↑

Properties

Label 20.0.868...000.3
Degree $20$
Signature $[0, 10]$
Discriminant $8.683\times 10^{37}$
Root discriminant \(78.87\)
Ramified primes $2,3,5,11$
Class number $318100$ (GRH)
Class group [10, 31810] (GRH)
Galois group $C_{20}$ (as 20T1)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 65*x^18 - 214*x^17 + 2035*x^16 - 5724*x^15 + 39824*x^14 - 96080*x^13 + 534824*x^12 - 1102236*x^11 + 5168310*x^10 - 9006872*x^9 + 36960084*x^8 - 53466672*x^7 + 196375269*x^6 - 224863132*x^5 + 733497061*x^4 - 597365054*x^3 + 1764047967*x^2 - 786154512*x + 2065660201)
 
gp: K = bnfinit(y^20 - 4*y^19 + 65*y^18 - 214*y^17 + 2035*y^16 - 5724*y^15 + 39824*y^14 - 96080*y^13 + 534824*y^12 - 1102236*y^11 + 5168310*y^10 - 9006872*y^9 + 36960084*y^8 - 53466672*y^7 + 196375269*y^6 - 224863132*y^5 + 733497061*y^4 - 597365054*y^3 + 1764047967*y^2 - 786154512*y + 2065660201, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^19 + 65*x^18 - 214*x^17 + 2035*x^16 - 5724*x^15 + 39824*x^14 - 96080*x^13 + 534824*x^12 - 1102236*x^11 + 5168310*x^10 - 9006872*x^9 + 36960084*x^8 - 53466672*x^7 + 196375269*x^6 - 224863132*x^5 + 733497061*x^4 - 597365054*x^3 + 1764047967*x^2 - 786154512*x + 2065660201);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 4*x^19 + 65*x^18 - 214*x^17 + 2035*x^16 - 5724*x^15 + 39824*x^14 - 96080*x^13 + 534824*x^12 - 1102236*x^11 + 5168310*x^10 - 9006872*x^9 + 36960084*x^8 - 53466672*x^7 + 196375269*x^6 - 224863132*x^5 + 733497061*x^4 - 597365054*x^3 + 1764047967*x^2 - 786154512*x + 2065660201)
 

\( x^{20} - 4 x^{19} + 65 x^{18} - 214 x^{17} + 2035 x^{16} - 5724 x^{15} + 39824 x^{14} + \cdots + 2065660201 \) 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:   \(86825139158850321116448000000000000000\) \(\medspace = 2^{20}\cdot 3^{10}\cdot 5^{15}\cdot 11^{16}\) 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:  \(78.87\)
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:  $2\cdot 3^{1/2}5^{3/4}11^{4/5}\approx 78.87371092461403$
Ramified primes:   \(2\), \(3\), \(5\), \(11\) 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:  \(660=2^{2}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{660}(1,·)$, $\chi_{660}(323,·)$, $\chi_{660}(647,·)$, $\chi_{660}(587,·)$, $\chi_{660}(529,·)$, $\chi_{660}(467,·)$, $\chi_{660}(203,·)$, $\chi_{660}(23,·)$, $\chi_{660}(229,·)$, $\chi_{660}(289,·)$, $\chi_{660}(421,·)$, $\chi_{660}(169,·)$, $\chi_{660}(301,·)$, $\chi_{660}(47,·)$, $\chi_{660}(287,·)$, $\chi_{660}(49,·)$, $\chi_{660}(181,·)$, $\chi_{660}(361,·)$, $\chi_{660}(443,·)$, $\chi_{660}(383,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{512}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{16}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{17}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{27\!\cdots\!83}a^{18}-\frac{18\!\cdots\!84}{27\!\cdots\!83}a^{17}+\frac{49\!\cdots\!02}{91\!\cdots\!61}a^{16}+\frac{22\!\cdots\!77}{27\!\cdots\!83}a^{15}+\frac{12\!\cdots\!99}{91\!\cdots\!61}a^{14}+\frac{27\!\cdots\!64}{27\!\cdots\!83}a^{13}+\frac{14\!\cdots\!93}{27\!\cdots\!83}a^{12}+\frac{76\!\cdots\!51}{27\!\cdots\!83}a^{11}+\frac{33\!\cdots\!65}{27\!\cdots\!83}a^{10}-\frac{17\!\cdots\!28}{91\!\cdots\!61}a^{9}-\frac{73\!\cdots\!73}{27\!\cdots\!83}a^{8}+\frac{36\!\cdots\!21}{91\!\cdots\!61}a^{7}-\frac{10\!\cdots\!60}{27\!\cdots\!83}a^{6}-\frac{11\!\cdots\!23}{27\!\cdots\!83}a^{5}-\frac{84\!\cdots\!87}{27\!\cdots\!83}a^{4}-\frac{11\!\cdots\!17}{27\!\cdots\!83}a^{3}-\frac{10\!\cdots\!35}{27\!\cdots\!83}a^{2}+\frac{93\!\cdots\!15}{27\!\cdots\!83}a-\frac{11\!\cdots\!37}{27\!\cdots\!83}$, $\frac{1}{23\!\cdots\!43}a^{19}-\frac{11\!\cdots\!75}{77\!\cdots\!81}a^{18}+\frac{30\!\cdots\!87}{23\!\cdots\!43}a^{17}+\frac{23\!\cdots\!95}{23\!\cdots\!43}a^{16}+\frac{14\!\cdots\!86}{23\!\cdots\!43}a^{15}+\frac{73\!\cdots\!55}{23\!\cdots\!43}a^{14}-\frac{32\!\cdots\!00}{77\!\cdots\!81}a^{13}+\frac{17\!\cdots\!27}{23\!\cdots\!43}a^{12}-\frac{10\!\cdots\!54}{77\!\cdots\!81}a^{11}+\frac{49\!\cdots\!87}{23\!\cdots\!43}a^{10}-\frac{59\!\cdots\!37}{77\!\cdots\!81}a^{9}+\frac{24\!\cdots\!61}{23\!\cdots\!43}a^{8}-\frac{63\!\cdots\!91}{23\!\cdots\!43}a^{7}+\frac{24\!\cdots\!33}{23\!\cdots\!43}a^{6}-\frac{34\!\cdots\!28}{77\!\cdots\!81}a^{5}-\frac{28\!\cdots\!07}{77\!\cdots\!81}a^{4}+\frac{12\!\cdots\!70}{77\!\cdots\!81}a^{3}-\frac{19\!\cdots\!27}{23\!\cdots\!43}a^{2}-\frac{88\!\cdots\!22}{23\!\cdots\!43}a+\frac{27\!\cdots\!56}{23\!\cdots\!43}$ 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

$C_{10}\times C_{31810}$, which has order $318100$ (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:   \( -1 \)  (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:   $\frac{13568180049644}{27\!\cdots\!83}a^{19}+\frac{134897714527690}{27\!\cdots\!83}a^{18}+\frac{104276651336804}{27\!\cdots\!83}a^{17}+\frac{71\!\cdots\!20}{27\!\cdots\!83}a^{16}-\frac{21\!\cdots\!12}{91\!\cdots\!61}a^{15}+\frac{18\!\cdots\!86}{27\!\cdots\!83}a^{14}-\frac{24\!\cdots\!80}{27\!\cdots\!83}a^{13}+\frac{30\!\cdots\!61}{27\!\cdots\!83}a^{12}-\frac{39\!\cdots\!92}{27\!\cdots\!83}a^{11}+\frac{34\!\cdots\!51}{27\!\cdots\!83}a^{10}-\frac{40\!\cdots\!08}{27\!\cdots\!83}a^{9}+\frac{90\!\cdots\!97}{91\!\cdots\!61}a^{8}-\frac{27\!\cdots\!28}{27\!\cdots\!83}a^{7}+\frac{53\!\cdots\!27}{91\!\cdots\!61}a^{6}-\frac{44\!\cdots\!16}{91\!\cdots\!61}a^{5}+\frac{66\!\cdots\!29}{27\!\cdots\!83}a^{4}-\frac{38\!\cdots\!96}{27\!\cdots\!83}a^{3}+\frac{17\!\cdots\!74}{27\!\cdots\!83}a^{2}-\frac{18\!\cdots\!94}{91\!\cdots\!61}a+\frac{40\!\cdots\!64}{27\!\cdots\!83}$, $\frac{11\!\cdots\!56}{77\!\cdots\!81}a^{19}-\frac{71\!\cdots\!10}{77\!\cdots\!81}a^{18}+\frac{32\!\cdots\!76}{77\!\cdots\!81}a^{17}+\frac{68\!\cdots\!10}{77\!\cdots\!81}a^{16}+\frac{10\!\cdots\!32}{77\!\cdots\!81}a^{15}+\frac{38\!\cdots\!24}{77\!\cdots\!81}a^{14}-\frac{13\!\cdots\!20}{77\!\cdots\!81}a^{13}+\frac{98\!\cdots\!69}{77\!\cdots\!81}a^{12}-\frac{37\!\cdots\!16}{77\!\cdots\!81}a^{11}+\frac{14\!\cdots\!23}{77\!\cdots\!81}a^{10}-\frac{52\!\cdots\!52}{77\!\cdots\!81}a^{9}+\frac{14\!\cdots\!64}{77\!\cdots\!81}a^{8}-\frac{45\!\cdots\!92}{77\!\cdots\!81}a^{7}+\frac{10\!\cdots\!39}{77\!\cdots\!81}a^{6}-\frac{27\!\cdots\!04}{77\!\cdots\!81}a^{5}+\frac{49\!\cdots\!61}{77\!\cdots\!81}a^{4}-\frac{10\!\cdots\!64}{77\!\cdots\!81}a^{3}+\frac{15\!\cdots\!22}{77\!\cdots\!81}a^{2}-\frac{20\!\cdots\!82}{77\!\cdots\!81}a+\frac{22\!\cdots\!80}{77\!\cdots\!81}$, $\frac{49\!\cdots\!20}{77\!\cdots\!81}a^{19}+\frac{13\!\cdots\!90}{77\!\cdots\!81}a^{18}-\frac{35\!\cdots\!50}{77\!\cdots\!81}a^{17}+\frac{87\!\cdots\!50}{77\!\cdots\!81}a^{16}-\frac{24\!\cdots\!00}{77\!\cdots\!81}a^{15}+\frac{27\!\cdots\!90}{77\!\cdots\!81}a^{14}-\frac{69\!\cdots\!60}{77\!\cdots\!81}a^{13}+\frac{52\!\cdots\!05}{77\!\cdots\!81}a^{12}-\frac{11\!\cdots\!40}{77\!\cdots\!81}a^{11}+\frac{66\!\cdots\!59}{77\!\cdots\!81}a^{10}-\frac{12\!\cdots\!30}{77\!\cdots\!81}a^{9}+\frac{60\!\cdots\!40}{77\!\cdots\!81}a^{8}-\frac{97\!\cdots\!40}{77\!\cdots\!81}a^{7}+\frac{39\!\cdots\!35}{77\!\cdots\!81}a^{6}-\frac{51\!\cdots\!74}{77\!\cdots\!81}a^{5}+\frac{18\!\cdots\!80}{77\!\cdots\!81}a^{4}-\frac{18\!\cdots\!90}{77\!\cdots\!81}a^{3}+\frac{54\!\cdots\!85}{77\!\cdots\!81}a^{2}-\frac{31\!\cdots\!40}{77\!\cdots\!81}a+\frac{73\!\cdots\!31}{77\!\cdots\!81}$, $\frac{63\!\cdots\!36}{77\!\cdots\!81}a^{19}-\frac{13\!\cdots\!00}{77\!\cdots\!81}a^{18}+\frac{68\!\cdots\!26}{77\!\cdots\!81}a^{17}-\frac{80\!\cdots\!40}{77\!\cdots\!81}a^{16}+\frac{25\!\cdots\!32}{77\!\cdots\!81}a^{15}-\frac{23\!\cdots\!66}{77\!\cdots\!81}a^{14}+\frac{55\!\cdots\!40}{77\!\cdots\!81}a^{13}-\frac{42\!\cdots\!36}{77\!\cdots\!81}a^{12}+\frac{78\!\cdots\!24}{77\!\cdots\!81}a^{11}-\frac{52\!\cdots\!36}{77\!\cdots\!81}a^{10}+\frac{75\!\cdots\!78}{77\!\cdots\!81}a^{9}-\frac{45\!\cdots\!76}{77\!\cdots\!81}a^{8}+\frac{51\!\cdots\!48}{77\!\cdots\!81}a^{7}-\frac{29\!\cdots\!96}{77\!\cdots\!81}a^{6}+\frac{24\!\cdots\!70}{77\!\cdots\!81}a^{5}-\frac{13\!\cdots\!19}{77\!\cdots\!81}a^{4}+\frac{75\!\cdots\!26}{77\!\cdots\!81}a^{3}-\frac{39\!\cdots\!63}{77\!\cdots\!81}a^{2}+\frac{11\!\cdots\!58}{77\!\cdots\!81}a-\frac{58\!\cdots\!32}{77\!\cdots\!81}$, $\frac{23\!\cdots\!60}{77\!\cdots\!81}a^{19}+\frac{14\!\cdots\!30}{77\!\cdots\!81}a^{18}-\frac{42\!\cdots\!20}{77\!\cdots\!81}a^{17}+\frac{92\!\cdots\!15}{77\!\cdots\!81}a^{16}-\frac{25\!\cdots\!20}{77\!\cdots\!81}a^{15}+\frac{28\!\cdots\!80}{77\!\cdots\!81}a^{14}-\frac{67\!\cdots\!00}{77\!\cdots\!81}a^{13}+\frac{52\!\cdots\!45}{77\!\cdots\!81}a^{12}-\frac{10\!\cdots\!80}{77\!\cdots\!81}a^{11}+\frac{66\!\cdots\!05}{77\!\cdots\!81}a^{10}-\frac{11\!\cdots\!00}{77\!\cdots\!81}a^{9}+\frac{59\!\cdots\!05}{77\!\cdots\!81}a^{8}-\frac{87\!\cdots\!40}{77\!\cdots\!81}a^{7}+\frac{38\!\cdots\!05}{77\!\cdots\!81}a^{6}-\frac{44\!\cdots\!24}{77\!\cdots\!81}a^{5}+\frac{17\!\cdots\!05}{77\!\cdots\!81}a^{4}-\frac{15\!\cdots\!00}{77\!\cdots\!81}a^{3}+\frac{53\!\cdots\!60}{77\!\cdots\!81}a^{2}-\frac{24\!\cdots\!70}{77\!\cdots\!81}a+\frac{71\!\cdots\!14}{77\!\cdots\!81}$, $\frac{18\!\cdots\!04}{23\!\cdots\!43}a^{19}+\frac{55\!\cdots\!80}{23\!\cdots\!43}a^{18}-\frac{11\!\cdots\!76}{23\!\cdots\!43}a^{17}+\frac{33\!\cdots\!65}{23\!\cdots\!43}a^{16}-\frac{26\!\cdots\!72}{77\!\cdots\!81}a^{15}+\frac{99\!\cdots\!46}{23\!\cdots\!43}a^{14}-\frac{22\!\cdots\!80}{23\!\cdots\!43}a^{13}+\frac{18\!\cdots\!16}{23\!\cdots\!43}a^{12}-\frac{36\!\cdots\!72}{23\!\cdots\!43}a^{11}+\frac{22\!\cdots\!86}{23\!\cdots\!43}a^{10}-\frac{38\!\cdots\!68}{23\!\cdots\!43}a^{9}+\frac{67\!\cdots\!42}{77\!\cdots\!81}a^{8}-\frac{28\!\cdots\!08}{23\!\cdots\!43}a^{7}+\frac{43\!\cdots\!72}{77\!\cdots\!81}a^{6}-\frac{48\!\cdots\!60}{77\!\cdots\!81}a^{5}+\frac{59\!\cdots\!24}{23\!\cdots\!43}a^{4}-\frac{48\!\cdots\!16}{23\!\cdots\!43}a^{3}+\frac{17\!\cdots\!34}{23\!\cdots\!43}a^{2}-\frac{26\!\cdots\!44}{77\!\cdots\!81}a+\frac{27\!\cdots\!29}{23\!\cdots\!43}$, $\frac{22\!\cdots\!44}{23\!\cdots\!43}a^{19}-\frac{13\!\cdots\!20}{23\!\cdots\!43}a^{18}+\frac{89\!\cdots\!44}{23\!\cdots\!43}a^{17}-\frac{40\!\cdots\!90}{23\!\cdots\!43}a^{16}+\frac{28\!\cdots\!84}{77\!\cdots\!81}a^{15}-\frac{42\!\cdots\!34}{23\!\cdots\!43}a^{14}-\frac{21\!\cdots\!80}{23\!\cdots\!43}a^{13}+\frac{35\!\cdots\!26}{23\!\cdots\!43}a^{12}-\frac{80\!\cdots\!16}{23\!\cdots\!43}a^{11}+\frac{15\!\cdots\!98}{23\!\cdots\!43}a^{10}-\frac{12\!\cdots\!88}{23\!\cdots\!43}a^{9}+\frac{69\!\cdots\!27}{77\!\cdots\!81}a^{8}-\frac{11\!\cdots\!88}{23\!\cdots\!43}a^{7}+\frac{55\!\cdots\!72}{77\!\cdots\!81}a^{6}-\frac{23\!\cdots\!68}{77\!\cdots\!81}a^{5}+\frac{91\!\cdots\!74}{23\!\cdots\!43}a^{4}-\frac{29\!\cdots\!76}{23\!\cdots\!43}a^{3}+\frac{30\!\cdots\!12}{23\!\cdots\!43}a^{2}-\frac{18\!\cdots\!08}{77\!\cdots\!81}a+\frac{34\!\cdots\!96}{23\!\cdots\!43}$, $\frac{38\!\cdots\!44}{23\!\cdots\!43}a^{19}+\frac{16\!\cdots\!10}{23\!\cdots\!43}a^{18}-\frac{13\!\cdots\!26}{23\!\cdots\!43}a^{17}+\frac{75\!\cdots\!15}{23\!\cdots\!43}a^{16}-\frac{21\!\cdots\!72}{77\!\cdots\!81}a^{15}+\frac{18\!\cdots\!76}{23\!\cdots\!43}a^{14}-\frac{12\!\cdots\!00}{23\!\cdots\!43}a^{13}+\frac{27\!\cdots\!01}{23\!\cdots\!43}a^{12}-\frac{10\!\cdots\!52}{23\!\cdots\!43}a^{11}+\frac{28\!\cdots\!09}{23\!\cdots\!43}a^{10}-\frac{10\!\cdots\!78}{23\!\cdots\!43}a^{9}+\frac{69\!\cdots\!02}{77\!\cdots\!81}a^{8}+\frac{77\!\cdots\!12}{23\!\cdots\!43}a^{7}+\frac{38\!\cdots\!37}{77\!\cdots\!81}a^{6}+\frac{33\!\cdots\!14}{77\!\cdots\!81}a^{5}+\frac{44\!\cdots\!84}{23\!\cdots\!43}a^{4}+\frac{72\!\cdots\!54}{23\!\cdots\!43}a^{3}+\frac{11\!\cdots\!79}{23\!\cdots\!43}a^{2}+\frac{56\!\cdots\!96}{77\!\cdots\!81}a+\frac{50\!\cdots\!36}{23\!\cdots\!43}$, $\frac{11\!\cdots\!94}{23\!\cdots\!43}a^{19}+\frac{34\!\cdots\!77}{23\!\cdots\!43}a^{18}-\frac{51\!\cdots\!96}{23\!\cdots\!43}a^{17}+\frac{21\!\cdots\!67}{23\!\cdots\!43}a^{16}-\frac{40\!\cdots\!12}{23\!\cdots\!43}a^{15}+\frac{21\!\cdots\!32}{77\!\cdots\!81}a^{14}-\frac{11\!\cdots\!42}{23\!\cdots\!43}a^{13}+\frac{11\!\cdots\!53}{23\!\cdots\!43}a^{12}-\frac{18\!\cdots\!76}{23\!\cdots\!43}a^{11}+\frac{15\!\cdots\!22}{23\!\cdots\!43}a^{10}-\frac{65\!\cdots\!70}{77\!\cdots\!81}a^{9}+\frac{13\!\cdots\!37}{23\!\cdots\!43}a^{8}-\frac{47\!\cdots\!80}{77\!\cdots\!81}a^{7}+\frac{29\!\cdots\!54}{77\!\cdots\!81}a^{6}-\frac{25\!\cdots\!20}{77\!\cdots\!81}a^{5}+\frac{41\!\cdots\!36}{23\!\cdots\!43}a^{4}-\frac{87\!\cdots\!64}{77\!\cdots\!81}a^{3}+\frac{12\!\cdots\!41}{23\!\cdots\!43}a^{2}-\frac{43\!\cdots\!34}{23\!\cdots\!43}a+\frac{61\!\cdots\!39}{77\!\cdots\!81}$ 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:  \( 140644.599182 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 140644.599182 \cdot 318100}{2\cdot\sqrt{86825139158850321116448000000000000000}}\cr\approx \mathstrut & 0.230214499275 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 65*x^18 - 214*x^17 + 2035*x^16 - 5724*x^15 + 39824*x^14 - 96080*x^13 + 534824*x^12 - 1102236*x^11 + 5168310*x^10 - 9006872*x^9 + 36960084*x^8 - 53466672*x^7 + 196375269*x^6 - 224863132*x^5 + 733497061*x^4 - 597365054*x^3 + 1764047967*x^2 - 786154512*x + 2065660201)
 
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 - 4*x^19 + 65*x^18 - 214*x^17 + 2035*x^16 - 5724*x^15 + 39824*x^14 - 96080*x^13 + 534824*x^12 - 1102236*x^11 + 5168310*x^10 - 9006872*x^9 + 36960084*x^8 - 53466672*x^7 + 196375269*x^6 - 224863132*x^5 + 733497061*x^4 - 597365054*x^3 + 1764047967*x^2 - 786154512*x + 2065660201, 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 - 4*x^19 + 65*x^18 - 214*x^17 + 2035*x^16 - 5724*x^15 + 39824*x^14 - 96080*x^13 + 534824*x^12 - 1102236*x^11 + 5168310*x^10 - 9006872*x^9 + 36960084*x^8 - 53466672*x^7 + 196375269*x^6 - 224863132*x^5 + 733497061*x^4 - 597365054*x^3 + 1764047967*x^2 - 786154512*x + 2065660201);
 
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 - 4*x^19 + 65*x^18 - 214*x^17 + 2035*x^16 - 5724*x^15 + 39824*x^14 - 96080*x^13 + 534824*x^12 - 1102236*x^11 + 5168310*x^10 - 9006872*x^9 + 36960084*x^8 - 53466672*x^7 + 196375269*x^6 - 224863132*x^5 + 733497061*x^4 - 597365054*x^3 + 1764047967*x^2 - 786154512*x + 2065660201);
 
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

\(\Q(\sqrt{5}) \), 4.0.18000.1, \(\Q(\zeta_{11})^+\), 10.10.669871503125.1

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 R R R $20$ R $20$ $20$ ${\href{/padicField/19.5.0.1}{5} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{5}$ ${\href{/padicField/29.5.0.1}{5} }^{4}$ ${\href{/padicField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/padicField/41.10.0.1}{10} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/padicField/59.10.0.1}{10} }^{2}$

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
\(2\) Copy content Toggle raw display Deg $20$$2$$10$$20$
\(3\) Copy content Toggle raw display 3.20.10.2$x^{20} + 162 x^{12} - 486 x^{10} + 1458 x^{8} - 19683 x^{2} + 118098$$2$$10$$10$20T1$[\ ]_{2}^{10}$
\(5\) Copy content Toggle raw display Deg $20$$4$$5$$15$
\(11\) Copy content Toggle raw display 11.5.4.4$x^{5} + 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} + 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} + 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} + 11$$5$$1$$4$$C_5$$[\ ]_{5}$