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Properties

Label 20.0.159...433.1
Degree $20$
Signature $(0, 10)$
Discriminant $1.591\times 10^{37}$
Root discriminant \(72.46\)
Ramified primes $11,17$
Class number $8194$ (GRH)
Class group [8194] (GRH)
Galois group $C_{20}$ (as 20T1)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 27*x^18 - 31*x^17 + 348*x^16 - 472*x^15 + 2986*x^14 - 4874*x^13 + 20558*x^12 - 40054*x^11 + 130294*x^10 - 168519*x^9 + 700706*x^8 - 32881*x^7 + 2845143*x^6 + 2390937*x^5 + 8994354*x^4 + 9962701*x^3 + 26015925*x^2 + 20544384*x + 83519437)
 
Copy content gp:K = bnfinit(y^20 - y^19 + 27*y^18 - 31*y^17 + 348*y^16 - 472*y^15 + 2986*y^14 - 4874*y^13 + 20558*y^12 - 40054*y^11 + 130294*y^10 - 168519*y^9 + 700706*y^8 - 32881*y^7 + 2845143*y^6 + 2390937*y^5 + 8994354*y^4 + 9962701*y^3 + 26015925*y^2 + 20544384*y + 83519437, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^19 + 27*x^18 - 31*x^17 + 348*x^16 - 472*x^15 + 2986*x^14 - 4874*x^13 + 20558*x^12 - 40054*x^11 + 130294*x^10 - 168519*x^9 + 700706*x^8 - 32881*x^7 + 2845143*x^6 + 2390937*x^5 + 8994354*x^4 + 9962701*x^3 + 26015925*x^2 + 20544384*x + 83519437);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - x^19 + 27*x^18 - 31*x^17 + 348*x^16 - 472*x^15 + 2986*x^14 - 4874*x^13 + 20558*x^12 - 40054*x^11 + 130294*x^10 - 168519*x^9 + 700706*x^8 - 32881*x^7 + 2845143*x^6 + 2390937*x^5 + 8994354*x^4 + 9962701*x^3 + 26015925*x^2 + 20544384*x + 83519437)
 

\( x^{20} - x^{19} + 27 x^{18} - 31 x^{17} + 348 x^{16} - 472 x^{15} + 2986 x^{14} - 4874 x^{13} + \cdots + 83519437 \) 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:  $(0, 10)$
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:   \(15914835482628690354834740122825579433\) \(\medspace = 11^{18}\cdot 17^{15}\) 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:  \(72.46\)
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:  $11^{9/10}17^{3/4}\approx 72.45862820705953$
Ramified primes:   \(11\), \(17\) 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:  \(\Q(\sqrt{17}) \)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_{20}$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(187=11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{187}(1,·)$, $\chi_{187}(67,·)$, $\chi_{187}(69,·)$, $\chi_{187}(135,·)$, $\chi_{187}(72,·)$, $\chi_{187}(137,·)$, $\chi_{187}(140,·)$, $\chi_{187}(13,·)$, $\chi_{187}(16,·)$, $\chi_{187}(21,·)$, $\chi_{187}(86,·)$, $\chi_{187}(152,·)$, $\chi_{187}(30,·)$, $\chi_{187}(98,·)$, $\chi_{187}(103,·)$, $\chi_{187}(169,·)$, $\chi_{187}(106,·)$, $\chi_{187}(183,·)$, $\chi_{187}(123,·)$, $\chi_{187}(149,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{512}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{30887}a^{11}+\frac{11}{30887}a^{9}+\frac{44}{30887}a^{7}+\frac{77}{30887}a^{5}+\frac{55}{30887}a^{3}+\frac{11}{30887}a+\frac{14665}{30887}$, $\frac{1}{30887}a^{12}+\frac{11}{30887}a^{10}+\frac{44}{30887}a^{8}+\frac{77}{30887}a^{6}+\frac{55}{30887}a^{4}+\frac{11}{30887}a^{2}+\frac{14665}{30887}a$, $\frac{1}{30887}a^{13}-\frac{77}{30887}a^{9}-\frac{407}{30887}a^{7}-\frac{792}{30887}a^{5}-\frac{594}{30887}a^{3}+\frac{14665}{30887}a^{2}-\frac{121}{30887}a-\frac{6880}{30887}$, $\frac{1}{30887}a^{14}-\frac{77