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Properties

Label 20.20.178...125.1
Degree $20$
Signature $(20, 0)$
Discriminant $1.784\times 10^{37}$
Root discriminant \(72.87\)
Ramified primes $5,19$
Class number $1$ (GRH)
Class group trivial (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 - 100*x^18 + 4250*x^16 - 101*x^15 - 100000*x^14 + 7575*x^13 + 1421875*x^12 - 227250*x^11 - 12505424*x^10 + 3471875*x^9 + 66677450*x^8 - 28406250*x^7 - 200058500*x^6 + 118907199*x^5 + 278040625*x^4 - 210961225*x^3 - 84750000*x^2 + 68478000*x - 6502099)
 
Copy content gp:K = bnfinit(y^20 - 100*y^18 + 4250*y^16 - 101*y^15 - 100000*y^14 + 7575*y^13 + 1421875*y^12 - 227250*y^11 - 12505424*y^10 + 3471875*y^9 + 66677450*y^8 - 28406250*y^7 - 200058500*y^6 + 118907199*y^5 + 278040625*y^4 - 210961225*y^3 - 84750000*y^2 + 68478000*y - 6502099, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 100*x^18 + 4250*x^16 - 101*x^15 - 100000*x^14 + 7575*x^13 + 1421875*x^12 - 227250*x^11 - 12505424*x^10 + 3471875*x^9 + 66677450*x^8 - 28406250*x^7 - 200058500*x^6 + 118907199*x^5 + 278040625*x^4 - 210961225*x^3 - 84750000*x^2 + 68478000*x - 6502099);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 100*x^18 + 4250*x^16 - 101*x^15 - 100000*x^14 + 7575*x^13 + 1421875*x^12 - 227250*x^11 - 12505424*x^10 + 3471875*x^9 + 66677450*x^8 - 28406250*x^7 - 200058500*x^6 + 118907199*x^5 + 278040625*x^4 - 210961225*x^3 - 84750000*x^2 + 68478000*x - 6502099)
 

\( x^{20} - 100 x^{18} + 4250 x^{16} - 101 x^{15} - 100000 x^{14} + 7575 x^{13} + 1421875 x^{12} + \cdots - 6502099 \) 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:  $(20, 0)$
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:   \(17843751288604107685387134552001953125\) \(\medspace = 5^{35}\cdot 19^{10}\) 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:  \(72.87\)
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:  $5^{7/4}19^{1/2}\approx 72.87428522161984$
Ramified primes:   \(5\), \(19\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_{20}$
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:  \(475=5^{2}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{475}(1,·)$, $\chi_{475}(322,·)$, $\chi_{475}(132,·)$, $\chi_{475}(134,·)$, $\chi_{475}(398,·)$, $\chi_{475}(208,·)$, $\chi_{475}(18,·)$, $\chi_{475}(227,·)$, $\chi_{475}(324,·)$, $\chi_{475}(286,·)$, $\chi_{475}(229,·)$, $\chi_{475}(96,·)$, $\chi_{475}(417,·)$, $\chi_{475}(419,·)$, $\chi_{475}(37,·)$, $\chi_{475}(39,·)$, $\chi_{475}(303,·)$, $\chi_{475}(113,·)$, $\chi_{475}(381,·)$, $\chi_{475}(191,·)$$\rbrace$
