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Properties

Label 20.0.221...125.2
Degree $20$
Signature $[0, 10]$
Discriminant $2.220\times 10^{34}$
Root discriminant \(52.16\)
Ramified primes $5,31$
Class number $400$ (GRH)
Class group [2, 2, 10, 10] (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 13*x^18 - 4*x^17 + 138*x^16 - 481*x^15 + 2045*x^14 - 4980*x^13 + 19301*x^12 - 36325*x^11 + 68890*x^10 - 104714*x^9 + 174168*x^8 - 44226*x^7 + 47983*x^6 - 14164*x^5 + 10616*x^4 + 445*x^3 + 2650*x^2 - 125*x + 625)
 
gp: K = bnfinit(y^20 - y^19 + 13*y^18 - 4*y^17 + 138*y^16 - 481*y^15 + 2045*y^14 - 4980*y^13 + 19301*y^12 - 36325*y^11 + 68890*y^10 - 104714*y^9 + 174168*y^8 - 44226*y^7 + 47983*y^6 - 14164*y^5 + 10616*y^4 + 445*y^3 + 2650*y^2 - 125*y + 625, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^19 + 13*x^18 - 4*x^17 + 138*x^16 - 481*x^15 + 2045*x^14 - 4980*x^13 + 19301*x^12 - 36325*x^11 + 68890*x^10 - 104714*x^9 + 174168*x^8 - 44226*x^7 + 47983*x^6 - 14164*x^5 + 10616*x^4 + 445*x^3 + 2650*x^2 - 125*x + 625);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - x^19 + 13*x^18 - 4*x^17 + 138*x^16 - 481*x^15 + 2045*x^14 - 4980*x^13 + 19301*x^12 - 36325*x^11 + 68890*x^10 - 104714*x^9 + 174168*x^8 - 44226*x^7 + 47983*x^6 - 14164*x^5 + 10616*x^4 + 445*x^3 + 2650*x^2 - 125*x + 625)
 

\( x^{20} - x^{19} + 13 x^{18} - 4 x^{17} + 138 x^{16} - 481 x^{15} + 2045 x^{14} - 4980 x^{13} + \cdots + 625 \) 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:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(22199191947851112787734405517578125\) \(\medspace = 5^{15}\cdot 31^{16}\) 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:  \(52.16\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $5^{3/4}31^{4/5}\approx 52.15751099959023$
Ramified primes:   \(5\), \(31\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:  $20$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(155=5\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{155}(64,·)$, $\chi_{155}(1,·)$, $\chi_{155}(2,·)$, $\chi_{155}(4,·)$, $\chi_{155}(97,·)$, $\chi_{155}(8,·)$, $\chi_{155}(128,·)$, $\chi_{155}(66,·)$, $\chi_{155}(78,·)$, $\chi_{155}(16,·)$, $\chi_{155}(132,·)$, $\chi_{155}(94,·)$, $\chi_{155}(32,·)$, $\chi_{155}(33,·)$, $\chi_{155}(101,·)$, $\chi_{155}(39,·)$, $\chi_{155}(109,·)$, $\chi_{155}(47,·)$, $\chi_{155}(126,·)$, $\chi_{155}(63,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{512}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5}a^{13}+\frac{2}{5}a^{12}+\frac{1}{5}a^{9}+\frac{1}{5}a^{7}+\frac{1}{5}a^{5}-\frac{2}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{5}a^{14}+\frac{1}{5}a^{12}+\frac{1}{5}a^{10}-\frac{2}{5}a^{9}+\frac{1}{5}a^{8}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}-\frac{2}{5}a^{3}-\frac{2}{5}a$, $\frac{1}{25}a^{15}+\frac{1}{25}a^{13}+\frac{1}{5}a^{12}+\frac{6}{25}a^{11}-\frac{12}{25}a^{10}+\frac{6}{25}a^{9}-\frac{7}{25}a^{8}+\frac{1}{25}a^{7}-\frac{2}{25}a^{6}-\frac{1}{5}a^{5}+\frac{3}{25}a^{4}-\frac{2}{5}a^{3}+\frac{3}{25}a^{2}+\frac{1}{5}a$, $\frac{1}{25}a^{16}+\frac{1}{25}a^{14}-\frac{4}{25}a^{12}-\frac{12}{25}a^{11}+\frac{6}{25}a^{10}-\frac{12}{25}a^{9}+\frac{1}{25}a^{8}-\frac{7}{25}a^{7}-\frac{1}{5}a^{6}-\frac{2}{25}a^{5}-\frac{2}{5}a^{4}+\frac{3}{25}a^{3}-\frac{2}{5}a^{2}-\frac{1}{5}a$, $\frac{1}{22\!\cdots\!25}a^{17}-\frac{294864270357121}{22\!\cdots\!25}a^{16}+\frac{10760498473873}{45\!\cdots\!05}a^{15}+\frac{909205899230709}{22\!\cdots\!25}a^{14}-\frac{210814021047053}{45\!\cdots\!05}a^{13}+\frac{10\!\cdots\!22}{22\!\cdots\!25}a^{12}+\frac{10\!\cdots\!67}{22\!\cdots\!25}a^{11}+\frac{45\!\cdots\!49}{22\!\cdots\!25}a^{10}-\frac{32\!\cdots\!48}{22\!\cdots\!25}a^{9}+\frac{504977962972054}{22\!\cdots\!25}a^{8}-\frac{15\!\cdots\!29}{22\!\cdots\!25}a^{7}+\frac{984027356852601}{45\!\cdots\!05}a^{6}-\frac{55\!\cdots\!73}{22\!\cdots\!25}a^{5}-\frac{563624833687709}{45\!\cdots\!05}a^{4}+\frac{11\!\cdots\!02}{22\!\cdots\!25}a^{3}+\frac{29\!\cdots\!97}{22\!\cdots\!25}a^{2}+\frac{22\!\cdots\!13}{45\!\cdots\!05}a+\frac{190931591142933}{900275927177101}$, $\frac{1}{22\!\cdots\!25}a^{18}+\frac{108192405962256}{22\!\cdots\!25}a^{16}+\frac{440880878229536}{22\!\cdots\!25}a^{15}+\frac{17\!\cdots\!36}{22\!\cdots\!25}a^{14}-\frac{19\!\cdots\!71}{22\!\cdots\!25}a^{13}-\frac{16\!\cdots\!64}{22\!\cdots\!25}a^{12}-\frac{93\!\cdots\!06}{22\!\cdots\!25}a^{11}+\frac{76\!\cdots\!09}{22\!\cdots\!25}a^{10}-\frac{10\!\cdots\!66}{22\!\cdots\!25}a^{9}+\frac{74\!\cdots\!08}{22\!\cdots\!25}a^{8}+\frac{43\!\cdots\!09}{22\!\cdots\!25}a^{7}+\frac{92\!\cdots\!03}{22\!\cdots\!25}a^{6}-\frac{54\!\cdots\!77}{22\!\cdots\!25}a^{5}+\frac{36\!\cdots\!48}{22\!\cdots\!25}a^{4}+\frac{429428454357341}{45\!\cdots\!05}a^{3}-\frac{17\!\cdots\!27}{22\!\cdots\!25}a^{2}-\frac{14\!\cdots\!82}{45\!\cdots\!05}a-\frac{68619080804835}{900275927177101}$, $\frac{1}{11\!\cdots\!25}a^{19}-\frac{1}{11\!\cdots\!25}a^{18}-\frac{2}{11\!\cdots\!25}a^{17}-\frac{533609215018744}{11\!\cdots\!25}a^{16}-\frac{129527128999212}{11\!\cdots\!25}a^{15}-\frac{10\!\cdots\!96}{11\!\cdots\!25}a^{14}+\frac{13\!\cdots\!89}{22\!\cdots\!25}a^{13}+\frac{72\!\cdots\!67}{22\!\cdots\!25}a^{12}-\frac{10\!\cdots\!69}{11\!\cdots\!25}a^{11}-\frac{10\!\cdots\!13}{22\!\cdots\!25}a^{10}+\frac{79\!\cdots\!34}{22\!\cdots\!25}a^{9}-\frac{11\!\cdots\!04}{11\!\cdots\!25}a^{8}-\frac{46\!\cdots\!12}{11\!\cdots\!25}a^{7}+\frac{37\!\cdots\!74}{11\!\cdots\!25}a^{6}+\frac{18\!\cdots\!88}{11\!\cdots\!25}a^{5}-\frac{46\!\cdots\!89}{11\!\cdots\!25}a^{4}+\frac{19\!\cdots\!71}{11\!\cdots\!25}a^{3}+\frac{95\!\cdots\!88}{22\!\cdots\!25}a^{2}-\frac{357946020788601}{45\!\cdots\!05}a-\frac{268705462053374}{900275927177101}$ 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

$C_{2}\times C_{2}\times C_{10}\times C_{10}$, which has order $400$ (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:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{34279214135504}{112534490897137625} a^{19} - \frac{33746819736904}{112534490897137625} a^{18} + \frac{438708656579752}{112534490897137625} a^{17} - \frac{134987278947616}{112534490897137625} a^{16} + \frac{4657061123692752}{112534490897137625} a^{15} - \frac{16532430736329149}{112534490897137625} a^{14} + \frac{13802449272393736}{22506898179427525} a^{13} - \frac{33611832457956384}{22506898179427525} a^{12} + \frac{651347367741984104}{112534490897137625} a^{11} - \frac{49034129077721512}{4501379635885505} a^{10} + \frac{454859818816663842}{22506898179427525} a^{9} - \frac{3533764481930165456}{112534490897137625} a^{8} + \frac{5877616099937095872}{112534490897137625} a^{7} - \frac{1492486849684316304}{112534490897137625} a^{6} + \frac{1619273651435864632}{112534490897137625} a^{5} - \frac{2003540773400225681}{112534490897137625} a^{4} + \frac{358256238326972864}{112534490897137625} a^{3} + \frac{3003466956584456}{22506898179427525} a^{2} + \frac{3577162892111824}{4501379635885505} a - \frac{33746819736904}{900275927177101} \)  (order $10$) 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:   $\frac{9199005007004}{11\!\cdots\!25}a^{19}-\frac{52556976583279}{11\!\cdots\!25}a^{18}+\frac{158646321311502}{11\!\cdots\!25}a^{17}-\frac{596889726796416}{11\!\cdots\!25}a^{16}+\frac{13\!\cdots\!52}{11\!\cdots\!25}a^{15}-\frac{10\!\cdots\!24}{11\!\cdots\!25}a^{14}+\frac{78\!\cdots\!61}{22\!\cdots\!25}a^{13}-\frac{26\!\cdots\!84}{22\!\cdots\!25}a^{12}+\frac{38\!\cdots\!04}{11\!\cdots\!25}a^{11}-\frac{46\!\cdots\!32}{45\!\cdots\!05}a^{10}+\frac{42\!\cdots\!92}{22\!\cdots\!25}a^{9}-\frac{38\!\cdots\!06}{11\!\cdots\!25}a^{8}+\frac{58\!\cdots\!97}{11\!\cdots\!25}a^{7}-\frac{75\!\cdots\!04}{11\!\cdots\!25}a^{6}+\frac{16\!\cdots\!32}{11\!\cdots\!25}a^{5}-\frac{20\!\cdots\!56}{11\!\cdots\!25}a^{4}+\frac{35\!\cdots\!39}{11\!\cdots\!25}a^{3}+\frac{19\!\cdots\!81}{22\!\cdots\!25}a^{2}-\frac{317669455741684}{900275927177101}a-\frac{44196906873779}{900275927177101}$, $\frac{13695749901067}{11\!\cdots\!25}a^{19}+\frac{10818431960093}{11\!\cdots\!25}a^{18}+\frac{176484631808006}{11\!\cdots\!25}a^{17}+\frac{240660561918182}{11\!\cdots\!25}a^{16}+\frac{20\!\cdots\!81}{11\!\cdots\!25}a^{15}-\frac{32\!\cdots\!77}{11\!\cdots\!25}a^{14}+\frac{38\!\cdots\!26}{22\!\cdots\!25}a^{13}-\frac{58\!\cdots\!61}{22\!\cdots\!25}a^{12}+\frac{18\!\cdots\!02}{11\!\cdots\!25}a^{11}-\frac{27\!\cdots\!44}{22\!\cdots\!25}a^{10}+\frac{97\!\cdots\!68}{22\!\cdots\!25}a^{9}-\frac{56\!\cdots\!68}{11\!\cdots\!25}a^{8}+\frac{13\!\cdots\!06}{11\!\cdots\!25}a^{7}+\frac{14\!\cdots\!88}{11\!\cdots\!25}a^{6}+\frac{32\!\cdots\!86}{11\!\cdots\!25}a^{5}+\frac{40\!\cdots\!17}{11\!\cdots\!25}a^{4}+\frac{84\!\cdots\!67}{11\!\cdots\!25}a^{3}+\frac{15\!\cdots\!68}{22\!\cdots\!25}a^{2}+\frac{89682462888563}{900275927177101}a+\frac{484703246834640}{900275927177101}$, $\frac{144500771880809}{11\!\cdots\!25}a^{19}+\frac{40297564990986}{11\!\cdots\!25}a^{18}+\frac{17\!\cdots\!87}{11\!\cdots\!25}a^{17}+\frac{17\!\cdots\!39}{11\!\cdots\!25}a^{16}+\frac{20\!\cdots\!12}{11\!\cdots\!25}a^{15}-\frac{44\!\cdots\!29}{11\!\cdots\!25}a^{14}+\frac{43\!\cdots\!57}{22\!\cdots\!25}a^{13}-\frac{76\!\cdots\!57}{22\!\cdots\!25}a^{12}+\frac{20\!\cdots\!54}{11\!\cdots\!25}a^{11}-\frac{41\!\cdots\!98}{22\!\cdots\!25}a^{10}+\frac{97\!\cdots\!21}{22\!\cdots\!25}a^{9}-\frac{53\!\cdots\!86}{11\!\cdots\!25}a^{8}+\frac{11\!\cdots\!12}{11\!\cdots\!25}a^{7}+\frac{17\!\cdots\!51}{11\!\cdots\!25}a^{6}+\frac{12\!\cdots\!22}{11\!\cdots\!25}a^{5}+\frac{49\!\cdots\!84}{11\!\cdots\!25}a^{4}+\frac{71\!\cdots\!09}{11\!\cdots\!25}a^{3}+\frac{11\!\cdots\!11}{22\!\cdots\!25}a^{2}+\frac{10\!\cdots\!51}{900275927177101}a+\frac{857308279509462}{900275927177101}$, $\frac{118223184162964}{11\!\cdots\!25}a^{19}-\frac{103405440354084}{11\!\cdots\!25}a^{18}+\frac{14\!\cdots\!72}{11\!\cdots\!25}a^{17}-\frac{244045083771211}{11\!\cdots\!25}a^{16}+\frac{15\!\cdots\!72}{11\!\cdots\!25}a^{15}-\frac{54\!\cdots\!14}{11\!\cdots\!25}a^{14}+\frac{46\!\cdots\!32}{22\!\cdots\!25}a^{13}-\frac{10\!\cdots\!56}{22\!\cdots\!25}a^{12}+\frac{21\!\cdots\!64}{11\!\cdots\!25}a^{11}-\frac{76\!\cdots\!97}{22\!\cdots\!25}a^{10}+\frac{14\!\cdots\!17}{22\!\cdots\!25}a^{9}-\frac{10\!\cdots\!36}{11\!\cdots\!25}a^{8}+\frac{16\!\cdots\!12}{11\!\cdots\!25}a^{7}+\frac{60\!\cdots\!31}{11\!\cdots\!25}a^{6}-\frac{23\!\cdots\!28}{11\!\cdots\!25}a^{5}+\frac{34\!\cdots\!49}{11\!\cdots\!25}a^{4}+\frac{10\!\cdots\!44}{11\!\cdots\!25}a^{3}+\frac{85\!\cdots\!76}{22\!\cdots\!25}a^{2}+\frac{268796472763286}{900275927177101}a+\frac{14\!\cdots\!29}{900275927177101}$, $\frac{61714558446633}{11\!\cdots\!25}a^{19}-\frac{158105136515648}{11\!\cdots\!25}a^{18}+\frac{873846937859124}{11\!\cdots\!25}a^{17}-\frac{14\!\cdots\!12}{11\!\cdots\!25}a^{16}+\frac{85\!\cdots\!74}{11\!\cdots\!25}a^{15}-\frac{42\!\cdots\!03}{11\!\cdots\!25}a^{14}+\frac{33\!\cdots\!62}{22\!\cdots\!25}a^{13}-\frac{98\!\cdots\!02}{22\!\cdots\!25}a^{12}+\frac{16\!\cdots\!38}{11\!\cdots\!25}a^{11}-\frac{79\!\cdots\!34}{22\!\cdots\!25}a^{10}+\frac{58\!\cdots\!30}{900275927177101}a^{9}-\frac{12\!\cdots\!92}{11\!\cdots\!25}a^{8}+\frac{19\!\cdots\!54}{11\!\cdots\!25}a^{7}-\frac{16\!\cdots\!98}{11\!\cdots\!25}a^{6}+\frac{30\!\cdots\!99}{11\!\cdots\!25}a^{5}-\frac{45\!\cdots\!27}{11\!\cdots\!25}a^{4}+\frac{11\!\cdots\!08}{11\!\cdots\!25}a^{3}+\frac{72\!\cdots\!82}{22\!\cdots\!25}a^{2}-\frac{632432762858203}{900275927177101}a-\frac{35\!\cdots\!15}{900275927177101}$, $\frac{248023891132}{22\!\cdots\!25}a^{19}-\frac{3224310584716}{22\!\cdots\!25}a^{18}+\frac{992095564528}{22\!\cdots\!25}a^{17}-\frac{34227296976216}{22\!\cdots\!25}a^{16}-\frac{20533178640532}{22\!\cdots\!25}a^{15}-\frac{101441771472988}{45\!\cdots\!05}a^{14}+\frac{247031795567472}{45\!\cdots\!05}a^{13}-\frac{47\!\cdots\!32}{22\!\cdots\!25}a^{12}+\frac{360378713814796}{900275927177101}a^{11}-\frac{40\!\cdots\!62}{22\!\cdots\!25}a^{10}+\frac{25\!\cdots\!48}{22\!\cdots\!25}a^{9}-\frac{43\!\cdots\!76}{22\!\cdots\!25}a^{8}+\frac{10\!\cdots\!32}{22\!\cdots\!25}a^{7}-\frac{11\!\cdots\!56}{22\!\cdots\!25}a^{6}-\frac{70\!\cdots\!03}{22\!\cdots\!25}a^{5}-\frac{26\!\cdots\!12}{22\!\cdots\!25}a^{4}-\frac{22074126310748}{45\!\cdots\!05}a^{3}-\frac{26290532459992}{900275927177101}a^{2}+\frac{1240119455660}{900275927177101}a-\frac{12\!\cdots\!31}{900275927177101}$, $\frac{611082793982}{22\!\cdots\!25}a^{19}-\frac{7944076321766}{22\!\cdots\!25}a^{18}+\frac{2444331175928}{22\!\cdots\!25}a^{17}-\frac{84329425569516}{22\!\cdots\!25}a^{16}-\frac{50732607264107}{22\!\cdots\!25}a^{15}-\frac{249932862738638}{45\!\cdots\!05}a^{14}+\frac{608638462806072}{45\!\cdots\!05}a^{13}-\frac{11\!\cdots\!82}{22\!\cdots\!25}a^{12}+\frac{887903299655846}{900275927177101}a^{11}-\frac{10\!\cdots\!37}{22\!\cdots\!25}a^{10}+\frac{63\!\cdots\!48}{22\!\cdots\!25}a^{9}-\frac{10\!\cdots\!76}{22\!\cdots\!25}a^{8}+\frac{27\!\cdots\!32}{22\!\cdots\!25}a^{7}-\frac{29\!\cdots\!06}{22\!\cdots\!25}a^{6}-\frac{17\!\cdots\!78}{22\!\cdots\!25}a^{5}-\frac{64\!\cdots\!12}{22\!\cdots\!25}a^{4}-\frac{54386368664398}{45\!\cdots\!05}a^{3}-\frac{64774776162092}{900275927177101}a^{2}+\frac{3055413969910}{900275927177101}a-\frac{37\!\cdots\!86}{900275927177101}$, $\frac{6702222049}{45\!\cdots\!05}a^{19}-\frac{87128886637}{45\!\cdots\!05}a^{18}+\frac{26808888196}{45\!\cdots\!05}a^{17}-\frac{924906642762}{45\!\cdots\!05}a^{16}-\frac{581386755294}{45\!\cdots\!05}a^{15}-\frac{2741208818041}{900275927177101}a^{14}+\frac{6675413160804}{900275927177101}a^{13}-\frac{129359587767749}{45\!\cdots\!05}a^{12}+\frac{48691643185985}{900275927177101}a^{11}-\frac{10\!\cdots\!29}{45\!\cdots\!05}a^{10}+\frac{701816479638986}{45\!\cdots\!05}a^{9}-\frac{11\!\cdots\!32}{45\!\cdots\!05}a^{8}+\frac{296412472339074}{45\!\cdots\!05}a^{7}-\frac{321592720577167}{45\!\cdots\!05}a^{6}-\frac{17\!\cdots\!11}{45\!\cdots\!05}a^{5}-\frac{71150789272184}{45\!\cdots\!05}a^{4}-\frac{596497762361}{900275927177101}a^{3}-\frac{3552177685970}{900275927177101}a^{2}+\frac{167555551225}{900275927177101}a+\frac{434165717085533}{900275927177101}$, $\frac{53922439434}{22\!\cdots\!25}a^{19}-\frac{700991712642}{22\!\cdots\!25}a^{18}+\frac{215689757736}{22\!\cdots\!25}a^{17}-\frac{7441296641892}{22\!\cdots\!25}a^{16}-\frac{4445978774109}{22\!\cdots\!25}a^{15}-\frac{22054277728506}{45\!\cdots\!05}a^{14}+\frac{53706749676264}{45\!\cdots\!05}a^{13}-\frac{10\!\cdots\!34}{22\!\cdots\!25}a^{12}+\frac{78349304497602}{900275927177101}a^{11}-\frac{88\!\cdots\!44}{22\!\cdots\!25}a^{10}+\frac{56\!\cdots\!76}{22\!\cdots\!25}a^{9}-\frac{93\!\cdots\!12}{22\!\cdots\!25}a^{8}+\frac{23\!\cdots\!84}{22\!\cdots\!25}a^{7}-\frac{25\!\cdots\!22}{22\!\cdots\!25}a^{6}-\frac{15\!\cdots\!36}{22\!\cdots\!25}a^{5}-\frac{572440617031344}{22\!\cdots\!25}a^{4}-\frac{4799097109626}{45\!\cdots\!05}a^{3}-\frac{5715778580004}{900275927177101}a^{2}+\frac{269612197170}{900275927177101}a-\frac{502917343734554}{900275927177101}$ 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:  \( 24173706.8324 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 24173706.8324 \cdot 400}{10\cdot\sqrt{22199191947851112787734405517578125}}\cr\approx \mathstrut & 0.622348063586 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 13*x^18 - 4*x^17 + 138*x^16 - 481*x^15 + 2045*x^14 - 4980*x^13 + 19301*x^12 - 36325*x^11 + 68890*x^10 - 104714*x^9 + 174168*x^8 - 44226*x^7 + 47983*x^6 - 14164*x^5 + 10616*x^4 + 445*x^3 + 2650*x^2 - 125*x + 625)
 
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 - x^19 + 13*x^18 - 4*x^17 + 138*x^16 - 481*x^15 + 2045*x^14 - 4980*x^13 + 19301*x^12 - 36325*x^11 + 68890*x^10 - 104714*x^9 + 174168*x^8 - 44226*x^7 + 47983*x^6 - 14164*x^5 + 10616*x^4 + 445*x^3 + 2650*x^2 - 125*x + 625, 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 - x^19 + 13*x^18 - 4*x^17 + 138*x^16 - 481*x^15 + 2045*x^14 - 4980*x^13 + 19301*x^12 - 36325*x^11 + 68890*x^10 - 104714*x^9 + 174168*x^8 - 44226*x^7 + 47983*x^6 - 14164*x^5 + 10616*x^4 + 445*x^3 + 2650*x^2 - 125*x + 625);
 
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 - x^19 + 13*x^18 - 4*x^17 + 138*x^16 - 481*x^15 + 2045*x^14 - 4980*x^13 + 19301*x^12 - 36325*x^11 + 68890*x^10 - 104714*x^9 + 174168*x^8 - 44226*x^7 + 47983*x^6 - 14164*x^5 + 10616*x^4 + 445*x^3 + 2650*x^2 - 125*x + 625);
 
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):

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 cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.5.923521.1, 10.10.2665284492003125.1

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]
 

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 $20$ ${\href{/padicField/11.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ $20$ ${\href{/padicField/29.10.0.1}{10} }^{2}$ R ${\href{/padicField/37.4.0.1}{4} }^{5}$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ $20$ $20$ $20$ ${\href{/padicField/59.10.0.1}{10} }^{2}$

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

# 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
\(5\) Copy content Toggle raw display 5.4.3.2$x^{4} + 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} + 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} + 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} + 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} + 5$$4$$1$$3$$C_4$$[\ ]_{4}$
\(31\) Copy content Toggle raw display 31.5.4.1$x^{5} + 31$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.4.1$x^{5} + 31$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.4.1$x^{5} + 31$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.4.1$x^{5} + 31$$5$$1$$4$$C_5$$[\ ]_{5}$