Properties

Label 20.20.589...249.1
Degree $20$
Signature $[20, 0]$
Discriminant $5.891\times 10^{34}$
Root discriminant \(54.77\)
Ramified primes $17,643$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2\times A_5$ (as 20T31)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 5*x^18 + 240*x^17 - 634*x^16 - 1660*x^15 + 7586*x^14 + 1448*x^13 - 34555*x^12 + 17062*x^11 + 80377*x^10 - 61496*x^9 - 107173*x^8 + 88980*x^7 + 85210*x^6 - 61692*x^5 - 38499*x^4 + 18788*x^3 + 7682*x^2 - 1660*x - 101)
 
gp: K = bnfinit(y^20 - 10*y^19 + 5*y^18 + 240*y^17 - 634*y^16 - 1660*y^15 + 7586*y^14 + 1448*y^13 - 34555*y^12 + 17062*y^11 + 80377*y^10 - 61496*y^9 - 107173*y^8 + 88980*y^7 + 85210*y^6 - 61692*y^5 - 38499*y^4 + 18788*y^3 + 7682*y^2 - 1660*y - 101, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 10*x^19 + 5*x^18 + 240*x^17 - 634*x^16 - 1660*x^15 + 7586*x^14 + 1448*x^13 - 34555*x^12 + 17062*x^11 + 80377*x^10 - 61496*x^9 - 107173*x^8 + 88980*x^7 + 85210*x^6 - 61692*x^5 - 38499*x^4 + 18788*x^3 + 7682*x^2 - 1660*x - 101);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 10*x^19 + 5*x^18 + 240*x^17 - 634*x^16 - 1660*x^15 + 7586*x^14 + 1448*x^13 - 34555*x^12 + 17062*x^11 + 80377*x^10 - 61496*x^9 - 107173*x^8 + 88980*x^7 + 85210*x^6 - 61692*x^5 - 38499*x^4 + 18788*x^3 + 7682*x^2 - 1660*x - 101)
 

\( x^{20} - 10 x^{19} + 5 x^{18} + 240 x^{17} - 634 x^{16} - 1660 x^{15} + 7586 x^{14} + 1448 x^{13} - 34555 x^{12} + 17062 x^{11} + 80377 x^{10} - 61496 x^{9} - 107173 x^{8} + \cdots - 101 \) Copy content Toggle raw display

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:  $[20, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(58908368812107185790916140058429249\) \(\medspace = 17^{10}\cdot 643^{8}\) 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:  \(54.77\)
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))
 
