Properties

Label 20.0.333...401.1
Degree $20$
Signature $[0, 10]$
Discriminant $3.330\times 10^{32}$
Root discriminant \(42.28\)
Ramified primes $19,293$
Class number $14$ (GRH)
Class group [14] (GRH)
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 + 3*x^18 - 19*x^17 + 21*x^16 - 159*x^15 - 7*x^14 + 354*x^13 + 1233*x^12 + 655*x^11 - 2385*x^10 + 2997*x^9 + 15528*x^8 + 15281*x^7 - 4769*x^6 - 14351*x^5 - 1363*x^4 + 4718*x^3 + 3265*x^2 - 9*x + 77)
 
gp: K = bnfinit(y^20 + 3*y^18 - 19*y^17 + 21*y^16 - 159*y^15 - 7*y^14 + 354*y^13 + 1233*y^12 + 655*y^11 - 2385*y^10 + 2997*y^9 + 15528*y^8 + 15281*y^7 - 4769*y^6 - 14351*y^5 - 1363*y^4 + 4718*y^3 + 3265*y^2 - 9*y + 77, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 3*x^18 - 19*x^17 + 21*x^16 - 159*x^15 - 7*x^14 + 354*x^13 + 1233*x^12 + 655*x^11 - 2385*x^10 + 2997*x^9 + 15528*x^8 + 15281*x^7 - 4769*x^6 - 14351*x^5 - 1363*x^4 + 4718*x^3 + 3265*x^2 - 9*x + 77);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + 3*x^18 - 19*x^17 + 21*x^16 - 159*x^15 - 7*x^14 + 354*x^13 + 1233*x^12 + 655*x^11 - 2385*x^10 + 2997*x^9 + 15528*x^8 + 15281*x^7 - 4769*x^6 - 14351*x^5 - 1363*x^4 + 4718*x^3 + 3265*x^2 - 9*x + 77)
 

\( x^{20} + 3 x^{18} - 19 x^{17} + 21 x^{16} - 159 x^{15} - 7 x^{14} + 354 x^{13} + 1233 x^{12} + 655 x^{11} - 2385 x^{10} + 2997 x^{9} + 15528 x^{8} + 15281 x^{7} - 4769 x^{6} + \cdots + 77 \) 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:   \(333025103817911062616373171479401\) \(\medspace = 19^{10}\cdot 293^{8}\) 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:  \(42.28\)
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:   \(19\), \(293\) 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 a CM field.
Reflex fields:  unavailable$^{512}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{87\!\cdots\!69}a^{19}+\frac{42\!\cdots\!02}{87\!\cdots\!69}a^{18}+\frac{23\!\cdots\!38}{87\!\cdots\!69}a^{17}-\frac{27\!\cdots\!36}{87\!\cdots\!69}a^{16}-\frac{28\!\cdots\!89}{87\!\cdots\!69}a^{15}+\frac{52\!\cdots\!95}{87\!\cdots\!69}a^{14}-\frac{38\!\cdots\!48}{87\!\cdots\!69}a^{13}+\frac{17\!\cdots\!86}{87\!\cdots\!69}a^{12}-\frac{29\!\cdots\!58}{87\!\cdots\!69}a^{11}-\frac{15\!\cdots\!94}{87\!\cdots\!69}a^{10}-\frac{50\!\cdots\!15}{87\!\cdots\!69}a^{9}-\frac{53\!\cdots\!94}{87\!\cdots\!69}a^{8}-\frac{27\!\cdots\!57}{87\!\cdots\!69}a^{7}-\frac{29\!\cdots\!21}{87\!\cdots\!69}a^{6}-\frac{29\!\cdots\!80}{87\!\cdots\!69}a^{5}-\frac{37\!\cdots\!58}{87\!\cdots\!69}a^{4}-\frac{57\!\cdots\!93}{87\!\cdots\!69}a^{3}-\frac{29\!\cdots\!78}{87\!\cdots\!69}a^{2}-\frac{25\!\cdots\!93}{87\!\cdots\!69}a+\frac{35\!\cdots\!59}{87\!\cdots\!69}$ 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

