Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 6*x^18 + 4*x^17 - 17*x^16 - 2*x^15 + 41*x^14 + 10*x^13 - 52*x^12 + 10*x^11 + 99*x^10 - 29*x^9 - 260*x^8 - 255*x^7 - 53*x^6 + 79*x^5 + 42*x^4 - 9*x^3 - 12*x^2 + 4*x + 1)
gp: K = bnfinit(y^20 - 5*y^19 + 6*y^18 + 4*y^17 - 17*y^16 - 2*y^15 + 41*y^14 + 10*y^13 - 52*y^12 + 10*y^11 + 99*y^10 - 29*y^9 - 260*y^8 - 255*y^7 - 53*y^6 + 79*y^5 + 42*y^4 - 9*y^3 - 12*y^2 + 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 5*x^19 + 6*x^18 + 4*x^17 - 17*x^16 - 2*x^15 + 41*x^14 + 10*x^13 - 52*x^12 + 10*x^11 + 99*x^10 - 29*x^9 - 260*x^8 - 255*x^7 - 53*x^6 + 79*x^5 + 42*x^4 - 9*x^3 - 12*x^2 + 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 5*x^19 + 6*x^18 + 4*x^17 - 17*x^16 - 2*x^15 + 41*x^14 + 10*x^13 - 52*x^12 + 10*x^11 + 99*x^10 - 29*x^9 - 260*x^8 - 255*x^7 - 53*x^6 + 79*x^5 + 42*x^4 - 9*x^3 - 12*x^2 + 4*x + 1)
\( x^{20} - 5 x^{19} + 6 x^{18} + 4 x^{17} - 17 x^{16} - 2 x^{15} + 41 x^{14} + 10 x^{13} - 52 x^{12} + \cdots + 1 \)
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: | $[4, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(6802209109663753569286129\)
\(\medspace = 11^{16}\cdot 23^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.44\) | 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: | $11^{4/5}23^{1/2}\approx 32.65713384043754$ | ||
| Ramified primes: |
\(11\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{6}a^{15}+\frac{1}{3}a^{14}+\frac{1}{6}a^{13}+\frac{1}{6}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}-\frac{1}{6}a^{7}+\frac{1}{6}a^{6}-\frac{1}{6}a^{5}-\frac{1}{6}a^{4}+\frac{1}{6}a^{3}+\frac{1}{6}a^{2}-\frac{1}{2}a-\frac{1}{6}$, $\frac{1}{6}a^{16}-\frac{1}{2}a^{14}-\frac{1}{6}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{9}-\frac{1}{6}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{6}a^{5}-\frac{1}{2}a^{4}-\frac{1}{6}a^{3}+\frac{1}{6}a^{2}-\frac{1}{6}a+\frac{1}{3}$, $\frac{1}{6}a^{17}-\frac{1}{6}a^{14}-\frac{1}{6}a^{13}-\frac{1}{2}a^{12}-\frac{1}{3}a^{10}-\frac{1}{6}a^{9}-\frac{1}{2}a^{8}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{6}a-\frac{1}{2}$, $\frac{1}{66}a^{18}+\frac{1}{66}a^{17}+\frac{1}{22}a^{16}+\frac{1}{33}a^{15}+\frac{1}{66}a^{14}+\frac{4}{33}a^{13}-\frac{5}{11}a^{12}+\frac{1}{3}a^{11}+\frac{3}{22}a^{10}+\frac{10}{33}a^{9}+\frac{1}{11}a^{8}-\frac{4}{33}a^{7}-\frac{16}{33}a^{6}+\frac{16}{33}a^{5}-\frac{4}{11}a^{4}-\frac{2}{11}a^{3}-\frac{23}{66}a^{2}-\frac{14}{33}a-\frac{1}{11}$, $\frac{1}{708721029658686}a^{19}-\frac{623409056021}{354360514829343}a^{18}+\frac{20580992271232}{354360514829343}a^{17}-\frac{2348113871967}{118120171609781}a^{16}-\frac{38178024855823}{708721029658686}a^{15}-\frac{3339155826536}{118120171609781}a^{14}-\frac{36355273812619}{708721029658686}a^{13}+\frac{11351871437107}{354360514829343}a^{12}+\frac{59454384728731}{236240343219562}a^{11}+\frac{166823411480533}{708721029658686}a^{10}-\frac{5837365437364}{354360514829343}a^{9}-\frac{111129620344925}{236240343219562}a^{8}-\frac{92000971561163}{236240343219562}a^{7}-\frac{79160165753137}{236240343219562}a^{6}+\frac{138614387324695}{708721029658686}a^{5}+\frac{105752501934741}{236240343219562}a^{4}-\frac{25158390028547}{118120171609781}a^{3}+\frac{147879424819111}{354360514829343}a^{2}+\frac{37089204766465}{236240343219562}a-\frac{11770584969662}{118120171609781}$