Ramified primes:   \(17\), \(643\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{9408446}a^{18}-\frac{9}{9408446}a^{17}+\frac{14237}{276719}a^{16}+\frac{48939}{553438}a^{15}+\frac{1135759}{4704223}a^{14}+\frac{116757}{9408446}a^{13}-\frac{5115}{9408446}a^{12}+\frac{1088145}{9408446}a^{11}-\frac{251121}{4704223}a^{10}+\frac{2014723}{9408446}a^{9}-\frac{4631487}{9408446}a^{8}+\frac{4273267}{9408446}a^{7}-\frac{1456826}{4704223}a^{6}-\frac{3479009}{9408446}a^{5}-\frac{2198962}{4704223}a^{4}+\frac{2138798}{4704223}a^{3}-\frac{13503}{276719}a^{2}+\frac{515256}{4704223}a+\frac{2054418}{4704223}$, $\frac{1}{13538753794}a^{19}+\frac{355}{6769376897}a^{18}-\frac{1570732895}{13538753794}a^{17}+\frac{9452985}{796397282}a^{16}+\frac{344916576}{6769376897}a^{15}-\frac{187682511}{6769376897}a^{14}-\frac{2159971203}{13538753794}a^{13}-\frac{928026701}{6769376897}a^{12}-\frac{66355325}{796397282}a^{11}+\frac{123873746}{6769376897}a^{10}-\frac{3248387037}{6769376897}a^{9}+\frac{1488946467}{6769376897}a^{8}+\frac{3169500153}{6769376897}a^{7}-\frac{1089183294}{6769376897}a^{6}+\frac{5453739921}{13538753794}a^{5}-\frac{565802887}{13538753794}a^{4}+\frac{3907779893}{13538753794}a^{3}-\frac{4849822129}{13538753794}a^{2}+\frac{120425110}{398198641}a+\frac{2497942933}{6769376897}$ 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:  $19$
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:   $\frac{1188849835}{13538753794}a^{19}-\frac{5515919713}{6769376897}a^{18}-\frac{253838361}{6769376897}a^{17}+\frac{8022089925}{398198641}a^{16}-\frac{286816040984}{6769376897}a^{15}-\frac{2040604847567}{13538753794}a^{14}+\frac{3596210237234}{6769376897}a^{13}+\frac{1749952899131}{6769376897}a^{12}-\frac{1876032387805}{796397282}a^{11}+\frac{5480051508203}{6769376897}a^{10}+\frac{65272381849733}{13538753794}a^{9}-\frac{23172199576828}{6769376897}a^{8}-\frac{64050106084993}{13538753794}a^{7}+\frac{29862148929159}{6769376897}a^{6}+\frac{26275895845829}{13538753794}a^{5}-\frac{15610808112304}{6769376897}a^{4}-\frac{1819947137033}{13538753794}a^{3}+\frac{5640050192977}{13538753794}a^{2}-\frac{48413901517}{796397282}a-\frac{6141517166}{6769376897}$, $\frac{4642757889}{13538753794}a^{19}-\frac{21933936023}{6769376897}a^{18}+\frac{5912536925}{13538753794}a^{17}+\frac{62177926349}{796397282}a^{16}-\frac{1196619165910}{6769376897}a^{15}-\frac{3733592537894}{6769376897}a^{14}+\frac{14284634203273}{6769376897}a^{13}+\frac{9454354833545}{13538753794}a^{12}-\frac{120527392996795}{13538753794}a^{11}+\frac{1493561871232}{398198641}a^{10}+\frac{238883577847645}{13538753794}a^{9}-\frac{179780741843373}{13538753794}a^{8}-\frac{234915053641545}{13538753794}a^{7}+\frac{111139258283945}{6769376897}a^{6}+\frac{51692850225527}{6769376897}a^{5}-\frac{118248246651225}{13538753794}a^{4}-\frac{5528899353283}{6769376897}a^{3}+\frac{23127712679415}{13538753794}a^{2}-\frac{1335609197738}{6769376897}a-\frac{114672059043}{6769376897}$, $\frac{312839038}{398198641}a^{19}-\frac{78727657355}{13538753794}a^{18}-\frac{180755914663}{13538753794}a^{17}+\frac{69052545927}{398198641}a^{16}-\frac{33518032855}{796397282}a^{15}-\frac{12869611162302}{6769376897}a^{14}+\frac{13001499920550}{6769376897}a^{13}+\frac{134435323645805}{13538753794}a^{12}-\frac{86408796584818}{6769376897}a^{11}-\frac{390196928