$C_{14}$, which has order $14$ (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:   $\frac{18\!\cdots\!83}{87\!\cdots\!69}a^{19}+\frac{62\!\cdots\!16}{87\!\cdots\!69}a^{18}+\frac{46\!\cdots\!52}{87\!\cdots\!69}a^{17}-\frac{31\!\cdots\!52}{87\!\cdots\!69}a^{16}+\frac{23\!\cdots\!41}{87\!\cdots\!69}a^{15}-\frac{26\!\cdots\!62}{87\!\cdots\!69}a^{14}-\frac{15\!\cdots\!24}{87\!\cdots\!69}a^{13}+\frac{80\!\cdots\!83}{87\!\cdots\!69}a^{12}+\frac{22\!\cdots\!14}{87\!\cdots\!69}a^{11}+\frac{16\!\cdots\!42}{87\!\cdots\!69}a^{10}-\frac{42\!\cdots\!64}{87\!\cdots\!69}a^{9}+\frac{47\!\cdots\!89}{87\!\cdots\!69}a^{8}+\frac{32\!\cdots\!28}{87\!\cdots\!69}a^{7}+\frac{30\!\cdots\!17}{87\!\cdots\!69}a^{6}-\frac{53\!\cdots\!87}{87\!\cdots\!69}a^{5}-\frac{18\!\cdots\!32}{87\!\cdots\!69}a^{4}-\frac{27\!\cdots\!24}{87\!\cdots\!69}a^{3}+\frac{20\!\cdots\!21}{87\!\cdots\!69}a^{2}-\frac{15\!\cdots\!41}{87\!\cdots\!69}a+\frac{57\!\cdots\!11}{87\!\cdots\!69}$, $\frac{19\!\cdots\!50}{14\!\cdots\!29}a^{19}-\frac{54\!\cdots\!69}{14\!\cdots\!29}a^{18}+\frac{53\!\cdots\!82}{14\!\cdots\!29}a^{17}-\frac{38\!\cdots\!80}{14\!\cdots\!29}a^{16}+\frac{49\!\cdots\!67}{14\!\cdots\!29}a^{15}-\frac{31\!\cdots\!00}{14\!\cdots\!29}a^{14}+\frac{46\!\cdots\!46}{14\!\cdots\!29}a^{13}+\frac{80\!\cdots\!50}{14\!\cdots\!29}a^{12}+\frac{20\!\cdots\!47}{14\!\cdots\!29}a^{11}+\frac{51\!\cdots\!92}{14\!\cdots\!29}a^{10}-\frac{54\!\cdots\!80}{14\!\cdots\!29}a^{9}+\frac{75\!\cdots\!99}{14\!\cdots\!29}a^{8}+\frac{30\!\cdots\!70}{14\!\cdots\!29}a^{7}+\frac{18\!\cdots\!70}{14\!\cdots\!29}a^{6}-\frac{22\!\cdots\!75}{14\!\cdots\!29}a^{5}-\frac{24\!\cdots\!12}{14\!\cdots\!29}a^{4}+\frac{11\!\cdots\!17}{14\!\cdots\!29}a^{3}+\frac{93\!\cdots\!68}{14\!\cdots\!29}a^{2}-\frac{10\!\cdots\!43}{14\!\cdots\!29}a-\frac{10\!\cdots\!94}{14\!\cdots\!29}$, $\frac{97\!\cdots\!12}{14\!\cdots\!29}a^{19}-\frac{20\!\cdots\!01}{14\!\cdots\!29}a^{18}+\frac{28\!\cdots\!97}{14\!\cdots\!29}a^{17}-\frac{18\!\cdots\!44}{14\!\cdots\!29}a^{16}+\frac{24\!\cdots\!37}{14\!\cdots\!29}a^{15}-\frac{15\!\cdots\!46}{14\!\cdots\!29}a^{14}+\frac{18\!\cdots\!21}{14\!\cdots\!29}a^{13}+\frac{37\!\cdots\!70}{14\!\cdots\!29}a^{12}+\frac{10\!\cdots\!92}{14\!\cdots\!29}a^{11}+\frac{35\!\cdots\!95}{14\!\cdots\!29}a^{10}-\frac{24\!\cdots\!61}{14\!\cdots\!29}a^{9}+\frac{35\!\cdots\!39}{14\!\cdots\!29}a^{8}+\frac{14\!\cdots\!02}{14\!\cdots\!29}a^{7}+\frac{10\!\cdots\!72}{14\!\cdots\!29}a^{6}-\frac{84\!\cdots\!41}{14\!\cdots\!29}a^{5}-\frac{10\!\cdots\!73}{14\!\cdots\!29}a^{4}+\frac{35\!\cdots\!87}{14\!\cdots\!29}a^{3}+\frac{34\!\cdots\!51}{14\!\cdots\!29}a^{2}-\frac{58\!\cdots\!43}{14\!\cdots\!29}a+\frac{29\!\cdots\!60}{14\!\cdots\!29}$, $\frac{28\!\cdots\!37}{87\!\cdots\!69}a^{19}-\frac{17\!\cdots\!18}{87\!\cdots\!69}a^{18}+\frac{11\!\cdots\!96}{87\!\cdots\!69}a^{17}-\frac{61\!\cdots\!07}{87\!\cdots\!69}a^{16}+\frac{10\!\cdots\!59}{87\!\cdots\!69}a^{15}-\frac{55\!\cdots\!57}{87\!\cdots\!69}a^{14}+\frac{37\!\cdots\!61}{87\!\cdots\!69}a^{13}+\frac{44\!\cdots\!32}{87\!\cdots\!69}a^{12}+\frac{34\!\cdots\!13}{87\!\cdots\!69}a^{11}-\frac{37\!\cdots\!70}{87\!\cdots\!69}a^{10}-\frac{44\!\cdots\!58}{87\!\cdots\!69}a^{9}+\frac{11\!\cdots\!75}{87\!\cdots\!69}a^{8}+\frac{33\!\cdots\!98}{87\!\cdots\!69}a^{7}+\frac{29\!\cdots\!90}{87\!\cdots\!69}a^{6}-\frac{12\!\cdots\!08}{87\!\cdots\!69}a^{5}-\frac{73\!\cdots\!85}{87\!\cdots\!69}a^{4}-\frac{78\!\cdots\!68}{87\!\cdots\!69}a^{3}+\frac{17\!\cdots\!58}{87\!\cdots\!69}a^{2}-\frac{22\!\cdots\!98}{87\!\cdots\!69}a-\frac{98\!\cdots\!11}{87\!\cdots\!69}$, $\frac{18\!\cdots\!58}{87\!\cdots\!69}a^{19}-\frac{10\!\cdots\!83}{87\!\cdots\!69}a^{18}+\frac{50\!\cdots\!22}{87\!\cdots\!69}a^{17}-\frac{37\!\cdots\!83}{87\!\cdots\!69}a^{16}+\frac{55\!\cdots\!61}{87\!\cdots\!69}a^{15}-\frac{30\!\cdots\!74}{87\!\cdots\!69}a^{14}+\frac{13\!\cdots\!34}{87\!\cdots\!69}a^{13}+\frac{72\!\cdots\!96}{87\!\cdots\!69}a^{12}+\frac{18\!\cdots\!23}{87\!\cdots\!69}a^{11}-\frac{14\!\cdots\!08}{87\!\cdots\!69}a^{10}-\frac{53\!\cdots\!24}{87\!\cdots\!69}a^{9}+\frac{79\!\cdots\!88}{87\!\cdots\!69}a^{8}+\frac{26\!\cdots\!18}{87\!\cdots\!69}a^{7}+\frac{10\!\cdots\!68}{87\!\cdots\!69}a^{6}-\frac{28\!\cdots\!81}{87\!\cdots\!69}a^{5}-\frac{22\!\cdots\!17}{87\!\cdots\!69}a^{4}+\frac{15\!\cdots\!69}{87\!\cdots\!69}a^{3}+\frac{11\!\cdots\!11}{87\!\cdots\!69}a^{2}-\frac{82\!\cdots\!28}{87\!\cdots\!69}a-\frac{43\!\cdots\!24}{87\!\cdots\!69}$, $\frac{20\!\cdots\!92}{87\!\cdots\!69}a^{19}-\frac{14\!\cdots\!52}{87\!\cdots\!69}a^{18}+\frac{59\!\cdots\!22}{87\!\cdots\!69}a^{17}-\frac{43\!