}{30887}a^{10}-\frac{407}{30887}a^{8}-\frac{792}{30887}a^{6}-\frac{594}{30887}a^{4}+\frac{14665}{30887}a^{3}-\frac{121}{30887}a^{2}-\frac{6880}{30887}a$, $\frac{1}{61774}a^{15}-\frac{1}{61774}a^{13}-\frac{1}{61774}a^{12}+\frac{15438}{30887}a^{10}-\frac{15185}{30887}a^{9}-\frac{22}{30887}a^{8}+\frac{3003}{61774}a^{7}-\frac{77}{61774}a^{6}+\frac{6127}{61774}a^{5}-\frac{16277}{61774}a^{4}-\frac{26179}{61774}a^{3}+\frac{9331}{61774}a^{2}-\frac{13697}{61774}a+\frac{24153}{61774}$, $\frac{1}{2502643205086}a^{16}-\frac{8372121}{2502643205086}a^{15}-\frac{38655455}{2502643205086}a^{14}+\frac{18497203}{1251321602543}a^{13}+\frac{28251677}{2502643205086}a^{12}+\frac{17102227}{1251321602543}a^{11}-\frac{161013994131}{1251321602543}a^{10}-\frac{541055715937}{1251321602543}a^{9}+\frac{1218489227347}{2502643205086}a^{8}-\frac{466725833993}{1251321602543}a^{7}-\frac{534465219149}{1251321602543}a^{6}-\frac{257990473602}{1251321602543}a^{5}-\frac{380266742709}{1251321602543}a^{4}+\frac{541726175575}{1251321602543}a^{3}+\frac{67473101116}{1251321602543}a^{2}-\frac{53033734868}{1251321602543}a+\frac{771452120321}{2502643205086}$, $\frac{1}{2502643205086}a^{17}-\frac{5587193}{2502643205086}a^{15}+\frac{29313715}{2502643205086}a^{14}+\frac{13754234}{1251321602543}a^{13}-\frac{14442998}{1251321602543}a^{12}-\frac{10163137}{1251321602543}a^{11}+\frac{388769291034}{1251321602543}a^{10}-\frac{1131462396497}{2502643205086}a^{9}+\frac{301946858209}{2502643205086}a^{8}-\frac{1152499577741}{2502643205086}a^{7}+\frac{282522920791}{2502643205086}a^{6}+\frac{1044374861419}{2502643205086}a^{5}-\frac{895913484533}{2502643205086}a^{4}-\frac{184068791665}{2502643205086}a^{3}-\frac{277154437689}{2502643205086}a^{2}+\frac{114140262026}{1251321602543}a+\frac{538442389146}{1251321602543}$, $\frac{1}{2502643205086}a^{18}+\frac{748541}{2502643205086}a^{15}-\frac{33433677}{2502643205086}a^{14}+\frac{1784145}{2502643205086}a^{13}-\frac{11187486}{1251321602543}a^{12}-\frac{6331065}{1251321602543}a^{11}+\frac{852715945333}{2502643205086}a^{10}-\frac{653896134655}{2502643205086}a^{9}+\frac{454585492758}{1251321602543}a^{8}-\frac{107368331880}{1251321602543}a^{7}-\frac{82706047459}{1251321602543}a^{6}+\frac{471139620551}{1251321602543}a^{5}-\frac{8179119867}{1251321602543}a^{4}-\frac{127564951569}{1251321602543}a^{3}-\frac{1206291573573}{2502643205086}a^{2}-\frac{716328211553}{2502643205086}a-\frac{342673681438}{1251321602543}$, $\frac{1}{2502643205086}a^{19}-\frac{7690924}{1251321602543}a^{15}-\frac{941029}{1251321602543}a^{14}+\frac{2003665}{1251321602543}a^{13}+\frac{14791057}{2502643205086}a^{12}-\frac{18927955}{2502643205086}a^{11}-\frac{1057310949435}{2502643205086}a^{10}+\frac{63415729885}{1251321602543}a^{9}+\frac{155000634833}{2502643205086}a^{8}-\frac{586995306289}{1251321602543}a^{7}+\frac{368816411947}{1251321602543}a^{6}-\frac{459350571569}{1251321602543}a^{5}-\frac{593839740802}{1251321602543}a^{4}-\frac{1005857634667}{2502643205086}a^{3}+\frac{647531067881}{2502643205086}a^{2}-\frac{456704347033}{1251321602543}a+\frac{179121068921}{2502643205086}$ 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:  $C_{8194}$, which has order $8194$ (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:  $C_{8194}$, which has order $8194$ (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);
 
Relative class number:   $8194$ (assuming GRH)

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:  $9$
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:   $\frac{45240}{1251321602543}a^{19}-\frac{36192}{1251321602543}a^{18}+\frac{1158144}{1251321602543}a^{17}-\frac{1121952}{1251321602543}a^{16}+\frac{14132976}{1251321602543}a^{15}-\frac{17082624}{1251321602543}a^{14}+\frac{115307712}{1251321602543}a^{13}-\frac{184575663}{1251321602543}a^{12}+\frac{764619336}{1251321602543}a^{11}-\frac{1596433372}{1251321602543}a^{10}+\frac{4751828640}{1251321602543}a^{9}-\frac{5923648690}{1251321602543}a^{8}+\frac{25398767472}{1251321602543}a^{7}+\frac{6124140160}{1251321602543}a^{6}+\frac{102995302176}{1251321602543}a^{5}+\frac{113041803265}{1251321602543}a^{4}+\frac{325531124568}{1251321602543}a^{3}+\frac{392171762252}{1251321602543}a^{2}+\frac{352923081264}{1251321602543}a+\frac{11244217726}{18676441829}$, $\frac{116364}{1251321602543}a^{17}+\frac{319124}{1251321602543}a^{16}+\frac{1978188}{1251321602543}a^{15}+\frac{5105984}{1251321602543}a^{14}+\frac{13847316}{1251321602543}a^{13}+\frac{33188896}{1251321602543}a^{12}+\frac{51432888}{1251321602543}a^{11}+\frac{112331648}{1251321602543}a^{10}+\frac{108800340}{1251321602543}a^{9}+\frac{210621840}{1251321602543}a^{8}+\frac{130560408}{1251321602543}a^{7}+\frac{16800391627}{1251321602543}a^{6}+\frac{33827573624}{1251321602543}a^{5}+\frac{99622867458}{1251321602543}a^{4}+\frac{168746186896}{1251321602543}a^{3}+\frac{149293886627}{1251321602543}a^{2}+\frac{168724426828}{1251321602543}a+\frac{33172518846}{1251321602543}$, $\frac{15185}{1251321602543}a^{19}-\frac{106295}{1251321602543}a^{18}+\frac{470735}{1251321602543}a^{17}-\frac{30370}{14059793287}a^{16}+\frac{7167320}{1251321602543}a^{15}-\frac{33194410}{1251321602543}a^{14}+\frac{75187201}{1251321602543}a^{13}-\frac{277627355}{1251321602543}a^{12}+\frac{623501633}{1251321602543}a^{11}-\frac{1917713650}{1251321602543}a^{10}+\frac{4446114165}{1251321602543}a^{9}-\frac{10575076960}{1251321602543}a^{8}+\frac{16979382116}{1251321602543}a^{7}-\frac{43163408055}{1251321602543}a^{6}+\frac{12797677902}{1251321602543}a^{5}-\frac{136566737865}{1251321602543}a^{4}-\frac{96854295134}{1251321602543}a^{3}-\frac{100292371735}{1251321602543}a^{2}-\frac{295654484982}{1251321602543}a+\frac{572594843848}{1251321602543}$, $\frac{15185}{1251321602543}a^{19}-\frac{106295}{1251321602543}a^{18}+\frac{470735}{1251321602543}a^{17}-\frac{30370}{14059793287}a^{16}+\frac{7167320}{1251321602543}a^{15}-\frac{33194410}{1251321602543}a^{14}+\frac{75187201}{1251321602543}a^{13}-\frac{277627355}{1251321602543}a^{12}+\frac{623501633}{1251321602543}a^{11}-\frac{1917713650}{1251321602543}a^{10}+\frac{4446114165}{1251321602543}a^{9}-\frac{10575076960}{1251321602543}a^{8}+\frac{16979382116}{1251321602543}a^{7}-\frac{43163408055}{1251321602543}a^{6}+\frac{12797677902}{1251321602543}a^{5}-\frac{136566737865}{1251321602543}a^{4}-\frac{96854295134}{1251321602543}a^{3}-\frac{100292371735}{1251321602543}a^{2}-\frac{295654484982}{1251321602543}a-\frac{678726758695}{1251321602543}$, $\frac{196460}{1251321602543}a^{19}-\frac{219433}{1251321602543}a^{18}+\frac{4232900}{1251321602543}a^{17}-\frac{6940522}{1251321602543}a^{16}+\frac{45162588}{1251321602543}a^{15}-\frac{93106202}{1251321602543}a^{14}+\frac{354602597}{1251321602543}a^{13}-\frac{820612795}{1251321602543}a^{12}+\frac{2409001268}{1251321602543}a^{11}-\frac{6005866618}{1251321602543}a^{10}+\frac{15773393227}{1251321602543}a^{9}-\frac{16419942825}{1251321602543}a^{8}+\frac{76118618321}{1251321602543}a^{7}-\frac{15540150986}{1251321602543}a^{6}+\frac{129145686420}{1251321602543}a^{5}-\frac{57137833007}{1251321602543}a^{4}+\frac{385303692418}{1251321602543}a^{3}+\frac{400781120177}{1251321602543}a^{2}+\frac{891830200023}{1251321602543}a-\frac{3780110370865}{1251321602543}$, $\frac{30685}{1251321602543}a^{19}-\frac{78445}{1251321602543}a^{18}+\frac{801829}{1251321602543}a^{17}-\frac{2066638}{1251321602543}a^{16}+\frac{10247532}{1251321602543}a^{15}-\frac{24230815}{1251321602543}a^{14}+\frac{85917150}{1251321602543}a^{13}-\frac{174615764}{1251321602543}a^{12}+\frac{519918206}{1251321602543}a^{11}-\frac{968114903}{1251321602543}a^{10}+\frac{2898375011}{1251321602543}a^{9}-\frac{2296877535}{1251321602543}a^{8}+\frac{12180181867}{1251321602543}a^{7}+\frac{2388919939}{1251321602543}a^{6}+\frac{28434384327}{1251321602543}a^{5}+\frac{7423782074}{1251321602543}a^{4}+\frac{98890624465}{1251321602543}a^{3}+\frac{74011404123}{1251321602543}a^{2}+\frac{174736547448}{1251321602543}a-\frac{162022264886}{1251321602543}$, $\frac{30370}{1251321602543}a^{19}-\frac{126281}{1251321602543}a^{18}+\frac{1057834}{1251321602543}a^{17}-\frac{3533174}{1251321602543}a^{16}+\frac{17535601}{1251321602543}a^{15}-\frac{740763}{18676441829}a^{14}+\frac{182563313}{1251321602543}a^{13}-\frac{474941100}{1251321602543}a^{12}+\frac{1448998613}{1251321602543}a^{11}-\frac{3612015969}{1251321602543}a^{10}+\frac{9782933024}{1251321602543}a^{9}-\frac{20785729442}{1251321602543}a^{8}+\frac{49015995391}{1251321602543}a^{7}-\frac{69406800209}{1251321602543}a^{6}+\frac{151161620570}{1251321602543}a^{5}-\frac{65984376841}{1251321602543}a^{4}+\frac{153750974733}{1251321602543}a^{3}+\frac{378634632470}{1251321602543}a^{2}-\frac{333963569267}{1251321602543}a+\frac{1987443145487}{1251321602543}$, $\frac{75295}{1251321602543}a^{19}-\frac{106295}{1251321602543}a^{18}+\frac{1861846}{1251321602543}a^{17}-\frac{2400186}{1251321602543}a^{16}+\frac{20537397}{1251321602543}a^{15}-\frac{25266712}{1251321602543}a^{14}+\frac{144793850}{1251321602543}a^{13}-\frac{202968863}{1251321602543}a^{12}+\frac{878011994}{1251321602543}a^{11}-\frac{1573695624}{1251321602543}a^{10}+\frac{5287909449}{1251321602543}a^{9}-\frac{1743500152}{1251321602543}a^{8}+\frac{19192106674}{1251321602543}a^{7}+\frac{47539664893}{1251321602543}a^{6}+\frac{69388324950}{1251321602543}a^{5}+\frac{178140636223}{1251321602543}a^{4}+\frac{453979952298}{1251321602543}a^{3}+\frac{259101586115}{1251321602543}a^{2}+\frac{818231096343}{1251321602543}a+\frac{2485452357348}{1251321602543}$, $\frac{9678}{1251321602543}a^{19}+\frac{1982}{14059793287}a^{18}-\frac{114702}{1251321602543}a^{17}+\frac{3645660}{1251321602543}a^{16}-\frac{4947224}{1251321602543}a^{15}+\frac{36766837}{1251321602543}a^{14}-\frac{66214002}{1251321602543}a^{13}+\frac{271717781}{1251321602543}a^{12}-\frac{553713785}{1251321602543}a^{11}+\frac{1835121525}{1251321602543}a^{10}-\frac{3926465255}{1251321602543}a^{9}+\frac{12546443885}{1251321602543}a^{8}-\frac{7051571996}{1251321602543}a^{7}+\frac{44005395622}{1251321602543}a^{6}+\frac{12857780809}{1251321602543}a^{5}+\frac{151072064343}{1251321602543}a^{4}+\frac{1887759053}{18676441829}a^{3}+\frac{265733641429}{1251321602543}a^{2}+\frac{434746099699}{1251321602543}a+\frac{2635047798633}{1251321602543}$ 