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 $1$
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: | $11$ | 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: |
$\frac{43384265848675}{354360514829343}a^{19}-\frac{281694018953563}{354360514829343}a^{18}+\frac{12\!\cdots\!15}{708721029658686}a^{17}-\frac{799154507508275}{708721029658686}a^{16}-\frac{276714351284190}{118120171609781}a^{15}+\frac{459718079812174}{118120171609781}a^{14}+\frac{10\!\cdots\!06}{354360514829343}a^{13}-\frac{444331670286251}{64429184514426}a^{12}-\frac{607965556612796}{354360514829343}a^{11}+\frac{39\!\cdots\!75}{354360514829343}a^{10}+\frac{813228528705091}{708721029658686}a^{9}-\frac{67\!\cdots\!38}{354360514829343}a^{8}-\frac{76\!\cdots\!23}{708721029658686}a^{7}+\frac{20\!\cdots\!03}{236240343219562}a^{6}+\frac{32\!\cdots\!43}{708721029658686}a^{5}-\frac{318765883449221}{236240343219562}a^{4}-\frac{10\!\cdots\!69}{708721029658686}a^{3}+\frac{187611993118633}{236240343219562}a^{2}-\frac{10348961100040}{354360514829343}a+\frac{26743392327853}{21476394838142}$, $\frac{85766132019805}{708721029658686}a^{19}-\frac{78979752835738}{118120171609781}a^{18}+\frac{765705997597027}{708721029658686}a^{17}-\frac{2438490361190}{32214592257213}a^{16}-\frac{500604460065069}{236240343219562}a^{15}+\frac{68426250881885}{64429184514426}a^{14}+\frac{15\!\cdots\!12}{354360514829343}a^{13}-\frac{293118783103007}{236240343219562}a^{12}-\frac{13\!\cdots\!23}{236240343219562}a^{11}+\frac{11\!\cdots\!75}{236240343219562}a^{10}+\frac{630549546690241}{64429184514426}a^{9}-\frac{32\!\cdots\!02}{354360514829343}a^{8}-\frac{18\!\cdots\!89}{708721029658686}a^{7}-\frac{11\!\cdots\!89}{708721029658686}a^{6}+\frac{12\!\cdots\!61}{708721029658686}a^{5}+\frac{684002193573091}{236240343219562}a^{4}-\frac{15\!\cdots\!65}{354360514829343}a^{3}-\frac{766090516689965}{354360514829343}a^{2}+\frac{232310367702664}{118120171609781}a+\frac{343048149295145}{236240343219562}$, $\frac{5988584610666}{118120171609781}a^{19}-\frac{45610853537485}{118120171609781}a^{18}+\frac{747315291190007}{708721029658686}a^{17}-\frac{122849382908485}{118120171609781}a^{16}-\frac{284770524621728}{354360514829343}a^{15}+\frac{17\!\cdots\!35}{708721029658686}a^{14}+\frac{144137649289585}{236240343219562}a^{13}-\frac{30\!\cdots\!49}{708721029658686}a^{12}-\frac{287515140544862}{354360514829343}a^{11}+\frac{851292152206935}{118120171609781}a^{10}+\frac{7564647347685}{236240343219562}a^{9}-\frac{27\!\cdots\!11}{236240343219562}a^{8}-\frac{87656826604553}{32214592257213}a^{7}+\frac{60\!\cdots\!05}{354360514829343}a^{6}+\frac{43\!\cdots\!41}{354360514829343}a^{5}-\frac{24\!\cdots\!42}{354360514829343}a^{4}-\frac{57\!\cdots\!20}{354360514829343}a^{3}-\frac{782059727430395}{118120171609781}a^{2}+\frac{976368574931941}{708721029658686}a+\frac{16\!\cdots\!19}{708721029658686}$, $\frac{2678371244079}{118120171609781}a^{19}-\frac{19357872165869}{118120171609781}a^{18}+\frac{11449566414548}{32214592257213}a^{17}-\frac{33932230853144}{354360514829343}a^{16}-\frac{343590243912665}{708721029658686}a^{15}+\frac{8405791884318}{118120171609781}a^{14}+\frac{456735165450537}{236240343219562}a^{13}-\frac{726206021918539}{708721029658686}a^{12}-\frac{14\!\cdots\!77}{354360514829343}a^{11}+\frac{432039146626442}{354360514829343}a^{10}+\frac{17\!\cdots\!79}{354360514829343}a^{9}-\frac{16\!\cdots\!66}{354360514829343}a^{8}-\frac{74\!\cdots\!45}{708721029658686}a^{7}+\frac{15\!\cdots\!87}{236240343219562}a^{6}+\frac{17\!\cdots\!69}{708721029658686}a^{5}+\frac{49\!\cdots\!61}{236240343219562}a^{4}+\frac{477776895759661}{708721029658686}a^{3}-\frac{54\!\cdots\!97}{708721029658686}a^{2}-\frac{32\!\cdots\!53}{708721029658686}a+\frac{18093420222117}{236240343219562}$, $\frac{49445820335897}{708721029658686}a^{19}-\frac{56451946067486}{118120171609781}a^{18}+\frac{832523428985657}{708721029658686}a^{17}-\frac{695026166493053}{708721029658686}a^{16}-\frac{178319334276057}{118120171609781}a^{15}+\frac{13\!\cdots\!62}{354360514829343}a^{14}-\frac{21800838389383}{118120171609781}a^{13}-\frac{17\!\cdots\!55}{354360514829343}a^{12}+\frac{346094146237201}{236240343219562}a^{11}+\frac{20\!\cdots\!27}{236240343219562}a^{10}-\frac{608681687115215}{236240343219562}a^{9}-\frac{92\!\cdots\!77}{708721029658686}a^{8}+\frac{571352738450389}{708721029658686}a^{7}+\frac{80\!\cdots\!73}{708721029658686}a^{6}-\frac{16\!\cdots\!01}{236240343219562}a^{5}-\frac{44\!\cdots\!73}{236240343219562}a^{4}-\frac{41\!\cdots\!11}{354360514829343}a^{3}+\frac{121986619481340}{118120171609781}a^{2}+\frac{13\!\cdots\!50}{354360514829343}a-\frac{605602725237178}{354360514829343}$, $\frac{141727228909697}{708721029658686}a^{19}-\frac{36116059001792}{32214592257213}a^{18}+\frac{12\!\cdots\!11}{708721029658686}a^{17}+\frac{9703373882255}{708721029658686}a^{16}-\frac{13\!\cdots\!42}{354360514829343}a^{15}+\frac{705303601105660}{354360514829343}a^{14}+\frac{27\!\cdots\!41}{354360514829343}a^{13}-\frac{10\!\cdots\!77}{354360514829343}a^{12}-\frac{24\!\cdots\!69}{236240343219562}a^{11}+\frac{169390507762153}{21476394838142}a^{10}+\frac{38\!\cdots\!29}{236240343219562}a^{9}-\frac{11\!\cdots\!01}{708721029658686}a^{8}-\frac{10\!\cdots\!37}{236240343219562}a^{7}-\frac{15\!\cdots\!15}{708721029658686}a^{6}+\frac{10\!\cdots\!51}{708721029658686}a^{5}+\frac{16\!\cdots\!63}{708721029658686}a^{4}+\frac{198545652547919}{32214592257213}a^{3}-\frac{10\!\cdots\!32}{354360514829343}a^{2}-\frac{21503981326587}{10738197419071}a+\frac{183392523865403}{354360514829343}$, $\frac{16256497606293}{118120171609781}a^{19}-\frac{265833681110086}{354360514829343}a^{18}+\frac{133564014122137}{118120171609781}a^{17}+\frac{12355445921753}{64429184514426}a^{16}-\frac{597644995090927}{236240343219562}a^{15}+\frac{31974925199105}{64429184514426}a^{14}+\frac{743930259872531}{118120171609781}a^{13}-\frac{351433252610051}{236240343219562}a^{12}-\frac{28\!\cdots\!50}{354360514829343}a^{11}+\frac{18\!\cdots\!69}{354360514829343}a^{10}+\frac{405096612747077}{32214592257213}a^{9}-\frac{72\!\cdots\!35}{708721029658686}a^{8}-\frac{40\!\cdots\!10}{118120171609781}a^{7}-\frac{65\!\cdots\!31}{354360514829343}a^{6}+\frac{33\!\cdots\!37}{354360514829343}a^{5}+\frac{19\!\cdots\!76}{118120171609781}a^{4}+\frac{11\!\cdots\!35}{354360514829343}a^{3}-\frac{838445445224437}{354360514829343}a^{2}-\frac{344102276439971}{354360514829343}a+\frac{354510941089303}{708721029658686}$, $\frac{40093469519365}{708721029658686}a^{19}-\frac{35637914020447}{118120171609781}a^{18}+\frac{348357745548445}{708721029658686}a^{17}-\frac{93512388887572}{354360514829343}a^{16}-\frac{12276713188177}{64429184514426}a^{15}-\frac{86642278830563}{236240343219562}a^{14}+\frac{420994260888664}{354360514829343}a^{13}+\frac{14\!\cdots\!99}{708721029658686}a^{12}-\frac{16\!\cdots\!67}{708721029658686}a^{11}-\frac{13\!\cdots\!61}{708721029658686}a^{10}+\frac{14\!\cdots\!67}{236240343219562}a^{9}+\frac{41140953970082}{32214592257213}a^{8}-\frac{10\!\cdots\!05}{708721029658686}a^{7}-\frac{11\!\cdots\!57}{708721029658686}a^{6}-\frac{11\!\cdots\!81}{708721029658686}a^{5}+\frac{481153490570377}{64429184514426}a^{4}+\frac{70776786953182}{118120171609781}a^{3}-\frac{33165500426068}{10738197419071}a^{2}-\frac{197046689254447}{118120171609781}a+\frac{744983994926957}{708721029658686}$, $\frac{68874928797578}{354360514829343}a^{19}-\frac{704441812787221}{708721029658686}a^{18}+\frac{407874518831150}{354360514829343}a^{17}+\frac{487118551782026}{354360514829343}a^{16}-\frac{16\!\cdots\!87}{354360514829343}a^{15}+\frac{337163907004015}{708721029658686}a^{14}+\frac{325480538519615}{32214592257213}a^{13}-\frac{706425885611885}{708721029658686}a^{12}-\frac{16\!\cdots\!37}{118120171609781}a^{11}+\frac{12\!\cdots\!29}{236240343219562}a^{10}+\frac{54\!\cdots\!71}{236240343219562}a^{9}-\frac{48\!\cdots\!65}{354360514829343}a^{8}-\frac{13\!\cdots\!79}{236240343219562}a^{7}-\frac{21\!\cdots\!33}{64429184514426}a^{6}+\frac{12\!\cdots\!27}{708721029658686}a^{5}+\frac{20\!\cdots\!95}{708721029658686}a^{4}+\frac{53\!\cdots\!39}{708721029658686}a^{3}-\frac{23\!\cdots\!83}{354360514829343}a^{2}-\frac{391265571356602}{118120171609781}a+\frac{418489658078633}{708721029658686}$, $\frac{60351094269523}{236240343219562}a^{19}-\frac{917993842107967}{708721029658686}a^{18}+\frac{197172748685754}{118120171609781}a^{17}+\frac{13118367735979}{21476394838142}a^{16}-\frac{13\!\cdots\!70}{354360514829343}a^{15}-\frac{8260087432190}{10738197419071}a^{14}+\frac{70\!\cdots\!59}{708721029658686}a^{13}+\frac{11\!\cdots\!75}{354360514829343}a^{12}-\frac{31\!\cdots\!29}{236240343219562}a^{11}+\frac{242566323998034}{118120171609781}a^{10}+\frac{16\!\cdots\!77}{64429184514426}a^{9}-\frac{849054807864913}{118120171609781}a^{8}-\frac{23\!\cdots\!87}{354360514829343}a^{7}-\frac{22\!\cdots\!84}{354360514829343}a^{6}-\frac{16\!\cdots\!05}{118120171609781}a^{5}+\frac{70\!\cdots\!23}{354360514829343}a^{4}+\frac{78\!