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11}a^{10}+\frac{5}{11}a^{8}-\frac{5}{11}a^{6}-\frac{1}{11}a^{5}-\frac{2}{11}a^{4}+\frac{3}{11}a^{3}+\frac{5}{11}a^{2}-\frac{4}{11}a-\frac{1}{11}$, $\frac{1}{11}a^{11}+\frac{5}{11}a^{9}-\frac{5}{11}a^{7}-\frac{1}{11}a^{6}-\frac{2}{11}a^{5}+\frac{3}{11}a^{4}+\frac{5}{11}a^{3}-\frac{4}{11}a^{2}-\frac{1}{11}a$, $\frac{1}{11}a^{12}+\frac{3}{11}a^{8}-\frac{1}{11}a^{7}+\frac{1}{11}a^{6}-\frac{3}{11}a^{5}+\frac{4}{11}a^{4}+\frac{3}{11}a^{3}-\frac{4}{11}a^{2}-\frac{2}{11}a+\frac{5}{11}$, $\frac{1}{200108161}a^{13}+\frac{6671470}{200108161}a^{12}-\frac{65}{200108161}a^{11}-\frac{71878}{200108161}a^{10}+\frac{1625}{200108161}a^{9}-\frac{89341000}{200108161}a^{8}-\frac{36402802}{200108161}a^{7}+\frac{86314489}{200108161}a^{6}+\frac{72880354}{200108161}a^{5}-\frac{76012471}{200108161}a^{4}-\frac{36667677}{200108161}a^{3}-\frac{80196297}{200108161}a^{2}+\frac{54778078}{200108161}a-\frac{29266262}{200108161}$, $\frac{1}{200108161}a^{14}-\frac{70}{200108161}a^{12}-\frac{3025952}{200108161}a^{11}+\frac{175}{18191651}a^{10}+\frac{93660756}{200108161}a^{9}-\frac{26250}{200108161}a^{8}-\frac{54050020}{200108161}a^{7}+\frac{54758703}{200108161}a^{6}+\frac{18146400}{200108161}a^{5}+\frac{35770802}{200108161}a^{4}-\frac{33622884}{200108161}a^{3}+\frac{18957276}{200108161}a^{2}+\frac{15431233}{200108161}a-\frac{156250}{200108161}$, $\frac{1}{200108161}a^{15}-\frac{9005978}{200108161}a^{12}-\frac{2625}{200108161}a^{11}-\frac{2328959}{200108161}a^{10}+\frac{87500}{200108161}a^{9}+\frac{22774528}{200108161}a^{8}-\frac{19372901}{200108161}a^{7}+\frac{38724149}{200108161}a^{6}+\frac{43733302}{200108161}a^{5}-\frac{60725413}{200108161}a^{4}-\frac{37332276}{200108161}a^{3}+\frac{41102253}{200108161}a^{2}-\frac{58704104}{200108161}a+\frac{79784827}{200108161}$, $\frac{1}{200108161}a^{16}-\frac{3000}{200108161}a^{12}-\frac{5584697}{200108161}a^{11}+\frac{10000}{18191651}a^{10}-\frac{41365230}{200108161}a^{9}-\frac{1687500}{200108161}a^{8}-\frac{64798003}{200108161}a^{7}-\frac{60166604}{200108161}a^{6}-\frac{2595723}{200108161}a^{5}-\frac{25558349}{200108161}a^{4}-\frac{13263717}{200108161}a^{3}-\frac{34708255}{200108161}a^{2}-\frac{3864399}{18191651}a-\frac{11718750}{200108161}$, $\frac{1}{200108161}a^{17}-\frac{1990797}{200108161}a^{12}-\frac{85000}{200108161}a^{11}-\frac{210556}{18191651}a^{10}+\frac{3187500}{200108161}a^{9}-\frac{70203820}{200108161}a^{8}-\frac{9516698}{200108161}a^{7}-\frac{71855661}{200108161}a^{6}+\frac{42816886}{200108161}a^{5}-\frac{36523083}{200108161}a^{4}-\frac{14634007}{200108161}a^{3}-\frac{28623263}{200108161}a^{2}+\frac{15523418}{200108161}a-\frac{5878274}{200108161}$, $\frac{1}{200108161}a^{18}-\frac{102000}{200108161}a^{12}-\frac{4376364}{200108161}a^{11}+\frac{382500}{18191651}a^{10}-\frac{505969}{200108161}a^{9}-\frac{68850000}{200108161}a^{8}+\frac{37735979}{200108161}a^{7}+\frac{722011}{18191651}a^{6}-\frac{51227632}{200108161}a^{5}+\frac{70389959}{200108161}a^{4}-\frac{73724320}{200108161}a^{3}-\frac{26508}{18191651}a^{2}+\frac{10743923}{200108161}a+\frac{69074483}{200108161}$, $\frac{1}{200108161}a^{19}+\frac{8666330}{200108161}a^{12}-\frac{2422500}{200108161}a^{11}-\frac{826616}{200108161}a^{10}+\frac{96900000}{200108161}a^{9}+\frac{90429664}{200108161}a^{8}+\frac{38215382}{200108161}a^{7}-\frac{77534196}{200108161}a^{6}+\frac{4402270}{18191651}a^{5}-\frac{45918469}{200108161}a^{4}-\frac{27241545}{200108161}a^{3}+\frac{82621885}{200108161}a^{2}-\frac{77999214}{200108161}a-\frac{1631346}{18191651}$ 