↑

Properties

Label 20.0.984...656.1
Degree $20$
Signature $[0, 10]$
Discriminant $9.842\times 10^{30}$
Root discriminant \(35.45\)
Ramified primes $2,3,7$
Class number $4$ (GRH)
Class group [4] (GRH)
Galois group $S_6$ (as 20T145)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 16*x^18 - 40*x^17 + 69*x^16 + 44*x^15 - 275*x^14 - 36*x^13 + 1240*x^12 - 2696*x^11 + 3576*x^10 - 3812*x^9 + 5907*x^8 - 12888*x^7 + 25347*x^6 - 36688*x^5 + 26208*x^4 + 6056*x^3 - 16823*x^2 - 1620*x + 7048)
 
gp: K = bnfinit(y^20 - 6*y^19 + 16*y^18 - 40*y^17 + 69*y^16 + 44*y^15 - 275*y^14 - 36*y^13 + 1240*y^12 - 2696*y^11 + 3576*y^10 - 3812*y^9 + 5907*y^8 - 12888*y^7 + 25347*y^6 - 36688*y^5 + 26208*y^4 + 6056*y^3 - 16823*y^2 - 1620*y + 7048, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 6*x^19 + 16*x^18 - 40*x^17 + 69*x^16 + 44*x^15 - 275*x^14 - 36*x^13 + 1240*x^12 - 2696*x^11 + 3576*x^10 - 3812*x^9 + 5907*x^8 - 12888*x^7 + 25347*x^6 - 36688*x^5 + 26208*x^4 + 6056*x^3 - 16823*x^2 - 1620*x + 7048);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 6*x^19 + 16*x^18 - 40*x^17 + 69*x^16 + 44*x^15 - 275*x^14 - 36*x^13 + 1240*x^12 - 2696*x^11 + 3576*x^10 - 3812*x^9 + 5907*x^8 - 12888*x^7 + 25347*x^6 - 36688*x^5 + 26208*x^4 + 6056*x^3 - 16823*x^2 - 1620*x + 7048)
 

\( x^{20} - 6 x^{19} + 16 x^{18} - 40 x^{17} + 69 x^{16} + 44 x^{15} - 275 x^{14} - 36 x^{13} + 1240 x^{12} + \cdots + 7048 \) 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:   \(9841893626795960030057286598656\) \(\medspace = 2^{20}\cdot 3^{24}\cdot 7^{16}\) 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:  \(35.45\)
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:  $2\cdot 3^{4/3}7^{4/5}\approx 41.045930044873074$
Ramified primes:   \(2\), \(3\), \(7\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{50}a^{18}-\frac{23}{50}a^{17}-\frac{9}{50}a^{16}-\frac{19}{50}a^{15}-\frac{7}{25}a^{14}-\frac{7}{25}a^{13}-\frac{13}{50}a^{12}+\frac{9}{50}a^{11}-\frac{1}{10}a^{10}-\frac{1}{10}a^{9}-\frac{9}{50}a^{8}+\frac{21}{50}a^{7}-\frac{3}{25}a^{6}-\frac{11}{25}a^{5}+\frac{17}{50}a^{4}-\frac{1}{2}a^{3}+\frac{11}{50}a^{2}+\frac{19}{50}a-\frac{11}{25}$, $\frac{1}{33\!\cdots\!00}a^{19}-\frac{18\!\cdots\!27}{33\!\cdots\!00}a^{18}+\frac{29\!\cdots\!83}{33\!\cdots\!00}a^{17}-\frac{23\!\cdots\!91}{25\!\cdots\!00}a^{16}+\frac{37\!\cdots\!03}{82\!\cdots\!00}a^{15}-\frac{16\!\cdots\!51}{41\!\cdots\!50}a^{14}+\frac{11\!\cdots\!93}{33\!\cdots\!00}a^{13}+\frac{81\!\cdots\!11}{33\!\cdots\!00}a^{12}-\frac{11\!\cdots\!91}{33\!\cdots\!00}a^{11}+\frac{13\!\cdots\!23}{66\!\cdots\!00}a^{10}+\frac{10\!\cdots\!61}{33\!\cdots\!00}a^{9}-\frac{14\!\cdots\!93}{33\!\cdots\!00}a^{8}+\frac{51\!\cdots\!97}{41\!\cdots\!75}a^{7}-\frac{65\!\cdots\!53}{20\!\cdots\!75}a^{6}-\frac{21\!\cdots\!49}{66\!\cdots\!00}a^{5}-\frac{89\!\cdots\!43}{33\!\cdots\!00}a^{4}-\frac{95\!\cdots\!89}{33\!\cdots\!00}a^{3}-\frac{38\!\cdots\!43}{13\!\cdots\!00}a^{2}+\frac{39\!\cdots\!51}{63\!\cdots\!00}a-\frac{13\!\cdots\!89}{41\!\cdots\!50}$ 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

$C_{4}$, which has order $4$ (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:   \( -\frac{9668050423477325909361}{3592748448391172994590000} a^{19} + \frac{43221287504412349927847}{3592748448391172994590000} a^{18} - \frac{90163814403327067717263}{3592748448391172994590000} a^{17} + \frac{254687174187033567475263}{3592748448391172994590000} a^{16} - \frac{72339605445982605248683}{898187112097793248647500} a^{15} - \frac{103784099614619368943289}{449093556048896624323750} a^{14} + \frac{1360013377083313834026027}{3592748448391172994590000} a^{13} + \frac{2304055130364308658024729}{3592748448391172994590000} a^{12} - \frac{8351958998289067532415849}{3592748448391172994590000} a^{11} + \frac{2727309247904786343346197}{718549689678234598918000} a^{10} - \frac{14736860065084919476834121}{3592748448391172994590000} a^{9} + \frac{16048526150470832842666773}{3592748448391172994590000} a^{8} - \frac{431727037031330674296462}{44909355604889662432375} a^{7} + \frac{4655473392779390412823233}{224546778024448312161875} a^{6} - \frac{27304882043660623441670871}{718549689678234598918000} a^{5} + \frac{156250663121329752702808323}{3592748448391172994590000} a^{4} - \frac{32557308383223031932872171}{3592748448391172994590000} a^{3} - \frac{3585810654861128031063741}{143709937935646919783600} a^{2} + \frac{5115145188353950355841657}{898187112097793248647500} a + \frac{5560244889818707672428229}{449093556048896624323750} \)  (order $4$) 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:   $\frac{29\!\cdots\!81}{33\!\cdots\!00}a^{19}-\frac{14\!\cdots\!87}{33\!\cdots\!00}a^{18}+\frac{32\!\cdots\!23}{33\!\cdots\!00}a^{17}-\frac{67\!\cdots\!71}{25\!\cdots\!00}a^{16}+\frac{31\!\cdots\!43}{82\!\cdots\!00}a^{15}+\frac{30\!\cdots\!69}{41\!\cdots\!50}a^{14}-\frac{55\!\cdots\!67}{33\!\cdots\!00}a^{13}-\frac{67\!\cdots\!09}{33\!\cdots\!00}a^{12}+\frac{28\!\cdots\!29}{33\!\cdots\!00}a^{11}-\frac{10\!\cdots\!37}{66\!\cdots\!00}a^{10}+\frac{61\!\cdots\!41}{33\!\cdots\!00}a^{9}-\frac{63\!\cdots\!33}{33\!\cdots\!00}a^{8}+\frac{16\!\cdots\!87}{41\!\cdots\!75}a^{7}-\frac{16\!\cdots\!18}{20\!\cdots\!75}a^{6}+\frac{10\!\cdots\!71}{66\!\cdots\!00}a^{5}-\frac{64\!\cdots\!83}{33\!\cdots\!00}a^{4}+\frac{23\!\cdots\!91}{33\!\cdots\!00}a^{3}+\frac{10\!\cdots\!33}{13\!\cdots\!00}a^{2}-\frac{18\!\cdots\!69}{63\!\cdots\!00}a-\frac{11\!\cdots\!09}{41\!\cdots\!50}$, $\frac{69\!\cdots\!87}{33\!\cdots\!00}a^{19}-\frac{31\!\cdots\!49}{33\!\cdots\!00}a^{18}+\frac{71\!\cdots\!21}{33\!\cdots\!00}a^{17}-\frac{15\!\cdots\!17}{25\!\cdots\!00}a^{16}+\frac{63\!\cdots\!61}{82\!\cdots\!00}a^{15}+\frac{61\!\cdots\!63}{41\!\cdots\!50}a^{14}-\frac{97\!\cdots\!09}{33\!\cdots\!00}a^{13}-\frac{13\!\cdots\!43}{33\!\cdots\!00}a^{12}+\frac{61\!\cdots\!83}{33\!\cdots\!00}a^{11}-\frac{21\!\cdots\!99}{66\!\cdots\!00}a^{10}+\frac{13\!\cdots\!07}{33\!\cdots\!00}a^{9}-\frac{19\!\cdots\!91}{33\!\cdots\!00}a^{8}+\frac{38\!\cdots\!39}{41\!\cdots\!75}a^{7}-\frac{37\!\cdots\!86}{20\!\cdots\!75}a^{6}+\frac{21\!\cdots\!37}{66\!\cdots\!00}a^{5}-\frac{14\!\cdots\!41}{33\!\cdots\!00}a^{4}+\frac{10\!\cdots\!57}{33\!\cdots\!00}a^{3}+\frac{43\!\cdots\!39}{13\!\cdots\!00}a^{2}-\frac{22\!\cdots\!63}{63\!\cdots\!00}a-\frac{24\!\cdots\!43}{41\!\cdots\!50}$, $\frac{20\!\cdots\!79}{33\!\cdots\!00}a^{19}+\frac{12\!\cdots\!67}{33\!\cdots\!00}a^{18}-\frac{39\!\cdots\!43}{33\!\cdots\!00}a^{17}+\frac{45\!\cdots\!11}{25\!\cdots\!00}a^{16}-\frac{60\!\cdots\!63}{82\!\cdots\!00}a^{15}+\frac{10\!\cdots\!71}{41\!\cdots\!50}a^{14}+\frac{12\!\cdots\!47}{33\!\cdots\!00}a^{13}-\frac{31\!\cdots\!31}{33\!\cdots\!00}a^{12}-\frac{38\!\cdots\!89}{33\!\cdots\!00}a^{11}+\frac{11\!\cdots\!17}{66\!\cdots\!00}a^{10}-\frac{73\!\cdots\!81}{33\!\cdots\!00}a^{9}+\frac{99\!\cdots\!53}{33\!\cdots\!00}a^{8}-\frac{15\!\cdots\!67}{41\!\cdots\!75}a^{7}+\frac{14\!\cdots\!88}{20\!\cdots\!75}a^{6}-\frac{12\!\cdots\!11}{66\!\cdots\!00}a^{5}+\frac{95\!\cdots\!03}{33\!\cdots\!00}a^{4}-\frac{46\!\cdots\!31}{33\!\cdots\!00}a^{3}-\frac{10\!\cdots\!53}{13\!\cdots\!00}a^{2}+\frac{37\!\cdots\!29}{63\!\cdots\!00}a+\frac{10\!\cdots\!69}{41\!\cdots\!50}$, $\frac{19\!\cdots\!61}{33\!\cdots\!00}a^{19}-\frac{77\!\cdots\!47}{33\!\cdots\!00}a^{18}+\frac{15\!\cdots\!63}{33\!\cdots\!00}a^{17}-\frac{36\!\cdots\!51}{25\!\cdots\!00}a^{16}+\frac{10\!\cdots\!83}{82\!\cdots\!00}a^{15}+\frac{19\!\cdots\!89}{41\!\cdots\!50}a^{14}-\frac{20\!\cdots\!27}{33\!\cdots\!00}a^{13}-\frac{43\!\cdots\!29}{33\!\cdots\!00}a^{12}+\frac{14\!\cdots\!49}{33\!\cdots\!00}a^{11}-\frac{46\!\cdots\!97}{66\!\cdots\!00}a^{10}+\frac{26\!\cdots\!21}{33\!\cdots\!00}a^{9}-\frac{31\!\cdots\!73}{33\!\cdots\!00}a^{8}+\frac{75\!\cdots\!97}{41\!\cdots\!75}a^{7}-\frac{78\!\cdots\!33}{20\!\cdots\!75}a^{6}+\frac{46\!\cdots\!51}{66\!\cdots\!00}a^{5}-\frac{25\!\cdots\!23}{33\!\cdots\!00}a^{4}+\frac{25\!\cdots\!71}{33\!\cdots\!00}a^{3}+\frac{64\!\cdots\!73}{13\!\cdots\!00}a^{2}-\frac{57\!\cdots\!89}{63\!\cdots\!00}a-\frac{92\!\cdots\!29}{41\!\cdots\!50}$, $\frac{54\!\cdots\!31}{16\!\cdots\!00}a^{19}-\frac{34\!\cdots\!37}{16\!\cdots\!00}a^{18}+\frac{72\!\cdots\!73}{16\!\cdots\!00}a^{17}-\frac{13\!\cdots\!21}{12\!\cdots\!00}a^{16}+\frac{41\!\cdots\!84}{20\!\cdots\!75}a^{15}+\frac{73\!\cdots\!69}{20\!\cdots\!75}a^{14}-\frac{15\!\cdots\!17}{16\!\cdots\!00}a^{13}-\frac{18\!\cdots\!59}{16\!\cdots\!00}a^{12}+\frac{68\!\cdots\!79}{16\!\cdots\!00}a^{11}-\frac{21\!\cdots\!87}{33\!\cdots\!00}a^{10}+\frac{12\!\cdots\!91}{16\!\cdots\!00}a^{9}-\frac{87\!\cdots\!83}{16\!\cdots\!00}a^{8}+\frac{13\!\cdots\!43}{82\!\cdots\!50}a^{7}-\frac{75\!\cdots\!86}{20\!\cdots\!75}a^{6}+\frac{21\!\cdots\!61}{33\!\cdots\!00}a^{5}-\frac{14\!\cdots\!33}{16\!\cdots\!00}a^{4}+\frac{11\!\cdots\!41}{16\!\cdots\!00}a^{3}+\frac{37\!\cdots\!39}{66\!\cdots\!00}a^{2}+\frac{10\!\cdots\!28}{15\!\cdots\!75}a-\frac{32\!\cdots\!84}{20\!\cdots\!75}$, $\frac{15\!\cdots\!21}{16\!\cdots\!00}a^{19}-\frac{72\!\cdots\!67}{16\!\cdots\!00}a^{18}+\frac{15\!\cdots\!43}{16\!\cdots\!00}a^{17}-\frac{34\!\cdots\!11}{12\!\cdots\!00}a^{16}+\frac{68\!\cdots\!44}{20\!\cdots\!75}a^{15}+\frac{15\!\cdots\!79}{20\!\cdots\!75}a^{14}-\frac{21\!\cdots\!47}{16\!\cdots\!00}a^{13}-\frac{33\!\cdots\!69}{16\!\cdots\!00}a^{12}+\frac{13\!\cdots\!89}{16\!\cdots\!00}a^{11}-\frac{48\!\cdots\!17}{33\!\cdots\!00}a^{10}+\frac{29\!\cdots\!81}{16\!\cdots\!00}a^{9}-\frac{34\!\cdots\!53}{16\!\cdots\!00}a^{8}+\frac{34\!\cdots\!43}{82\!\cdots\!50}a^{7}-\frac{17\!\cdots\!76}{20\!\cdots\!75}a^{6}+\frac{51\!\cdots\!11}{33\!\cdots\!00}a^{5}-\frac{30\!\cdots\!03}{16\!\cdots\!00}a^{4}+\frac{14\!\cdots\!31}{16\!\cdots\!00}a^{3}+\frac{15\!\cdots\!73}{66\!\cdots\!00}a^{2}+\frac{26\!\cdots\!98}{15\!\cdots\!75}a-\frac{97\!\cdots\!44}{20\!\cdots\!75}$, $\frac{50\!\cdots\!73}{33\!\cdots\!00}a^{19}-\frac{22\!\cdots\!71}{33\!\cdots\!00}a^{18}+\frac{49\!\cdots\!59}{33\!\cdots\!00}a^{17}-\frac{10\!\cdots\!43}{25\!\cdots\!00}a^{16}+\frac{40\!\cdots\!19}{82\!\cdots\!00}a^{15}+\frac{52\!\cdots\!27}{41\!\cdots\!50}a^{14}-\frac{75\!\cdots\!11}{33\!\cdots\!00}a^{13}-\frac{10\!\cdots\!97}{33\!\cdots\!00}a^{12}+\frac{45\!\cdots\!57}{33\!\cdots\!00}a^{11}-\frac{15\!\cdots\!21}{66\!\cdots\!00}a^{10}+\frac{83\!\cdots\!53}{33\!\cdots\!00}a^{9}-\frac{81\!\cdots\!89}{33\!\cdots\!00}a^{8}+\frac{22\!\cdots\!66}{41\!\cdots\!75}a^{7}-\frac{25\!\cdots\!44}{20\!\cdots\!75}a^{6}+\frac{15\!\cdots\!03}{66\!\cdots\!00}a^{5}-\frac{88\!\cdots\!39}{33\!\cdots\!00}a^{4}+\frac{16\!\cdots\!03}{33\!\cdots\!00}a^{3}+\frac{21\!\cdots\!33}{13\!\cdots\!00}a^{2}-\frac{23\!\cdots\!77}{63\!\cdots\!00}a-\frac{32\!\cdots\!97}{41\!\cdots\!50}$, $\frac{81\!\cdots\!01}{33\!\cdots\!00}a^{19}-\frac{21\!\cdots\!27}{33\!\cdots\!00}a^{18}+\frac{35\!\cdots\!83}{33\!\cdots\!00}a^{17}-\frac{11\!\cdots\!91}{25\!\cdots\!00}a^{16}-\frac{38\!\cdots\!47}{82\!\cdots\!00}a^{15}+\frac{82\!\cdots\!49}{41\!\cdots\!50}a^{14}+\frac{17\!\cdots\!93}{33\!\cdots\!00}a^{13}-\frac{19\!\cdots\!89}{33\!\cdots\!00}a^{12}+\frac{31\!\cdots\!09}{33\!\cdots\!00}a^{11}-\frac{97\!\cdots\!77}{66\!\cdots\!00}a^{10}+\frac{40\!\cdots\!61}{33\!\cdots\!00}a^{9}-\frac{89\!\cdots\!93}{33\!\cdots\!00}a^{8}+\frac{48\!\cdots\!99}{82\!\cdots\!50}a^{7}-\frac{21\!\cdots\!53}{20\!\cdots\!75}a^{6}+\frac{11\!\cdots\!71}{66\!\cdots\!00}a^{5}-\frac{37\!\cdots\!43}{33\!\cdots\!00}a^{4}-\frac{13\!\cdots\!89}{33\!\cdots\!00}a^{3}-\frac{10\!\cdots\!67}{26\!\cdots\!40}a^{2}+\frac{85\!\cdots\!01}{63\!\cdots\!00}a+\frac{10\!\cdots\!11}{41\!\cdots\!50}$, $\frac{13\!\cdots\!21}{16\!\cdots\!00}a^{19}-\frac{57\!\cdots\!67}{16\!\cdots\!00}a^{18}+\frac{11\!\cdots\!43}{16\!\cdots\!00}a^{17}-\frac{26\!\cdots\!11}{12\!\cdots\!00}a^{16}+\frac{83\!\cdots\!63}{41\!\cdots\!50}a^{15}+\frac{15\!\cdots\!29}{20\!\cdots\!75}a^{14}-\frac{16\!\cdots\!47}{16\!\cdots\!00}a^{13}-\frac{33\!\cdots\!69}{16\!\cdots\!00}a^{12}+\frac{10\!\cdots\!89}{16\!\cdots\!00}a^{11}-\frac{34\!\cdots\!17}{33\!\cdots\!00}a^{10}+\frac{17\!\cdots\!81}{16\!\cdots\!00}a^{9}-\frac{19\!\cdots\!53}{16\!\cdots\!00}a^{8}+\frac{11\!\cdots\!09}{41\!\cdots\!75}a^{7}-\frac{11\!\cdots\!01}{20\!\cdots\!75}a^{6}+\frac{34\!\cdots\!11}{33\!\cdots\!00}a^{5}-\frac{18\!\cdots\!03}{16\!\cdots\!00}a^{4}+\frac{18\!\cdots\!31}{16\!\cdots\!00}a^{3}+\frac{47\!\cdots\!53}{66\!\cdots\!00}a^{2}-\frac{33\!\cdots\!29}{31\!\cdots\!50}a-\frac{66\!\cdots\!44}{20\!\cdots\!75}$ 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:  \( 78497371.7756 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 78497371.7756 \cdot 4}{4\cdot\sqrt{9841893626795960030057286598656}}\cr\approx \mathstrut & 2.39946525872 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 16*x^18 - 40*x^17 + 69*x^16 + 44*x^15 - 275*x^14 - 36*x^13 + 1240*x^12 - 2696*x^11 + 3576*x^10 - 3812*x^9 + 5907*x^8 - 12888*x^7 + 25347*x^6 - 36688*x^5 + 26208*x^4 + 6056*x^3 - 16823*x^2 - 1620*x + 7048)
 
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 - 6*x^19 + 16*x^18 - 40*x^17 + 69*x^16 + 44*x^15 - 275*x^14 - 36*x^13 + 1240*x^12 - 2696*x^11 + 3576*x^10 - 3812*x^9 + 5907*x^8 - 12888*x^7 + 25347*x^6 - 36688*x^5 + 26208*x^4 + 6056*x^3 - 16823*x^2 - 1620*x + 7048, 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 - 6*x^19 + 16*x^18 - 40*x^17 + 69*x^16 + 44*x^15 - 275*x^14 - 36*x^13 + 1240*x^12 - 2696*x^11 + 3576*x^10 - 3812*x^9 + 5907*x^8 - 12888*x^7 + 25347*x^6 - 36688*x^5 + 26208*x^4 + 6056*x^3 - 16823*x^2 - 1620*x + 7048);
 
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 - 6*x^19 + 16*x^18 - 40*x^17 + 69*x^16 + 44*x^15 - 275*x^14 - 36*x^13 + 1240*x^12 - 2696*x^11 + 3576*x^10 - 3812*x^9 + 5907*x^8 - 12888*x^7 + 25347*x^6 - 36688*x^5 + 26208*x^4 + 6056*x^3 - 16823*x^2 - 1620*x + 7048);
 
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

$S_6$ (as 20T145):

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 720
The 11 conjugacy class representatives for $S_6$
Character table for $S_6$

Intermediate fields

\(\Q(\sqrt{-1}) \), 10.4.196073702927424.1

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 6 siblings: 6.0.12446784.1, 6.4.63011844.2
Degree 10 sibling: 10.4.196073702927424.1
Degree 12 siblings: deg 12, deg 12
Degree 15 siblings: deg 15, deg 15
Degree 20 siblings: deg 20, deg 20
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: data not computed
Degree 45 sibling: data not computed
Minimal sibling: 6.0.12446784.1

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 ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.1.0.1}{1} }^{4}$ R ${\href{/padicField/11.6.0.1}{6} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ ${\href{/padicField/17.5.0.1}{5} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.5.0.1}{5} }^{4}$ ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.5.0.1}{5} }^{4}$ ${\href{/padicField/41.3.0.1}{3} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.5.0.1}{5} }^{4}$ ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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
\(2\) Copy content Toggle raw display 2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
\(3\) Copy content Toggle raw display 3.2.0.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $18$$3$$6$$24$
\(7\) Copy content Toggle raw display 7.10.8.1$x^{10} + 30 x^{9} + 375 x^{8} + 2520 x^{7} + 9810 x^{6} + 22370 x^{5} + 29640 x^{4} + 24780 x^{3} + 21465 x^{2} + 33300 x + 33934$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
7.10.8.1$x^{10} + 30 x^{9} + 375 x^{8} + 2520 x^{7} + 9810 x^{6} + 22370 x^{5} + 29640 x^{4} + 24780 x^{3} + 21465 x^{2} + 33300 x + 33934$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$