\cdots\!05}{708721029658686}a^{3}-\frac{105172412636186}{354360514829343}a^{2}-\frac{839776022444847}{236240343219562}a-\frac{9240116363708}{354360514829343}$, $\frac{165426770033405}{354360514829343}a^{19}-\frac{17\!\cdots\!17}{708721029658686}a^{18}+\frac{749316097744751}{236240343219562}a^{17}+\frac{915634682424023}{708721029658686}a^{16}-\frac{171985095032197}{21476394838142}a^{15}+\frac{283041576186637}{708721029658686}a^{14}+\frac{43\!\cdots\!57}{236240343219562}a^{13}+\frac{12\!\cdots\!27}{708721029658686}a^{12}-\frac{82\!\cdots\!17}{354360514829343}a^{11}+\frac{58\!\cdots\!57}{708721029658686}a^{10}+\frac{51\!\cdots\!15}{118120171609781}a^{9}-\frac{631379604124724}{32214592257213}a^{8}-\frac{81\!\cdots\!93}{708721029658686}a^{7}-\frac{73\!\cdots\!29}{708721029658686}a^{6}-\frac{12\!\cdots\!85}{708721029658686}a^{5}+\frac{21\!\cdots\!37}{64429184514426}a^{4}+\frac{34\!\cdots\!43}{236240343219562}a^{3}-\frac{135305064593141}{32214592257213}a^{2}-\frac{10\!\cdots\!83}{236240343219562}a+\frac{658709981020511}{236240343219562}$
| 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: | \( 31805.2169747 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 31805.2169747 \cdot 1}{2\cdot\sqrt{6802209109663753569286129}}\cr\approx \mathstrut & 0.236974877376 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 6*x^18 + 4*x^17 - 17*x^16 - 2*x^15 + 41*x^14 + 10*x^13 - 52*x^12 + 10*x^11 + 99*x^10 - 29*x^9 - 260*x^8 - 255*x^7 - 53*x^6 + 79*x^5 + 42*x^4 - 9*x^3 - 12*x^2 + 4*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 - 5*x^19 + 6*x^18 + 4*x^17 - 17*x^16 - 2*x^15 + 41*x^14 + 10*x^13 - 52*x^12 + 10*x^11 + 99*x^10 - 29*x^9 - 260*x^8 - 255*x^7 - 53*x^6 + 79*x^5 + 42*x^4 - 9*x^3 - 12*x^2 + 4*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 - 5*x^19 + 6*x^18 + 4*x^17 - 17*x^16 - 2*x^15 + 41*x^14 + 10*x^13 - 52*x^12 + 10*x^11 + 99*x^10 - 29*x^9 - 260*x^8 - 255*x^7 - 53*x^6 + 79*x^5 + 42*x^4 - 9*x^3 - 12*x^2 + 4*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 - 5*x^19 + 6*x^18 + 4*x^17 - 17*x^16 - 2*x^15 + 41*x^14 + 10*x^13 - 52*x^12 + 10*x^11 + 99*x^10 - 29*x^9 - 260*x^8 - 255*x^7 - 53*x^6 + 79*x^5 + 42*x^4 - 9*x^3 - 12*x^2 + 4*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
$C_2^4:C_5$ (as 20T17):
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 80 |
| The 8 conjugacy class representatives for $C_2^4:C_5$ |
| Character table for $C_2^4:C_5$ |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.6.113395848049.1 x2, 10.2.113395848049.1 |
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 10 siblings: | data not computed |
| Degree 16 sibling: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.2.113395848049.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{4}$ | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | ${\href{/padicField/5.5.0.1}{5} }^{4}$ | ${\href{/padicField/7.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/13.5.0.1}{5} }^{4}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{10}$ | ${\href{/padicField/47.5.0.1}{5} }^{4}$ | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
|
\(23\)
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |