}+\frac{936443073880}{9805336707}a^{11}-\frac{982820325628}{9805336707}a^{10}-\frac{169540849114}{3268445569}a^{9}+\frac{1728014595617}{9805336707}a^{8}-\frac{1208648501539}{9805336707}a^{7}+\frac{392125921933}{9805336707}a^{6}+\frac{2719370533}{9805336707}a^{5}-\frac{435023782172}{9805336707}a^{4}+\frac{479463224723}{9805336707}a^{3}-\frac{104985453733}{9805336707}a^{2}-\frac{36329263789}{3268445569}a+\frac{33282383602}{9805336707}$, $\frac{172017781387}{9805336707}a^{19}-\frac{426017436974}{9805336707}a^{18}-\frac{395297292700}{9805336707}a^{17}+\frac{2201749660025}{9805336707}a^{16}-\frac{819641672651}{3268445569}a^{15}+\frac{2151117657610}{9805336707}a^{14}+\frac{95052037754}{9805336707}a^{13}-\frac{15606916269946}{9805336707}a^{12}+\frac{10121085087596}{3268445569}a^{11}-\frac{6450407473873}{9805336707}a^{10}-\frac{11682895728697}{3268445569}a^{9}+\frac{38408024630465}{9805336707}a^{8}-\frac{968874081236}{516070353}a^{7}+\frac{1665262884084}{3268445569}a^{6}+\frac{7253476418581}{9805336707}a^{5}-\frac{13251050047180}{9805336707}a^{4}+\frac{7029208295795}{9805336707}a^{3}+\frac{473494097507}{3268445569}a^{2}-\frac{665668470556}{3268445569}a+\frac{332921442905}{9805336707}$, $\frac{13766649052}{3268445569}a^{19}-\frac{112582352671}{9805336707}a^{18}-\frac{24253025787}{3268445569}a^{17}+\frac{559212756589}{9805336707}a^{16}-\frac{711649044170}{9805336707}a^{15}+\frac{624982786199}{9805336707}a^{14}-\frac{73554168101}{9805336707}a^{13}-\frac{3781473185713}{9805336707}a^{12}+\frac{8209581441275}{9805336707}a^{11}-\frac{3066449659841}{9805336707}a^{10}-\frac{8500436495621}{9805336707}a^{9}+\frac{3751633324062}{3268445569}a^{8}-\frac{6056041336213}{9805336707}a^{7}+\frac{1806807640318}{9805336707}a^{6}+\frac{1612755772684}{9805336707}a^{5}-\frac{3638182251350}{9805336707}a^{4}+\frac{769782363022}{3268445569}a^{3}+\frac{43410758204}{3268445569}a^{2}-\frac{615562973821}{9805336707}a+\frac{147609390002}{9805336707}$, $\frac{95171342473}{9805336707}a^{19}-\frac{4100372381}{172023451}a^{18}-\frac{222473948365}{9805336707}a^{17}+\frac{1212090543974}{9805336707}a^{16}-\frac{1340301908990}{9805336707}a^{15}+\frac{1172473721783}{9805336707}a^{14}+\frac{75590020741}{9805336707}a^{13}-\frac{8632591054046}{9805336707}a^{12}+\frac{16632483081293}{9805336707}a^{11}-\frac{3316684904791}{9805336707}a^{10}-\frac{339971454186}{172023451}a^{9}+\frac{20984444863585}{9805336707}a^{8}-\frac{9965958074035}{9805336707}a^{7}+\frac{2569711579421}{9805336707}a^{6}+\frac{4147624595456}{9805336707}a^{5}-\frac{2433301893691}{3268445569}a^{4}+\frac{1270027755006}{3268445569}a^{3}+\frac{814945669774}{9805336707}a^{2}-\frac{1109451209768}{9805336707}a+\frac{63448231278}{3268445569}$, $\frac{11450269156}{3268445569}a^{19}-\frac{93059638858}{9805336707}a^{18}-\frac{62040789071}{9805336707}a^{17}+\frac{154543838138}{3268445569}a^{16}-\frac{583623757993}{9805336707}a^{15}+\frac{170584441087}{3268445569}a^{14}-\frac{56353835887}{9805336707}a^{13}-\frac{3143432250313}{9805336707}a^{12}+\frac{2257344734084}{3268445569}a^{11}-\frac{2441787700124}{9805336707}a^{10}-\frac{7074657440225}{9805336707}a^{9}+\frac{3067971791477}{3268445569}a^{8}-\frac{1634454506746}{3268445569}a^{7}+\frac{1473121769531}{9805336707}a^{6}+\frac{1331188654126}{9805336707}a^{5}-\frac{990842908469}{3268445569}a^{4}+\frac{619285740138}{3268445569}a^{3}+\frac{130152797213}{9805336707}a^{2}-\frac{505085831059}{9805336707}a+\frac{108616207267}{9805336707}$, $\frac{197222390117}{9805336707}a^{19}-\frac{161927337085}{3268445569}a^{18}-\frac{460116927784}{9805336707}a^{17}+\frac{2518870611994}{9805336707}a^{16}-\frac{928040206250}{3268445569}a^{15}+\frac{808343900893}{3268445569}a^{14}+\frac{144760698422}{9805336707}a^{13}-\frac{17895975346867}{9805336707}a^{12}+\frac{34570692919394}{9805336707}a^{11}-\frac{6897417461830}{9805336707}a^{10}-\frac{40324540513889}{9805336707}a^{9}+\frac{14496020583276}{3268445569}a^{8}-\frac{6824159864677}{3268445569}a^{7}+\frac{5410103999242}{9805336707}a^{6}+\frac{8435976969571}{9805336707}a^{5}-\frac{15102260460430}{9805336707}a^{4}+\frac{7835580422876}{9805336707}a^{3}+\frac{579061638138}{3268445569}a^{2}-\frac{752087552295}{3268445569}a+\frac{114866649593}{3268445569}$, $\frac{35295053032}{3268445569}a^{19}-\frac{262894537667}{9805336707}a^{18}-\frac{80747616548}{3268445569}a^{17}+\frac{452516524569}{3268445569}a^{16}-\frac{1519416099661}{9805336707}a^{15}+\frac{1328095039124}{9805336707}a^{14}+\frac{51262153780}{9805336707}a^{13}-\frac{9607122466814}{9805336707}a^{12}+\frac{18741493253380}{9805336707}a^{11}-\frac{1345206910787}{3268445569}a^{10}-\frac{7198442329594}{3268445569}a^{9}+\frac{23705335792177}{9805336707}a^{8}-\frac{11353799160703}{9805336707}a^{7}+\frac{3110972436412}{9805336707}a^{6}+\frac{4422756161153}{9805336707}a^{5}-\frac{8139433176601}{9805336707}a^{4}+\frac{4301075985935}{9805336707}a^{3}+\frac{899295584987}{9805336707}a^{2}-\frac{1233179172157}{9805336707}a+\frac{197222390117}{9805336707}$, $\frac{6805660780}{9805336707}a^{19}-\frac{15365057539}{9805336707}a^{18}-\frac{18365414302}{9805336707}a^{17}+\frac{81206723590}{9805336707}a^{16}-\frac{80050037485}{9805336707}a^{15}+\frac{75821601050}{9805336707}a^{14}+\frac{7244836966}{9805336707}a^{13}-\frac{200318825187}{3268445569}a^{12}+\frac{1062650495791}{9805336707}a^{11}-\frac{76260848807}{9805336707}a^{10}-\frac{1267689538234}{9805336707}a^{9}+\frac{1149080955023}{9805336707}a^{8}-\frac{551821045657}{9805336707}a^{7}+\frac{255351518459}{9805336707}a^{6}+\frac{62725143027}{3268445569}a^{5}-\frac{399302012498}{9805336707}a^{4}+\frac{159220624379}{9805336707}a^{3}+\frac{27248247198}{3268445569}a^{2}-\frac{20250797033}{9805336707}a-\frac{3091253119}{3268445569}$, $\frac{22099137755}{3268445569}a^{19}-\frac{171086660161}{9805336707}a^{18}-\frac{7274164741}{516070353}a^{17}+\frac{290160868098}{3268445569}a^{16}-\frac{342394024404}{3268445569}a^{15}+\frac{894792998285}{9805336707}a^{14}-\frac{19930491211}{9805336707}a^{13}-\frac{6049094549347}{9805336707}a^{12}+\frac{12319299862063}{9805336707}a^{11}-\frac{3445940109793}{9805336707}a^{10}-\frac{13660342293574}{9805336707}a^{9}+\frac{16189705864219}{9805336707}a^{8}-\frac{427285064122}{516070353}a^{7}+\frac{2224020882188}{9805336707}a^{6}+\frac{2812595525692}{9805336707}a^{5}-\frac{5460000277861}{9805336707}a^{4}+\frac{3126992042774}{9805336707}a^{3}+\frac{140436265770}{3268445569}a^{2}-\frac{15785187520}{172023451}a+\frac{180095856454}{9805336707}$, $\frac{177192450}{3268445569}a^{19}+\frac{5210928730}{9805336707}a^{18}-\frac{15896634889}{9805336707}a^{17}-\frac{10923331402}{9805336707}a^{16}+\frac{70043599240}{9805336707}a^{15}-\frac{69418490485}{9805336707}a^{14}+\frac{3822744050}{516070353}a^{13}-\frac{41128701884}{9805336707}a^{12}-\frac{160627716387}{3268445569}a^{11}+\frac{993545440291}{9805336707}a^{10}-\frac{9161635853}{516070353}a^{9}-\frac{1092269012734}{9805336707}a^{8}+\frac{1030007628623}{9805336707}a^{7}-\frac{506636970907}{9805336707}a^{6}+\frac{269704106909}{9805336707}a^{5}+\frac{45183090477}{3268445569}a^{4}-\frac{356244247148}{9805336707}a^{3}+\frac{152310118829}{9805336707}a^{2}+\frac{24413167998}{3268445569}a-\frac{25803514940}{9805336707}$, $\frac{11190161196}{3268445569}a^{19}-\frac{82615007164}{9805336707}a^{18}-\frac{76701581621}{9805336707}a^{17}+\frac{425731203101}{9805336707}a^{16}-\frac{479115636460}{9805336707}a^{15}+\frac{427947586328}{9805336707}a^{14}+\frac{4505994555}{3268445569}a^{13}-\frac{3028244401564}{9805336707}a^{12}+\frac{1962093414907}{3268445569}a^{11}-\frac{435546288189}{3268445569}a^{10}-\frac{2227976470523}{3268445569}a^{9}+\frac{7442223708662}{9805336707}a^{8}-\frac{3691916983945}{9805336707}a^{7}+\frac{355662704314}{3268445569}a^{6}+\frac{1323252063118}{9805336707}a^{5}-\frac{840943574383}{3268445569}a^{4}+\frac{1359788724136}{9805336707}a^{3}+\frac{87118737626}{3268445569}a^{2}-\frac{373070174813}{9805336707}a+\frac{23596074195}{3268445569}$
| 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: | \( 62751.6666339 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{6}\cdot 62751.6666339 \cdot 1}{2\cdot\sqrt{11149939773041205947265625}}\cr\approx \mathstrut & 0.148005612270 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - x^18 + 14*x^17 - 21*x^16 + 20*x^15 - 6*x^14 - 91*x^13 + 224*x^12 - 130*x^11 - 184*x^10 + 330*x^9 - 224*x^8 + 85*x^7 + 27*x^6 - 99*x^5 + 81*x^4 - 13*x^3 - 16*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^20 - 3*x^19 - x^18 + 14*x^17 - 21*x^16 + 20*x^15 - 6*x^14 - 91*x^13 + 224*x^12 - 130*x^11 - 184*x^10 + 330*x^9 - 224*x^8 + 85*x^7 + 27*x^6 - 99*x^5 + 81*x^4 - 13*x^3 - 16*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^20 - 3*x^19 - x^18 + 14*x^17 - 21*x^16 + 20*x^15 - 6*x^14 - 91*x^13 + 224*x^12 - 130*x^11 - 184*x^10 + 330*x^9 - 224*x^8 + 85*x^7 + 27*x^6 - 99*x^5 + 81*x^4 - 13*x^3 - 16*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^20 - 3*x^19 - x^18 + 14*x^17 - 21*x^16 + 20*x^15 - 6*x^14 - 91*x^13 + 224*x^12 - 130*x^11 - 184*x^10 + 330*x^9 - 224*x^8 + 85*x^7 + 27*x^6 - 99*x^5 + 81*x^4 - 13*x^3 - 16*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
$C_2^8.D_5^2:C_2^2$ (as 20T760):
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 solvable group of order 102400 |
| The 130 conjugacy class representatives for $C_2^8.D_5^2:C_2^2$ |
| Character table for $C_2^8.D_5^2:C_2^2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 10.6.34424253125.1 |
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 20 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 20.6.37006650106723762538974609375.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{3}$ | ${\href{/padicField/11.5.0.1}{5} }^{4}$ | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{10}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| Deg $20$ | $2$ | $10$ | $10$ | |||
|
\(97\)
| 97.4.2.1 | $x^{4} + 10474 x^{3} + 27919909 x^{2} + 2585716380 x + 183392493$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 97.8.0.1 | $x^{8} + 65 x^{3} + x^{2} + 32 x + 5$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 97.8.0.1 | $x^{8} + 65 x^{3} + x^{2} + 32 x + 5$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(3319\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |