Properties

Label 20.4.250...000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2.500\times 10^{25}$
Root discriminant \(18.62\)
Ramified primes $2,5$
Class number $1$
Class group trivial
Galois group $C_2\times A_5$ (as 20T31)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^18 - 10*x^17 + 5*x^16 + 50*x^15 + 55*x^14 - 70*x^13 - 180*x^12 - 90*x^11 + 88*x^10 + 250*x^9 + 290*x^8 + 60*x^7 - 195*x^6 - 100*x^5 + 30*x^4 + 20*x^3 + 15*x^2 - 10*x + 1)
 
gp: K = bnfinit(y^20 - 5*y^18 - 10*y^17 + 5*y^16 + 50*y^15 + 55*y^14 - 70*y^13 - 180*y^12 - 90*y^11 + 88*y^10 + 250*y^9 + 290*y^8 + 60*y^7 - 195*y^6 - 100*y^5 + 30*y^4 + 20*y^3 + 15*y^2 - 10*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 5*x^18 - 10*x^17 + 5*x^16 + 50*x^15 + 55*x^14 - 70*x^13 - 180*x^12 - 90*x^11 + 88*x^10 + 250*x^9 + 290*x^8 + 60*x^7 - 195*x^6 - 100*x^5 + 30*x^4 + 20*x^3 + 15*x^2 - 10*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 5*x^18 - 10*x^17 + 5*x^16 + 50*x^15 + 55*x^14 - 70*x^13 - 180*x^12 - 90*x^11 + 88*x^10 + 250*x^9 + 290*x^8 + 60*x^7 - 195*x^6 - 100*x^5 + 30*x^4 + 20*x^3 + 15*x^2 - 10*x + 1)
 

\( x^{20} - 5 x^{18} - 10 x^{17} + 5 x^{16} + 50 x^{15} + 55 x^{14} - 70 x^{13} - 180 x^{12} - 90 x^{11} + 88 x^{10} + 250 x^{9} + 290 x^{8} + 60 x^{7} - 195 x^{6} - 100 x^{5} + 30 x^{4} + 20 x^{3} + \cdots + 1 \) 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:  $[4, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(25000000000000000000000000\) \(\medspace = 2^{24}\cdot 5^{26}\) 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:  \(18.62\)
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:   \(2\), \(5\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{14\!\cdots\!69}a^{19}+\frac{41\!\cdots\!78}{14\!\cdots\!69}a^{18}+\frac{69\!\cdots\!45}{14\!\cdots\!69}a^{17}+\frac{48\!\cdots\!08}{14\!\cdots\!69}a^{16}+\frac{18\!\cdots\!88}{14\!\cdots\!69}a^{15}-\frac{65\!\cdots\!66}{14\!\cdots\!69}a^{14}+\frac{39\!\cdots\!15}{14\!\cdots\!69}a^{13}+\frac{52\!\cdots\!74}{14\!\cdots\!69}a^{12}+\frac{56\!\cdots\!62}{14\!\cdots\!69}a^{11}+\frac{96\!\cdots\!15}{20\!\cdots\!67}a^{10}-\frac{28\!\cdots\!77}{14\!\cdots\!69}a^{9}-\frac{68\!\cdots\!67}{14\!\cdots\!69}a^{8}-\frac{41\!\cdots\!05}{14\!\cdots\!69}a^{7}-\frac{51\!\cdots\!55}{14\!\cdots\!69}a^{6}+\frac{32\!\cdots\!65}{14\!\cdots\!69}a^{5}-\frac{19\!\cdots\!51}{14\!\cdots\!69}a^{4}+\frac{35\!\cdots\!81}{14\!\cdots\!69}a^{3}+\frac{29\!\cdots\!70}{14\!\cdots\!69}a^{2}-\frac{11\!\cdots\!45}{14\!\cdots\!69}a-\frac{32\!\cdots\!96}{14\!\cdots\!69}$ 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 $1$