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:  Trivial group, which has order $1$ (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:  $C_{2}\times C_{2}$, which has order $4$ (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);
 

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:  $19$
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:   $\frac{101}{18191651}a^{15}-\frac{7575}{18191651}a^{13}+\frac{227250}{18191651}a^{11}-\frac{7076}{18191651}a^{10}-\frac{3471875}{18191651}a^{9}+\frac{353800}{18191651}a^{8}+\frac{28406250}{18191651}a^{7}-\frac{6191500}{18191651}a^{6}-\frac{119306250}{18191651}a^{5}+\frac{44225000}{18191651}a^{4}+\frac{220937500}{18191651}a^{3}-\frac{110562500}{18191651}a^{2}-\frac{118359375}{18191651}a+\frac{26033349}{18191651}$, $\frac{101}{200108161}a^{19}-\frac{100}{200108161}a^{18}-\frac{9191}{200108161}a^{17}+\frac{8091}{200108161}a^{16}+\frac{348349}{200108161}a^{15}-\frac{271856}{200108161}a^{14}-\frac{7103324}{200108161}a^{13}+\frac{4944295}{200108161}a^{12}+\frac{83925560}{200108161}a^{11}-\frac{53227464}{200108161}a^{10}-\frac{574929955}{200108161}a^{9}+\frac{31455850}{18191651}a^{8}+\frac{195242414}{18191651}a^{7}-\frac{1288202000}{200108161}a^{6}-\frac{3718966765}{200108161}a^{5}+\frac{2226242724}{200108161}a^{4}+\frac{1915550551}{200108161}a^{3}-\frac{688921084}{200108161}a^{2}-\frac{210606531}{200108161}a+\frac{144404151}{200108161}$, $\frac{3024}{200108161}a^{13}-\frac{15679}{200108161}a^{12}-\frac{196560}{200108161}a^{11}+\frac{940740}{200108161}a^{10}+\frac{4914000}{200108161}a^{9}-\frac{21166650}{200108161}a^{8}-\frac{58968000}{200108161}a^{7}+\frac{219506000}{200108161}a^{6}+\frac{343980000}{200108161}a^{5}-\frac{1028934375}{200108161}a^{4}-\frac{859950000}{200108161}a^{3}+\frac{1763887500}{200108161}a^{2}+\frac{614250000}{200108161}a-\frac{489968750}{200108161}$, $\frac{56}{200108161}a^{19}-\frac{5041}{200108161}a^{17}+\frac{190196}{200108161}a^{15}-\frac{2531}{200108161}a^{14}-\frac{3905276}{200108161}a^{13}+\frac{14681}{18191651}a^{12}+\frac{47402881}{200108161}a^{11}-\frac{4009271}{200108161}a^{10}-\frac{345680255}{200108161}a^{9}+\frac{48855351}{200108161}a^{8}+\frac{1469142100}{200108161}a^{7}-\frac{302160831}{200108161}a^{6}-\frac{3336664625}{200108161}a^{5}+\frac{878434855}{200108161}a^{4}+\frac{3338959375}{200108161}a^{3}-\frac{959983100}{200108161}a^{2}-\frac{893878125}{200108161}a+\frac{19459000}{18191651}$, $\frac{56}{200108161}a^{19}-\frac{5320}{200108161}a^{17}+\frac{212800}{200108161}a^{15}-\frac{4655000}{200108161}a^{13}+\frac{60515000}{200108161}a^{11}-\frac{43225000}{18191651}a^{9}+\frac{199500000}{18191651}a^{7}-\frac{831041}{200108161}a^{6}-\frac{498750000}{18191651}a^{5}+\frac{24931230}{200108161}a^{4}+\frac{6234375000}{200108161}a^{3}-\frac{186984225}{200108161}a^{2}-\frac{2078125000}{200108161}a+\frac{207760250}{200108161}$, $\frac{100}{200108161}a^{18}-\frac{404}{200108161}a^{17}-\frac{9000}{200108161}a^{16}+\frac{34340}{200108161}a^{15}+\frac{337500}{200108161}a^{14}-\frac{1212000}{200108161}a^{13}-\frac{6796696}{200108161}a^{12}+\frac{22977500}{200108161}a^{11}+\frac{78739260}{200108161}a^{10}-\frac{252500000}{200108161}a^{9}-\frac{47087000}{18191651}a^{8}+\frac{146534436}{18191651}a^{7}+\frac{1741065000}{200108161}a^{6}-\frac{5600132860}{200108161}a^{5}-\frac{182500000}{18191651}a^{4}+\frac{8734074449}{200108161}a^{3}-\frac{1362570896}{200108161}a^{2}-\frac{3064821735}{200108161}a+\frac{59155360}{18191651}$, $\frac{279}{200108161}a^{17}-\frac{23715}{200108161}a^{15}+\frac{830025}{200108161}a^{13}-\frac{15414750}{200108161}a^{11}+\frac{14821875}{18191651}a^{9}-\frac{308549}{200108161}a^{8}-\frac{88931250}{18191651}a^{7}+\frac{12341960}{200108161}a^{6}+\frac{3112593750}{200108161}a^{5}-\frac{154274500}{200108161}a^{4}-\frac{4446562500}{200108161}a^{3}+\frac{617098000}{200108161}a^{2}+\frac{1852734375}{200108161}a-\frac{385686250}{200108161}$, $\frac{45}{200108161}a^{19}-\frac{4275}{200108161}a^{17}-\frac{909}{200108161}a^{16}+\frac{171000}{200108161}a^{15}+\frac{68175}{200108161}a^{14}-\frac{3740625}{200108161}a^{13}-\frac{2045250}{200108161}a^{12}+\frac{48691809}{200108161}a^{11}+\frac{2840625}{18191651}a^{10}-\frac{385262325}{200108161}a^{9}-\frac{255656250}{200108161}a^{8}+\frac{1819161000}{200108161}a^{7}+\frac{1070164791}{200108161}a^{6}-\frac{4806618750}{200108161}a^{5}-\frac{1898885381}{200108161}a^{4}+\frac{6004828125}{200108161}a^{3}+\frac{620989120}{200108161}a^{2}-\frac{1904222016}{200108161}a-\frac{11717800}{200108161}$, $\frac{2531}{200108161}a^{14}-\frac{177170}{200108161}a^{12}+\frac{559}{200108161}a^{11}+\frac{442925}{18191651}a^{10}-\frac{2795}{18191651}a^{9}-\frac{66438750}{200108161}a^{8}+\frac{55900}{18191651}a^{7}+\frac{465071250}{200108161}a^{6}-\frac{489125}{18191651}a^{5}-\frac{1550237500}{200108161}a^{4}+\frac{1746875}{18191651}a^{3}+\frac{1937796875}{200108161}a^{2}-\frac{1746875}{18191651}a-\frac{395468750}{200108161}$, $\frac{148}{200108161}a^{19}+\frac{52}{200108161}a^{18}-\frac{13993}{200108161}a^{17}-\frac{5646}{200108161}a^{16}+\frac{554382}{200108161}a^{15}+\frac{239121}{200108161}a^{14}-\frac{11921345}{200108161}a^{13}-\frac{5089759}{200108161}a^{12}+\frac{150556971}{200108161}a^{11}+\frac{57318013}{200108161}a^{10}-\frac{1126970672}{200108161}a^{9}-\frac{319156957}{200108161}a^{8}+\frac{4776297921}{200108161}a^{7}+\frac{635563611}{200108161}a^{6}-\frac{10044950336}{200108161}a^{5}+\frac{350567643}{200108161}a^{4}+\frac{7259673505}{200108161}a^{3}+\frac{24039031}{200108161}a^{2}-\frac{1521292643}{200108161}a+\frac{157407607}{200108161}$, $\frac{247}{200108161}a^{19}-\frac{1010}{200108161}a^{18}-\frac{21931}{200108161}a^{17}+\frac{89284}{200108161}a^{16}+\frac{807099}{200108161}a^{15}-\frac{3287550}{200108161}a^{14}-\frac{15814140}{200108161}a^{13}+\frac{65170250}{200108161}a^{12}+\frac{175174941}{200108161}a^{11}-\frac{749215914}{200108161}a^{10}-\frac{1053238300}{200108161}a^{9}+\frac{4992020845}{200108161}a^{8}+\frac{2716345250}{200108161}a^{7}-\frac{17997248259}{200108161}a^{6}+\frac{1259550000}{200108161}a^{5}+\frac{28827585067}{200108161}a^{4}-\frac{14456099115}{200108161}a^{3}-\frac{9503412683}{200108161}a^{2}+\frac{5811125016}{200108161}a-\frac{315263665}{200108161}$, $\frac{74}{200108161}a^{18}+\frac{250}{200108161}a^{17}-\frac{4438}{200108161}a^{16}-\frac{20139}{200108161}a^{15}+\frac{77545}{200108161}a^{14}+\frac{674577}{200108161}a^{13}+\frac{312600}{200108161}a^{12}-\frac{12310466}{200108161}a^{11}-\frac{26113711}{200108161}a^{10}+\frac{134402745}{200108161}a^{9}+\frac{327674948}{200108161}a^{8}-\frac{892560007}{200108161}a^{7}-\frac{1725588670}{200108161}a^{6}+\frac{3377072245}{200108161}a^{5}+\frac{3571580250}{200108161}a^{4}-\frac{5967016648}{200108161}a^{3}-\frac{1240330422}{200108161}a^{2}+\frac{344270}{25951}a-\frac{466625636}{200108161}$, $\frac{190}{200108161}a^{19}-\frac{683}{200108161}a^{18}-\frac{1595}{18191651}a^{17}+\frac{57632}{200108161}a^{16}+\frac{677964}{200108161}a^{15}-\frac{183804}{18191651}a^{14}-\frac{14195425}{200108161}a^{13}+\frac{38263700}{200108161}a^{12}+\frac{174285941}{200108161}a^{11}-\frac{424051564}{200108161}a^{10}-\frac{1268443565}{200108161}a^{9}+\frac{2791415700}{200108161}a^{8}+\frac{5268591045}{200108161}a^{7}-\frac{10513492438}{200108161}a^{6}-\frac{11319376825}{200108161}a^{5}+\frac{20366755687}{200108161}a^{4}+\frac{10383286729}{200108161}a^{3}-\frac{16056938995}{200108161}a^{2}-\frac{3604802572}{200108161}a+\frac{3656510205}{200108161}$, $\frac{200}{200108161}a^{19}-\frac{17879}{200108161}a^{17}-\frac{2645}{200108161}a^{16}+\frac{663604}{200108161}a^{15}+\frac{17400}{18191651}a^{14}-\frac{13213307}{200108161}a^{13}-\frac{5494917}{200108161}a^{12}+\frac{152402495}{200108161}a^{11}+\frac{79487856}{200108161}a^{10}-\frac{1030046924}{200108161}a^{9}-\frac{608314742}{200108161}a^{8}+\frac{3984906205}{200108161}a^{7}+\frac{214784740}{18191651}a^{6}-\frac{8504041825}{200108161}a^{5}-\frac{3913546227}{200108161}a^{4}+\frac{9427799375}{200108161}a^{3}+\frac{747416040}{200108161}a^{2}-\frac{2794750210}{200108161}a+\frac{323759756}{200108161}$, $\frac{315}{200108161}a^{19}-\frac{631}{200108161}a^{18}-\frac{28790}{200108161}a^{17}+\frac{58046}{200108161}a^{16}+\frac{1095772}{200108161}a^{15}-\frac{2239195}{200108161}a^{14}-\frac{22394198}{200108161}a^{13}+\frac{46822108}{200108161}a^{12}+\frac{263411238}{200108161}a^{11}-\frac{571437478}{200108161}a^{10}-\frac{1762051617}{200108161}a^{9}+\frac{4064300149}{200108161}a^{8}+\frac{6055980066}{200108161}a^{7}-\frac{15721852397}{200108161}a^{6}-\frac{7448699510}{200108161}a^{5}+\frac{27363310558}{200108161}a^{4}-\frac{3486443541}{200108161}a^{3}-\frac{11196929024}{200108161}a^{2}+\frac{1876340922}{200108161}a-\frac{82087066}{200108161}$, $\frac{79}{200108161}a^{19}-\frac{148}{200108161}a^{18}-\frac{653}{18191651}a^{17}+\frac{11988}{200108161}a^{16}+\frac{275153}{200108161}a^{15}-\frac{401020}{200108161}a^{14}-\frac{5768000}{200108161}a^{13}+\frac{7160944}{200108161}a^{12}+\frac{71934091}{200108161}a^{11}-\frac{73608888}{200108161}a^{10}-\frac{543405169}{200108161}a^{9}+\frac{436182245}{200108161}a^{8}+\frac{2423972069}{200108161}a^{7}-\frac{1392006636}{200108161}a^{6}-\frac{5936552414}{200108161}a^{5}+\frac{1942197627}{200108161}a^{4}+\frac{6911712057}{200108161}a^{3}-\frac{501599607}{200108161}a^{2}-\frac{2977715867}{200108161}a+\frac{290557653}{200108161}$, $\frac{1}{18191651}a^{19}-\frac{21}{200108161}a^{18}-\frac{381}{200108161}a^{17}+\frac{2540}{200108161}a^{16}-\frac{15442}{200108161}a^{15}-\frac{118823}{200108161}a^{14}+\frac{1139379}{200108161}a^{13}+\frac{2817414}{200108161}a^{12}-\frac{27771615}{200108161}a^{11}-\frac{36123712}{200108161}a^{10}+\frac{351251178}{200108161}a^{9}+\frac{236841706}{200108161}a^{8}-\frac{2471654518}{200108161}a^{7}-\frac{551664947}{200108161}a^{6}+\frac{9298622930}{200108161}a^{5}-\frac{1222686639}{200108161}a^{4}-\frac{15745942222}{200108161}a^{3}+\frac{5780801129}{200108161}a^{2}+\frac{6450203543}{200108161}a-\frac{1114840296}{200108161}$, $\frac{1}{200108161}a^{19}+\frac{525}{200108161}a^{18}-\frac{1374}{200108161}a^{17}-\frac{47250}{200108161}a^{16}+\frac{111404}{200108161}a^{15}+\frac{1777923}{200108161}a^{14}-\frac{3857850}{200108161}a^{13}-\frac{36106014}{200108161}a^{12}+\frac{72535460}{200108161}a^{11}+\frac{425101351}{200108161}a^{10}-\frac{795257800}{200108161}a^{9}-\frac{2882139993}{200108161}a^{8}+\frac{460395539}{18191651}a^{7}+\frac{10332133990}{200108161}a^{6}-\frac{17283631265}{200108161}a^{5}-\frac{14943977100}{200108161}a^{4}+\frac{24675848192}{200108161}a^{3}-\frac{78256704}{200108161}a^{2}-\frac{3094002255}{200108161}a+\frac{2477786}{18191651}$, $\frac{303}{200108161}a^{19}+\frac{505}{200108161}a^{18}-\frac{29410}{200108161}a^{17}-\frac{47325}{200108161}a^{16}+\frac{1205636}{200108161}a^{15}+\frac{1843764}{200108161}a^{14}-\frac{27139645}{200108161}a^{13}-\frac{38534237}{200108161}a^{12}+\frac{365227582}{200108161}a^{11}+\frac{464350784}{200108161}a^{10}-\frac{2997447107}{200108161}a^{9}-\frac{3214502395}{200108161}a^{8}+\frac{14639680815}{200108161}a^{7}+\frac{11863548800}{200108161}a^{6}-\frac{39236476600}{200108161}a^{5}-\frac{18605700578}{200108161}a^{4}+\frac{46731755495}{200108161}a^{3}+\frac{3065470169}{200108161}a^{2}-\frac{941402194}{18191651}a+\frac{1158944205}{200108161}$ 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:  \( 1261324189610 \) (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)
 