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:  \( 3338983.62101 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 3338983.62101 \cdot 8194}{2\cdot\sqrt{15914835482628690354834740122825579433}}\cr\approx \mathstrut & 0.328834866285 \end{aligned}\] (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 + 27*x^18 - 31*x^17 + 348*x^16 - 472*x^15 + 2986*x^14 - 4874*x^13 + 20558*x^12 - 40054*x^11 + 130294*x^10 - 168519*x^9 + 700706*x^8 - 32881*x^7 + 2845143*x^6 + 2390937*x^5 + 8994354*x^4 + 9962701*x^3 + 26015925*x^2 + 20544384*x + 83519437) 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 + 27*x^18 - 31*x^17 + 348*x^16 - 472*x^15 + 2986*x^14 - 4874*x^13 + 20558*x^12 - 40054*x^11 + 130294*x^10 - 168519*x^9 + 700706*x^8 - 32881*x^7 + 2845143*x^6 + 2390937*x^5 + 8994354*x^4 + 9962701*x^3 + 26015925*x^2 + 20544384*x + 83519437, 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 + 27*x^18 - 31*x^17 + 348*x^16 - 472*x^15 + 2986*x^14 - 4874*x^13 + 20558*x^12 - 40054*x^11 + 130294*x^10 - 168519*x^9 + 700706*x^8 - 32881*x^7 + 2845143*x^6 + 2390937*x^5 + 8994354*x^4 + 9962701*x^3 + 26015925*x^2 + 20544384*x + 83519437); 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 + 27*x^18 - 31*x^17 + 348*x^16 - 472*x^15 + 2986*x^14 - 4874*x^13 + 20558*x^12 - 40054*x^11 + 130294*x^10 - 168519*x^9 + 700706*x^8 - 32881*x^7 + 2845143*x^6 + 2390937*x^5 + 8994354*x^4 + 9962701*x^3 + 26015925*x^2 + 20544384*x + 83519437); 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

$C_{20}$ (as 20T1):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
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 cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{-374 -22 \sqrt{17}})\), \(\Q(\zeta_{11})^+\), 10.10.304358957700017.1

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 ${\href{/padicField/2.5.0.1}{5} }^{4}$ $20$ $20$ $20$ R ${\href{/padicField/13.10.0.1}{10} }^{2}$ R ${\href{/padicField/19.5.0.1}{5} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{5}$ $20$ $20$ $20$ $20$ ${\href{/padicField/43.1.0.1}{1} }^{20}$ ${\href{/padicField/47.5.0.1}{5} }^{4}$ ${\href{/padicField/53.10.0.1}{10} }^{2}$ ${\href{/padicField/59.10.0.1}{10} }^{2}$

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
\(11\) Copy content Toggle raw display 11.2.10.18a1.7$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241653 x^{10} + 2355135020 x^{9} + 1953240660 x^{8} + 1157466240 x^{7} + 496075680 x^{6} + 154293888 x^{5} + 34538880 x^{4} + 5429760 x^{3} + 569600 x^{2} + 35950 x + 1057$$10$$2$$18$20T1$$[\ ]_{10}^{2}$$
\(17\) Copy content Toggle raw display 17.5.4.15a1.2$x^{20} + 4 x^{16} + 56 x^{15} + 6 x^{12} + 168 x^{11} + 1176 x^{10} + 4 x^{8} + 168 x^{7} + 2352 x^{6} + 10976 x^{5} + x^{4} + 56 x^{3} + 1193 x^{2} + 10976 x + 38416$$4$$5$$15$20T1$$[\ ]_{4}^{5}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)