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:  $11$
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:   $\frac{16\!\cdots\!64}{849025159696231}a^{19}+\frac{614431251688500}{849025159696231}a^{18}-\frac{78\!\cdots\!46}{849025159696231}a^{17}-\frac{19\!\cdots\!05}{849025159696231}a^{16}+\frac{685795625953880}{849025159696231}a^{15}+\frac{80\!\cdots\!25}{849025159696231}a^{14}+\frac{11\!\cdots\!54}{849025159696231}a^{13}-\frac{66\!\cdots\!45}{849025159696231}a^{12}-\frac{31\!\cdots\!76}{849025159696231}a^{11}-\frac{38\!\cdots\!60}{121289308528033}a^{10}+\frac{35\!\cdots\!26}{849025159696231}a^{9}+\frac{41\!\cdots\!25}{849025159696231}a^{8}+\frac{62\!\cdots\!36}{849025159696231}a^{7}+\frac{34\!\cdots\!70}{849025159696231}a^{6}-\frac{17\!\cdots\!60}{849025159696231}a^{5}-\frac{22\!\cdots\!65}{849025159696231}a^{4}-\frac{39\!\cdots\!24}{849025159696231}a^{3}+\frac{12\!\cdots\!75}{849025159696231}a^{2}+\frac{27\!\cdots\!16}{849025159696231}a-\frac{41\!\cdots\!42}{849025159696231}$, $\frac{13\!\cdots\!86}{14\!\cdots\!69}a^{19}+\frac{65\!\cdots\!07}{14\!\cdots\!69}a^{18}-\frac{64\!\cdots\!03}{14\!\cdots\!69}a^{17}-\frac{16\!\cdots\!67}{14\!\cdots\!69}a^{16}-\frac{11\!\cdots\!05}{14\!\cdots\!69}a^{15}+\frac{66\!\cdots\!55}{14\!\cdots\!69}a^{14}+\frac{10\!\cdots\!55}{14\!\cdots\!69}a^{13}-\frac{44\!\cdots\!46}{14\!\cdots\!69}a^{12}-\frac{26\!\cdots\!70}{14\!\cdots\!69}a^{11}-\frac{35\!\cdots\!56}{20\!\cdots\!67}a^{10}+\frac{76\!\cdots\!78}{14\!\cdots\!69}a^{9}+\frac{34\!\cdots\!25}{14\!\cdots\!69}a^{8}+\frac{55\!\cdots\!62}{14\!\cdots\!69}a^{7}+\frac{33\!\cdots\!97}{14\!\cdots\!69}a^{6}-\frac{11\!\cdots\!78}{14\!\cdots\!69}a^{5}-\frac{20\!\cdots\!11}{14\!\cdots\!69}a^{4}-\frac{58\!\cdots\!72}{14\!\cdots\!69}a^{3}+\frac{53\!\cdots\!48}{14\!\cdots\!69}a^{2}+\frac{23\!\cdots\!02}{14\!\cdots\!69}a-\frac{13\!\cdots\!05}{14\!\cdots\!69}$, $\frac{64\!\cdots\!02}{14\!\cdots\!69}a^{19}+\frac{47\!\cdots\!81}{14\!\cdots\!69}a^{18}-\frac{29\!\cdots\!07}{14\!\cdots\!69}a^{17}-\frac{87\!\cdots\!13}{14\!\cdots\!69}a^{16}-\frac{25\!\cdots\!25}{14\!\cdots\!69}a^{15}+\frac{31\!\cdots\!58}{14\!\cdots\!69}a^{14}+\frac{58\!\cdots\!17}{14\!\cdots\!69}a^{13}-\frac{82\!\cdots\!85}{14\!\cdots\!69}a^{12}-\frac{13\!\cdots\!68}{14\!\cdots\!69}a^{11}-\frac{21\!\cdots\!61}{20\!\cdots\!67}a^{10}-\frac{28\!\cdots\!14}{14\!\cdots\!69}a^{9}+\frac{16\!\cdots\!38}{14\!\cdots\!69}a^{8}+\frac{30\!\cdots\!06}{14\!\cdots\!69}a^{7}+\frac{22\!\cdots\!16}{14\!\cdots\!69}a^{6}-\frac{73\!\cdots\!69}{14\!\cdots\!69}a^{5}-\frac{98\!\cdots\!14}{14\!\cdots\!69}a^{4}-\frac{39\!\cdots\!25}{14\!\cdots\!69}a^{3}+\frac{12\!\cdots\!81}{14\!\cdots\!69}a^{2}+\frac{95\!\cdots\!29}{14\!\cdots\!69}a+\frac{66\!\cdots\!70}{14\!\cdots\!69}$, $\frac{11\!\cdots\!72}{14\!\cdots\!69}a^{19}+\frac{79\!\cdots\!19}{14\!\cdots\!69}a^{18}-\frac{54\!\cdots\!27}{14\!\cdots\!69}a^{17}-\frac{15\!\cdots\!75}{14\!\cdots\!69}a^{16}-\frac{41\!\cdots\!14}{14\!\cdots\!69}a^{15}+\frac{57\!\cdots\!27}{14\!\cdots\!69}a^{14}+\frac{10\!\cdots\!44}{14\!\cdots\!69}a^{13}-\frac{17\!\cdots\!99}{14\!\cdots\!69}a^{12}-\frac{23\!\cdots\!21}{14\!\cdots\!69}a^{11}-\frac{37\!\cdots\!96}{20\!\cdots\!67}a^{10}-\frac{56\!\cdots\!82}{14\!\cdots\!69}a^{9}+\frac{27\!\cdots\!98}{14\!\cdots\!69}a^{8}+\frac{53\!\cdots\!20}{14\!\cdots\!69}a^{7}+\frac{41\!\cdots\!31}{14\!\cdots\!69}a^{6}+\frac{14\!\cdots\!17}{14\!\cdots\!69}a^{5}-\frac{13\!\cdots\!31}{14\!\cdots\!69}a^{4}-\frac{57\!\cdots\!27}{14\!\cdots\!69}a^{3}-\frac{79\!\cdots\!90}{14\!\cdots\!69}a^{2}+\frac{13\!\cdots\!62}{14\!\cdots\!69}a-\frac{81\!\cdots\!41}{14\!\cdots\!69}$, $\frac{61\!\cdots\!73}{14\!\cdots\!69}a^{19}+\frac{28\!\cdots\!13}{14\!\cdots\!69}a^{18}-\frac{27\!\cdots\!40}{14\!\cdots\!69}a^{17}-\frac{75\!\cdots\!49}{14\!\cdots\!69}a^{16}-\frac{11\!\cdots\!54}{14\!\cdots\!69}a^{15}+\frac{28\!\cdots\!70}{14\!\cdots\!69}a^{14}+\frac{48\!\cdots\!83}{14\!\cdots\!69}a^{13}-\frac{13\!\cdots\!39}{14\!\cdots\!69}a^{12}-\frac{10\!\cdots\!98}{14\!\cdots\!69}a^{11}-\frac{16\!\cdots\!09}{20\!\cdots\!67}a^{10}-\frac{23\!\cdots\!87}{14\!\cdots\!69}a^{9}+\frac{12\!\cdots\!52}{14\!\cdots\!69}a^{8}+\frac{24\!\cdots\!15}{14\!\cdots\!69}a^{7}+\frac{18\!\cdots\!93}{14\!\cdots\!69}a^{6}+\frac{99\!\cdots\!86}{14\!\cdots\!69}a^{5}-\frac{39\!\cdots\!26}{14\!\cdots\!69}a^{4}-\frac{16\!\cdots\!60}{14\!\cdots\!69}a^{3}-\frac{40\!\cdots\!30}{14\!\cdots\!69}a^{2}+\frac{56\!\cdots\!31}{14\!\cdots\!69}a-\frac{15\!\cdots\!99}{14\!\cdots\!69}$, $\frac{28\!\cdots\!68}{14\!\cdots\!69}a^{19}+\frac{27\!\cdots\!76}{14\!\cdots\!69}a^{18}-\frac{13\!\cdots\!36}{14\!\cdots\!69}a^{17}-\frac{41\!\cdots\!75}{14\!\cdots\!69}a^{16}-\frac{18\!\cdots\!69}{14\!\cdots\!69}a^{15}+\frac{14\!\cdots\!03}{14\!\cdots\!69}a^{14}+\frac{28\!\cdots\!07}{14\!\cdots\!69}a^{13}+\frac{48\!\cdots\!35}{14\!\cdots\!69}a^{12}-\frac{60\!\cdots\!98}{14\!\cdots\!69}a^{11}-\frac{10\!\cdots\!82}{20\!\cdots\!67}a^{10}-\frac{22\!\cdots\!49}{14\!\cdots\!69}a^{9}+\frac{70\!\cdots\!96}{14\!\cdots\!69}a^{8}+\frac{14\!\cdots\!42}{14\!\cdots\!69}a^{7}+\frac{12\!\cdots\!81}{14\!\cdots\!69}a^{6}+\frac{13\!\cdots\!84}{14\!\cdots\!69}a^{5}-\frac{41\!\cdots\!17}{14\!\cdots\!69}a^{4}-\frac{21\!\cdots\!09}{14\!\cdots\!69}a^{3}-\frac{24\!\cdots\!17}{14\!\cdots\!69}a^{2}+\frac{51\!\cdots\!13}{14\!\cdots\!69}a-\frac{11\!\cdots\!32}{14\!\cdots\!69}$, $\frac{12\!\cdots\!46}{849025159696231}a^{19}+\frac{542397357934018}{849025159696231}a^{18}-\frac{61\!\cdots\!30}{849025159696231}a^{17}-\frac{15\!\cdots\!32}{849025159696231}a^{16}-\frac{12349785668606}{849025159696231}a^{15}+\frac{63\!\cdots\!27}{849025159696231}a^{14}+\frac{95\!\cdots\!71}{849025159696231}a^{13}-\frac{49\!\cdots\!87}{849025159696231}a^{12}-\frac{24\!\cdots\!90}{849025159696231}a^{11}-\frac{30\!\cdots\!76}{121289308528033}a^{10}+\frac{24\!\cdots\!22}{849025159696231}a^{9}+\frac{32\!\cdots\!46}{849025159696231}a^{8}+\frac{50\!\cdots\!82}{849025159696231}a^{7}+\frac{28\!\cdots\!43}{849025159696231}a^{6}-\frac{13\!\cdots\!61}{849025159696231}a^{5}-\frac{18\!\cdots\!28}{849025159696231}a^{4}-\frac{32\!\cdots\!49}{849025159696231}a^{3}+\frac{10\!\cdots\!68}{849025159696231}a^{2}+\frac{21\!\cdots\!73}{849025159696231}a-\frac{34\!\cdots\!14}{849025159696231}$, $\frac{77\!\cdots\!33}{14\!\cdots\!69}a^{19}+\frac{70\!\cdots\!30}{14\!\cdots\!69}a^{18}-\frac{35\!\cdots\!45}{14\!\cdots\!69}a^{17}-\frac{11\!\cdots\!16}{14\!\cdots\!69}a^{16}-\frac{47\!\cdots\!48}{14\!\cdots\!69}a^{15}+\frac{38\!\cdots\!81}{14\!\cdots\!69}a^{14}+\frac{77\!\cdots\!24}{14\!\cdots\!69}a^{13}+\frac{61\!\cdots\!03}{14\!\cdots\!69}a^{12}-\frac{16\!\cdots\!66}{14\!\cdots\!69}a^{11}-\frac{29\!\cdots\!30}{20\!\cdots\!67}a^{10}-\frac{59\!\cdots\!27}{14\!\cdots\!69}a^{9}+\frac{19\!\cdots\!65}{14\!\cdots\!69}a^{8}+\frac{39\!\cdots\!06}{14\!\cdots\!69}a^{7}+\frac{33\!\cdots\!17}{14\!\cdots\!69}a^{6}+\frac{30\!\cdots\!60}{14\!\cdots\!69}a^{5}-\frac{12\!\cdots\!01}{14\!\cdots\!69}a^{4}-\frac{63\!\cdots\!49}{14\!\cdots\!69}a^{3}-\frac{63\!\cdots\!09}{14\!\cdots\!69}a^{2}+\frac{10\!\cdots\!30}{14\!\cdots\!69}a+\frac{22\!\cdots\!99}{14\!\cdots\!69}$, $\frac{449685012060553}{89\!\cdots\!29}a^{19}+\frac{486324839543875}{89\!\cdots\!29}a^{18}-\frac{17\!\cdots\!13}{89\!\cdots\!29}a^{17}-\frac{62\!\cdots\!81}{89\!\cdots\!29}a^{16}-\frac{48\!\cdots\!57}{89\!\cdots\!29}a^{15}+\frac{16\!\cdots\!51}{89\!\cdots\!29}a^{14}+\frac{43\!\cdots\!18}{89\!\cdots\!29}a^{13}+\frac{19\!\cdots\!62}{89\!\cdots\!29}a^{12}-\frac{54\!\cdots\!95}{89\!\cdots\!29}a^{11}-\frac{11\!\cdots\!03}{89\!\cdots\!29}a^{10}-\frac{11\!\cdots\!55}{89\!\cdots\!29}a^{9}-\frac{13\!\cdots\!06}{89\!\cdots\!29}a^{8}+\frac{16\!\cdots\!29}{89\!\cdots\!29}a^{7}+\frac{27\!\cdots\!85}{89\!\cdots\!29}a^{6}+\frac{24\!\cdots\!81}{89\!\cdots\!29}a^{5}+\frac{16\!\cdots\!44}{89\!\cdots\!29}a^{4}+\frac{57\!\cdots\!70}{89\!\cdots\!29}a^{3}-\frac{34\!\cdots\!50}{89\!\cdots\!29}a^{2}-\frac{34\!\cdots\!57}{89\!\cdots\!29}a-\frac{44\!\cdots\!47}{89\!\cdots\!29}$, $\frac{47\!\cdots\!88}{14\!\cdots\!69}a^{19}+\frac{25\!\cdots\!39}{14\!\cdots\!69}a^{18}-\frac{22\!\cdots\!18}{14\!\cdots\!69}a^{17}-\frac{59\!\cdots\!05}{14\!\cdots\!69}a^{16}-\frac{74\!\cdots\!87}{14\!\cdots\!69}a^{15}+\frac{23\!\cdots\!40}{14\!\cdots\!69}a^{14}+\frac{38\!\cdots\!78}{14\!\cdots\!69}a^{13}-\frac{13\!\cdots\!40}{14\!\cdots\!69}a^{12}-\frac{92\!\cdots\!62}{14\!\cdots\!69}a^{11}-\frac{13\!\cdots\!27}{20\!\cdots\!67}a^{10}-\frac{67\!\cdots\!04}{14\!\cdots\!69}a^{9}+\frac{11\!\cdots\!19}{14\!\cdots\!69}a^{8}+\frac{20\!\cdots\!29}{14\!\cdots\!69}a^{7}+\frac{13\!\cdots\!68}{14\!\cdots\!69}a^{6}-\frac{22\!\cdots\!66}{14\!\cdots\!69}a^{5}-\frac{61\!\cdots\!79}{14\!\cdots\!69}a^{4}-\frac{18\!\cdots\!85}{14\!\cdots\!69}a^{3}+\frac{47\!\cdots\!85}{14\!\cdots\!69}a^{2}+\frac{85\!\cdots\!69}{14\!\cdots\!69}a-\frac{61\!\cdots\!19}{14\!\cdots\!69}$, $\frac{23\!\cdots\!36}{89\!\cdots\!29}a^{19}+\frac{11\!\cdots\!84}{89\!\cdots\!29}a^{18}-\frac{11\!\cdots\!09}{89\!\cdots\!29}a^{17}-\frac{28\!\cdots\!15}{89\!\cdots\!29}a^{16}-\frac{27\!\cdots\!86}{89\!\cdots\!29}a^{15}+\frac{11\!\cdots\!52}{89\!\cdots\!29}a^{14}+\frac{18\!\cdots\!21}{89\!\cdots\!29}a^{13}-\frac{70\!\cdots\!25}{89\!\cdots\!29}a^{12}-\frac{45\!\cdots\!64}{89\!\cdots\!29}a^{11}-\frac{43\!\cdots\!46}{89\!\cdots\!29}a^{10}-\frac{13\!\cdots\!36}{89\!\cdots\!29}a^{9}+\frac{57\!\cdots\!59}{89\!\cdots\!29}a^{8}+\frac{96\!\cdots\!48}{89\!\cdots\!29}a^{7}+\frac{62\!\cdots\!16}{89\!\cdots\!29}a^{6}-\frac{14\!\cdots\!34}{89\!\cdots\!29}a^{5}-\frac{31\!\cdots\!12}{89\!\cdots\!29}a^{4}-\frac{87\!\cdots\!88}{89\!\cdots\!29}a^{3}+\frac{13\!\cdots\!50}{89\!\cdots\!29}a^{2}+\frac{37\!\cdots\!09}{89\!\cdots\!29}a-\frac{44\!\cdots\!73}{89\!\cdots\!29}$ Copy content Toggle raw display
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:  \( 94707.1867115 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 94707.1867115 \cdot 1}{2\cdot\sqrt{25000000000000000000000000}}\cr\approx \mathstrut & 0.368079699377 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^18 - 10*x^17 + 5*x^16 + 50*x^15 + 55*x^14 - 70*x^13 - 180*x^12 - 90*x^11 + 88*x^10 + 250*x^9 + 290*x^8 + 60*x^7 - 195*x^6 - 100*x^5 + 30*x^4 + 20*x^3 + 15*x^2 - 10*x + 1)
 
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 - 5*x^18 - 10*x^17 + 5*x^16 + 50*x^15 + 55*x^14 - 70*x^13 - 180*x^12 - 90*x^11 + 88*x^10 + 250*x^9 + 290*x^8 + 60*x^7 - 195*x^6 - 100*x^5 + 30*x^4 + 20*x^3 + 15*x^2 - 10*x + 1, 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 - 5*x^18 - 10*x^17 + 5*x^16 + 50*x^15 + 55*x^14 - 70*x^13 - 180*x^12 - 90*x^11 + 88*x^10 + 250*x^9 + 290*x^8 + 60*x^7 - 195*x^6 - 100*x^5 + 30*x^4 + 20*x^3 + 15*x^2 - 10*x + 1);
 
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 - 5*x^18 - 10*x^17 + 5*x^16 + 50*x^15 + 55*x^14 - 70*x^13 - 180*x^12 - 90*x^11 + 88*x^10 + 250*x^9 + 290*x^8 + 60*x^7 - 195*x^6 - 100*x^5 + 30*x^4 + 20*x^3 + 15*x^2 - 10*x + 1);
 
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

\(\Q(\sqrt{5}) \), 10.2.1000000000000.1

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: 10.2.5000000000000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.10.0.1}{10} }^{2}$ R ${\href{/padicField/7.6.0.1}{6} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.3.0.1}{3} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.10.0.1}{10} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.5.0.1}{5} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{10}$ ${\href{/padicField/29.3.0.1}{3} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.5.0.1}{5} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{10}$ ${\href{/padicField/41.3.0.1}{3} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.10.0.1}{10} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }^{2}$ ${\href{/padicField/59.5.0.1}{5} }^{4}$

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

# 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 2.8.12.16$x^{8} + 8 x^{7} + 32 x^{6} + 78 x^{5} + 137 x^{4} + 186 x^{3} + 128 x^{2} - 10 x + 7$$4$$2$$12$$A_4\times C_2$$[2, 2]^{6}$
2.12.12.11$x^{12} + 28 x^{10} + 40 x^{9} + 356 x^{8} + 896 x^{7} + 2720 x^{6} + 6656 x^{5} + 12464 x^{4} + 19456 x^{3} + 26304 x^{2} + 19840 x + 5824$$2$$6$$12$$A_4 \times C_2$$[2, 2]^{6}$
\(5\) Copy content Toggle raw display 5.10.13.2$x^{10} + 10 x^{4} + 5$$10$$1$$13$$D_{10}$$[3/2]_{2}^{2}$
5.10.13.2$x^{10} + 10 x^{4} + 5$$10$$1$$13$$D_{10}$$[3/2]_{2}^{2}$