Unit signature rank:  \( 18 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 1261324189610 \cdot 1}{2\cdot\sqrt{17843751288604107685387134552001953125}}\cr\approx \mathstrut & 0.156550176524534 \end{aligned}\] (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 - 100*x^18 + 4250*x^16 - 101*x^15 - 100000*x^14 + 7575*x^13 + 1421875*x^12 - 227250*x^11 - 12505424*x^10 + 3471875*x^9 + 66677450*x^8 - 28406250*x^7 - 200058500*x^6 + 118907199*x^5 + 278040625*x^4 - 210961225*x^3 - 84750000*x^2 + 68478000*x - 6502099) 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 - 100*x^18 + 4250*x^16 - 101*x^15 - 100000*x^14 + 7575*x^13 + 1421875*x^12 - 227250*x^11 - 12505424*x^10 + 3471875*x^9 + 66677450*x^8 - 28406250*x^7 - 200058500*x^6 + 118907199*x^5 + 278040625*x^4 - 210961225*x^3 - 84750000*x^2 + 68478000*x - 6502099, 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 - 100*x^18 + 4250*x^16 - 101*x^15 - 100000*x^14 + 7575*x^13 + 1421875*x^12 - 227250*x^11 - 12505424*x^10 + 3471875*x^9 + 66677450*x^8 - 28406250*x^7 - 200058500*x^6 + 118907199*x^5 + 278040625*x^4 - 210961225*x^3 - 84750000*x^2 + 68478000*x - 6502099); 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 - 100*x^18 + 4250*x^16 - 101*x^15 - 100000*x^14 + 7575*x^13 + 1421875*x^12 - 227250*x^11 - 12505424*x^10 + 3471875*x^9 + 66677450*x^8 - 28406250*x^7 - 200058500*x^6 + 118907199*x^5 + 278040625*x^4 - 210961225*x^3 - 84750000*x^2 + 68478000*x - 6502099); 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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{190 -38 \sqrt{5}})\), 5.5.390625.1, \(\Q(\zeta_{25})^+\)

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 $20$ $20$ R ${\href{/padicField/7.4.0.1}{4} }^{5}$ ${\href{/padicField/11.5.0.1}{5} }^{4}$ $20$ $20$ R $20$ ${\href{/padicField/29.5.0.1}{5} }^{4}$ ${\href{/padicField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/padicField/41.10.0.1}{10} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/padicField/59.5.0.1}{5} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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
\(5\) Copy content Toggle raw display 5.1.20.35a1.1$x^{20} + 20 x^{16} + 5$$20$$1$$35$not computednot computed
\(19\) Copy content Toggle raw display 19.5.2.5a1.1$x^{10} + 10 x^{6} + 34 x^{5} + 25 x^{2} + 189 x + 289$$2$$5$$5$$C_{10}$$$[\ ]_{2}^{5}$$
19.5.2.5a1.1$x^{10} + 10 x^{6} + 34 x^{5} + 25 x^{2} + 189 x + 289$$2$$5$$5$$C_{10}$$$[\ ]_{2}^{5}$$

Spectrum of ring of integers

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