798225}{13538753794}a^{10}+\frac{515322371876967}{13538753794}a^{9}+\frac{678355942209087}{13538753794}a^{8}-\frac{788852313175563}{13538753794}a^{7}-\frac{354115872216953}{6769376897}a^{6}+\frac{606964898815671}{13538753794}a^{5}+\frac{200999079297537}{6769376897}a^{4}-\frac{203894935558745}{13538753794}a^{3}-\frac{2753216542935}{398198641}a^{2}+\frac{9776416803854}{6769376897}a+\frac{1215283806465}{13538753794}$, $\frac{3834104485}{13538753794}a^{19}-\frac{39450575833}{13538753794}a^{18}+\frac{17541476887}{6769376897}a^{17}+\frac{51127854795}{796397282}a^{16}-\frac{2689437576455}{13538753794}a^{15}-\frac{4583932277045}{13538753794}a^{14}+\frac{14233468642402}{6769376897}a^{13}-\frac{5109251997280}{6769376897}a^{12}-\frac{52837032158277}{6769376897}a^{11}+\frac{111731426067241}{13538753794}a^{10}+\frac{175533895621785}{13538753794}a^{9}-\frac{277329770114873}{13538753794}a^{8}-\frac{63915905182431}{6769376897}a^{7}+\frac{309365381823413}{13538753794}a^{6}+\frac{21559805687761}{13538753794}a^{5}-\frac{80602323949048}{6769376897}a^{4}+\frac{7427897686444}{6769376897}a^{3}+\frac{32312097348149}{13538753794}a^{2}-\frac{2415929437783}{6769376897}a-\frac{170247835124}{6769376897}$, $\frac{7368292021}{6769376897}a^{19}-\frac{118098532697}{13538753794}a^{18}-\frac{169776679133}{13538753794}a^{17}+\frac{96467952716}{398198641}a^{16}-\frac{2807518766905}{13538753794}a^{15}-\frac{32035247962857}{13538753794}a^{14}+\frac{310602831535}{81070382}a^{13}+\frac{139826345283339}{13538753794}a^{12}-\frac{278154743111831}{13538753794}a^{11}-\frac{168657443155672}{6769376897}a^{10}+\frac{716272578630043}{13538753794}a^{9}+\frac{256935675188862}{6769376897}a^{8}-\frac{489462669630567}{6769376897}a^{7}-\frac{517104637317389}{13538753794}a^{6}+\frac{40437408645635}{796397282}a^{5}+\frac{152699951272385}{6769376897}a^{4}-\frac{106048874302505}{6769376897}a^{3}-\frac{75824411225043}{13538753794}a^{2}+\frac{17595055694673}{13538753794}a+\frac{565206467941}{6769376897}$, $\frac{112260403}{796397282}a^{19}-\frac{10514664905}{13538753794}a^{18}-\frac{31845040427}{6769376897}a^{17}+\frac{11766745646}{398198641}a^{16}+\frac{42048490273}{796397282}a^{15}-\frac{2926704540009}{6769376897}a^{14}-\frac{2343226225435}{13538753794}a^{13}+\frac{41728212037265}{13538753794}a^{12}-\frac{9158457260215}{13538753794}a^{11}-\frac{77328745151734}{6769376897}a^{10}+\frac{36553823221625}{6769376897}a^{9}+\frac{156806479343464}{6769376897}a^{8}-\frac{166523007128167}{13538753794}a^{7}-\frac{174396561598088}{6769376897}a^{6}+\frac{81728857046666}{6769376897}a^{5}+\frac{98382758912220}{6769376897}a^{4}-\frac{66672185545825}{13538753794}a^{3}-\frac{2593440663519}{796397282}a^{2}+\frac{4200449902870}{6769376897}a+\frac{539266086837}{13538753794}$, $\frac{312839038}{398198641}a^{19}-\frac{78727657355}{13538753794}a^{18}-\frac{180755914663}{13538753794}a^{17}+\frac{69052545927}{398198641}a^{16}-\frac{33518032855}{796397282}a^{15}-\frac{12869611162302}{6769376897}a^{14}+\frac{13001499920550}{6769376897}a^{13}+\frac{134435323645805}{13538753794}a^{12}-\frac{86408796584818}{6769376897}a^{11}-\frac{390196928798225}{13538753794}a^{10}+\frac{515322371876967}{13538753794}a^{9}+\frac{678355942209087}{13538753794}a^{8}-\frac{788852313175563}{13538753794}a^{7}-\frac{354115872216953}{6769376897}a^{6}+\frac{606964898815671}{13538753794}a^{5}+\frac{200999079297537}{6769376897}a^{4}-\frac{203894935558745}{13538753794}a^{3}-\frac{2753216542935}{398198641}a^{2}+\frac{9776416803854}{6769376897}a+\frac{1201745052671}{13538753794}$, $\frac{3834104485}{13538753794}a^{19}-\frac{16698704691}{6769376897}a^{18}-\frac{19395544285}{13538753794}a^{17}+\frac{50300418283}{796397282}a^{16}-\frac{1342060254819}{13538753794}a^{15}-\frac{3547970663425}{6769376897}a^{14}+\frac{18149936658795}{13538753794}a^{13}+\frac{10579370726524}{6769376897}a^{12}-\frac{82457863301595}{13538753794}a^{11}-\frac{22085799165509}{13538753794}a^{10}+\frac{90604061903215}{6769376897}a^{9}-\frac{3519171785961}{13538753794}a^{8}-\frac{210940770558567}{13538753794}a^{7}+\frac{16284160815297}{13538753794}a^{6}+\frac{62841426756592}{6769376897}a^{5}-\frac{1972299751082}{6769376897}a^{4}-\frac{15947470954469}{6769376897}a^{3}-\frac{1252538522563}{13538753794}a^{2}+\frac{786837810277}{6769376897}a+\frac{10541868277}{13538753794}$, $\frac{4146384384}{6769376897}a^{19}-\frac{2367641015}{398198641}a^{18}+\frac{15989669601}{6769376897}a^{17}+\frac{110236667655}{796397282}a^{16}-\frac{2376231680734}{6769376897}a^{15}-\frac{12018587602445}{13538753794}a^{14}+\frac{26890565212975}{6769376897}a^{13}+\frac{1579547063646}{6769376897}a^{12}-\frac{106950290486943}{6769376897}a^{11}+\frac{69425293714220}{6769376897}a^{10}+\frac{197252860053256}{6769376897}a^{9}-\frac{199759528336014}{6769376897}a^{8}-\frac{352365063429495}{13538753794}a^{7}+\frac{232048787767059}{6769376897}a^{6}+\frac{130257049896979}{13538753794}a^{5}-\frac{725850687214}{40535191}a^{4}-\frac{2362951792899}{13538753794}a^{3}+\frac{23843411436514}{6769376897}a^{2}-\frac{6221726677677}{13538753794}a-\frac{451215721045}{13538753794}$, $\frac{11936813192}{6769376897}a^{19}-\frac{205283492429}{13538753794}a^{18}-\frac{74272015667}{6769376897}a^{17}+\frac{314325497589}{796397282}a^{16}-\frac{7696562862841}{13538753794}a^{15}-\frac{22988890847771}{6769376897}a^{14}+\frac{54230052235116}{6769376897}a^{13}+\frac{75998458958603}{6769376897}a^{12}-\frac{507897434137419}{13538753794}a^{11}-\frac{111189653273974}{6769376897}a^{10}+\frac{11\!\cdots\!09}{13538753794}a^{9}+\frac{154393043534045}{13538753794}a^{8}-\frac{13\!\cdots\!13}{13538753794}a^{7}-\frac{47957530745787}{6769376897}a^{6}+\frac{862818298885641}{13538753794}a^{5}+\frac{88936259781683}{13538753794}a^{4}-\frac{229321624189893}{13538753794}a^{3}-\frac{34501372602653}{13538753794}a^{2}+\frac{12867161497523}{13538753794}a+\frac{715399068821}{13538753794}$, $\frac{315431465}{796397282}a^{19}-\frac{29290738942}{6769376897}a^{18}+\frac{39903578012}{6769376897}a^{17}+\frac{72141989699}{796397282}a^{16}-\frac{267554551055}{796397282}a^{15}-\frac{2505458118453}{6769376897}a^{14}+\frac{23144341403154}{6769376897}a^{13}-\frac{16438120586520}{6769376897}a^{12}-\frac{82916789048726}{6769376897}a^{11}+\frac{121330306074761}{6769376897}a^{10}+\frac{253960672657099}{13538753794}a^{9}-\frac{580854245761703}{13538753794}a^{8}-\frac{69672286903945}{6769376897}a^{7}+\frac{328085645104736}{6769376897}a^{6}-\frac{16792757432147}{6769376897}a^{5}-\frac{353282272578695}{13538753794}a^{4}+\frac{26766080544971}{6769376897}a^{3}+\frac{4342106570765}{796397282}a^{2}-\frac{11938798368571}{13538753794}a-\frac{391241705058}{6769376897}$, $\frac{7368292021}{6769376897}a^{19}-\frac{161896564101}{13538753794}a^{18}+\frac{224405603503}{13538753794}a^{17}+\frac{99871138790}{398198641}a^{16}-\frac{12667983785449}{13538753794}a^{15}-\frac{6984282495982}{6769376897}a^{14}+\frac{3800994832855}{398198641}a^{13}-\frac{45270574148163}{6769376897}a^{12}-\frac{233237144060213}{6769376897}a^{11}+\frac{674441766913295}{13538753794}a^{10}+\frac{361015687866890}{6769376897}a^{9}-\frac{810113319191129}{6769376897}a^{8}-\frac{405270135533951}{13538753794}a^{7}+\frac{107644137757545}{796397282}a^{6}-\frac{86290701887859}{13538753794}a^{5}-\frac{490451439765329}{6769376897}a^{4}+\frac{73762547976492}{6769376897}a^{3}+\frac{101712916971756}{6769376897}a^{2}-\frac{32724670976587}{13538753794}a-\frac{1072252255807}{6769376897}$, $\frac{3606005227}{6769376897}a^{19}-\frac{56949437669}{13538753794}a^{18}-\frac{90007222779}{13538753794}a^{17}+\frac{46979735749}{398198641}a^{16}-\frac{594809252956}{6769376897}a^{15}-\frac{15886135766297}{13538753794}a^{14}+\frac{11868141082790}{6769376897}a^{13}+\frac{71748740257127}{13538753794}a^{12}-\frac{65166389880551}{6769376897}a^{11}-\frac{90673728427501}{6769376897}a^{10}+\frac{342205697128627}{13538753794}a^{9}+\frac{288345500775895}{13538753794}a^{8}-\frac{476496638127929}{13538753794}a^{7}-\frac{147478336672966}{6769376897}a^{6}+\frac{170381186802839}{6769376897}a^{5}+\frac{172192750964731}{13538753794}a^{4}-\frac{53619772046849}{6769376897}a^{3}-\frac{41767522118645}{13538753794}a^{2}+\frac{9268144845235}{13538753794}a+\frac{278706250230}{6769376897}$, $\frac{3488929427}{6769376897}a^{19}-\frac{26569234875}{6769376897}a^{18}-\frac{106058363467}{13538753794}a^{17}+\frac{45630377062}{398198641}a^{16}-\frac{703235728463}{13538753794}a^{15}-\frac{98485279211}{81070382}a^{14}+\frac{9963591784193}{6769376897}a^{13}+\frac{81658864880575}{13538753794}a^{12}-\frac{61382696057292}{6769376897}a^{11}-\frac{112222244977664}{6769376897}a^{10}+\frac{348839656936079}{13538753794}a^{9}+\frac{373840302166721}{13538753794}a^{8}-\frac{514541566468781}{13538753794}a^{7}-\frac{382966379986347}{13538753794}a^{6}+\frac{11311330681702}{398198641}a^{5}+\frac{218050894935669}{13538753794}a^{4}-\frac{126274365851461}{13538753794}a^{3}-\frac{25641114356014}{6769376897}a^{2}+\frac{11834766962641}{13538753794}a+\frac{598458031559}{13538753794}$, $\frac{2484616119}{6769376897}a^{19}-\frac{54921209619}{13538753794}a^{18}+\frac{2344303805}{398198641}a^{17}+\frac{33397228549}{398198641}a^{16}-\frac{4340005318951}{13538753794}a^{15}-\frac{4351087717479}{13538753794}a^{14}+\frac{43558935712381}{13538753794}a^{13}-\frac{16705090986317}{6769376897}a^{12}-\frac{76412097291835}{6769376897}a^{11}+\frac{233183664062209}{13538753794}a^{10}+\frac{226220607979375}{13538753794}a^{9}-\frac{545735949641995}{13538753794}a^{8}-\frac{56343726326457}{6769376897}a^{7}+\frac{303052446993603}{6769376897}a^{6}-\frac{21062977836921}{6769376897}a^{5}-\frac{321566265703913}{13538753794}a^{4}+\frac{3044876378219}{796397282}a^{3}+\frac{66232472674141}{13538753794}a^{2}-\frac{5442889810018}{6769376897}a-\frac{40433984151}{796397282}$, $\frac{29604688}{40535191}a^{19}-\frac{81250051087}{13538753794}a^{18}-\frac{95193927145}{13538753794}a^{17}+\frac{64756360587}{398198641}a^{16}-\frac{1173783576928}{6769376897}a^{15}-\frac{10274532042632}{6769376897}a^{14}+\frac{19181965736557}{6769376897}a^{13}+\frac{41077325487543}{6769376897}a^{12}-\frac{97318302610880}{6769376897}a^{11}-\frac{10313955440177}{796397282}a^{10}+\frac{478015162971675}{13538753794}a^{9}+\frac{235848835546503}{13538753794}a^{8}-\frac{312817505379806}{6769376897}a^{7}-\frac{223308235732079}{13538753794}a^{6}+\frac{421916579989783}{13538753794}a^{5}+\frac{66885766967103}{6769376897}a^{4}-\frac{124440120042145}{13538753794}a^{3}-\frac{34521305091859}{13538753794}a^{2}+\frac{9343098677691}{13538753794}a+\frac{272674038439}{6769376897}$, $\frac{394103}{9408446}a^{18}-\frac{3546927}{9408446}a^{17}-\frac{96785}{553438}a^{16}+\frac{2751758}{276719}a^{15}-\frac{156817905}{9408446}a^{14}-\frac{820960217}{9408446}a^{13}+\frac{1101925593}{4704223}a^{12}+\frac{2751506487}{9408446}a^{11}-\frac{11078637143}{9408446}a^{10}-\frac{1709287465}{4704223}a^{9}+\frac{27559983403}{9408446}a^{8}-\frac{625446935}{9408446}a^{7}-\frac{18222648276}{4704223}a^{6}+\frac{4840892897}{9408446}a^{5}+\frac{24116839715}{9408446}a^{4}-\frac{3684752947}{9408446}a^{3}-\frac{182347541}{276719}a^{2}+\frac{434233866}{4704223}a+\frac{87516393}{9408446}$, $\frac{2235460443}{6769376897}a^{19}-\frac{38230833893}{13538753794}a^{18}-\frac{30236216993}{13538753794}a^{17}+\frac{29508046370}{398198641}a^{16}-\frac{690960832953}{6769376897}a^{15}-\frac{4397601737632}{6769376897}a^{14}+\frac{9943201118885}{6769376897}a^{13}+\frac{15333329885113}{6769376897}a^{12}-\frac{2787188481747}{398198641}a^{11}-\frac{51900377292847}{13538753794}a^{10}+\frac{221372139693327}{13538753794}a^{9}+\frac{51030645865241}{13538753794}a^{8}-\frac{139133586667298}{6769376897}a^{7}-\frac{41012205386281}{13538753794}a^{6}+\frac{181399255730675}{13538753794}a^{5}+\frac{13851899189961}{6769376897}a^{4}-\frac{50991754014427}{13538753794}a^{3}-\frac{8608133134159}{13538753794}a^{2}+\frac{177647277053}{796397282}a+\frac{77865737982}{6769376897}$, $\frac{5386204733}{6769376897}a^{19}-\frac{85900466337}{13538753794}a^{18}-\frac{63345748349}{6769376897}a^{17}+\frac{140396343947}{796397282}a^{16}-\frac{1968844629291}{13538753794}a^{15}-\frac{11677475098175}{6769376897}a^{14}+\frac{18469691062927}{6769376897}a^{13}+\frac{51309529509387}{6769376897}a^{12}-\frac{198292039999313}{13538753794}a^{11}-\frac{125840832165811}{6769376897}a^{10}+\frac{512603837606145}{13538753794}a^{9}+\frac{391244839966921}{13538753794}a^{8}-\frac{706230711388401}{13538753794}a^{7}-\frac{198776686936366}{6769376897}a^{6}+\frac{501839626388805}{13538753794}a^{5}+\frac{13737546505879}{796397282}a^{4}-\frac{157463004807565}{13538753794}a^{3}-\frac{57265464089663}{13538753794}a^{2}+\frac{13611258444923}{13538753794}a+\frac{801079320403}{13538753794}$ 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:  \( 94964848909.5 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 94964848909.5 \cdot 1}{2\cdot\sqrt{58908368812107185790916140058429249}}\cr\approx \mathstrut & 0.205137142031 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 5*x^18 + 240*x^17 - 634*x^16 - 1660*x^15 + 7586*x^14 + 1448*x^13 - 34555*x^12 + 17062*x^11 + 80377*x^10 - 61496*x^9 - 107173*x^8 + 88980*x^7 + 85210*x^6 - 61692*x^5 - 38499*x^4 + 18788*x^3 + 7682*x^2 - 1660*x - 101)
 
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 - 10*x^19 + 5*x^18 + 240*x^17 - 634*x^16 - 1660*x^15 + 7586*x^14 + 1448*x^13 - 34555*x^12 + 17062*x^11 + 80377*x^10 - 61496*x^9 - 107173*x^8 + 88980*x^7 + 85210*x^6 - 61692*x^5 - 38499*x^4 + 18788*x^3 + 7682*x^2 - 1660*x - 101, 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 - 10*x^19 + 5*x^18 + 240*x^17 - 634*x^16 - 1660*x^15 + 7586*x^14 + 1448*x^13 - 34555*x^12 + 17062*x^11 + 80377*x^10 - 61496*x^9 - 107173*x^8 + 88980*x^7 + 85210*x^6 - 61692*x^5 - 38499*x^4 + 18788*x^3 + 7682*x^2 - 1660*x - 101);
 
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 - 10*x^19 + 5*x^18 + 240*x^17 - 634*x^16 - 1660*x^15 + 7586*x^14 + 1448*x^13 - 34555*x^12 + 17062*x^11 + 80377*x^10 - 61496*x^9 - 107173*x^8 + 88980*x^7 + 85210*x^6 - 61692*x^5 - 38499*x^4 + 18788*x^3 + 7682*x^2 - 1660*x - 101);
 
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

$C_2\times A_5$ (as 20T31):

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 120
The 10 conjugacy class representatives for $C_2\times A_5$
Character table for $C_2\times A_5$

Intermediate fields

\(\Q(\sqrt{17}) \), 10.10.14277086054271121.1

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 sibling: data not computed
Degree 12 siblings: data not computed
Degree 20 sibling: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: data not computed
Minimal sibling: 10.10.242710462922609057.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.5.0.1}{5} }^{4}$ ${\href{/padicField/3.10.0.1}{10} }^{2}$ ${\href{/padicField/5.10.0.1}{10} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.10.0.1}{10} }^{2}$ ${\href{/padicField/13.3.0.1}{3} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ R ${\href{/padicField/19.5.0.1}{5} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{10}$ ${\href{/padicField/29.6.0.1}{6} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.10.0.1}{10} }^{2}$ ${\href{/padicField/41.10.0.1}{10} }^{2}$ ${\href{/padicField/43.5.0.1}{5} }^{4}$ ${\href{/padicField/47.5.0.1}{5} }^{4}$ ${\href{/padicField/53.3.0.1}{3} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.3.0.1}{3} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(17\) Copy content Toggle raw display 17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(643\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$