\cdots\!23}{87\!\cdots\!69}a^{16}+\frac{70\!\cdots\!94}{87\!\cdots\!69}a^{15}-\frac{35\!\cdots\!06}{87\!\cdots\!69}a^{14}+\frac{21\!\cdots\!68}{87\!\cdots\!69}a^{13}+\frac{77\!\cdots\!57}{87\!\cdots\!69}a^{12}+\frac{19\!\cdots\!36}{87\!\cdots\!69}a^{11}-\frac{57\!\cdots\!52}{87\!\cdots\!69}a^{10}-\frac{60\!\cdots\!94}{87\!\cdots\!69}a^{9}+\frac{96\!\cdots\!38}{87\!\cdots\!69}a^{8}+\frac{27\!\cdots\!55}{87\!\cdots\!69}a^{7}+\frac{71\!\cdots\!59}{87\!\cdots\!69}a^{6}-\frac{35\!\cdots\!68}{87\!\cdots\!69}a^{5}-\frac{24\!\cdots\!65}{87\!\cdots\!69}a^{4}+\frac{20\!\cdots\!33}{87\!\cdots\!69}a^{3}+\frac{14\!\cdots\!29}{87\!\cdots\!69}a^{2}-\frac{80\!\cdots\!45}{87\!\cdots\!69}a-\frac{51\!\cdots\!20}{87\!\cdots\!69}$, $\frac{39\!\cdots\!54}{87\!\cdots\!69}a^{19}+\frac{51\!\cdots\!06}{87\!\cdots\!69}a^{18}+\frac{15\!\cdots\!05}{87\!\cdots\!69}a^{17}-\frac{59\!\cdots\!75}{87\!\cdots\!69}a^{16}-\frac{43\!\cdots\!51}{87\!\cdots\!69}a^{15}-\frac{58\!\cdots\!02}{87\!\cdots\!69}a^{14}-\frac{76\!\cdots\!38}{87\!\cdots\!69}a^{13}+\frac{76\!\cdots\!44}{87\!\cdots\!69}a^{12}+\frac{66\!\cdots\!87}{87\!\cdots\!69}a^{11}+\frac{10\!\cdots\!81}{87\!\cdots\!69}a^{10}-\frac{18\!\cdots\!23}{87\!\cdots\!69}a^{9}+\frac{22\!\cdots\!20}{87\!\cdots\!69}a^{8}+\frac{68\!\cdots\!24}{87\!\cdots\!69}a^{7}+\frac{15\!\cdots\!35}{87\!\cdots\!69}a^{6}+\frac{11\!\cdots\!10}{87\!\cdots\!69}a^{5}-\frac{30\!\cdots\!83}{87\!\cdots\!69}a^{4}-\frac{83\!\cdots\!93}{87\!\cdots\!69}a^{3}-\frac{40\!\cdots\!95}{87\!\cdots\!69}a^{2}-\frac{47\!\cdots\!17}{87\!\cdots\!69}a+\frac{69\!\cdots\!37}{87\!\cdots\!69}$, $\frac{70\!\cdots\!93}{87\!\cdots\!69}a^{19}-\frac{19\!\cdots\!38}{87\!\cdots\!69}a^{18}+\frac{20\!\cdots\!40}{87\!\cdots\!69}a^{17}-\frac{13\!\cdots\!95}{87\!\cdots\!69}a^{16}+\frac{18\!\cdots\!88}{87\!\cdots\!69}a^{15}-\frac{11\!\cdots\!36}{87\!\cdots\!69}a^{14}+\frac{20\!\cdots\!95}{87\!\cdots\!69}a^{13}+\frac{27\!\cdots\!53}{87\!\cdots\!69}a^{12}+\frac{77\!\cdots\!19}{87\!\cdots\!69}a^{11}+\frac{21\!\cdots\!50}{87\!\cdots\!69}a^{10}-\frac{18\!\cdots\!38}{87\!\cdots\!69}a^{9}+\frac{27\!\cdots\!94}{87\!\cdots\!69}a^{8}+\frac{10\!\cdots\!89}{87\!\cdots\!69}a^{7}+\frac{71\!\cdots\!87}{87\!\cdots\!69}a^{6}-\frac{67\!\cdots\!48}{87\!\cdots\!69}a^{5}-\frac{85\!\cdots\!14}{87\!\cdots\!69}a^{4}+\frac{34\!\cdots\!61}{87\!\cdots\!69}a^{3}+\frac{28\!\cdots\!68}{87\!\cdots\!69}a^{2}-\frac{37\!\cdots\!03}{87\!\cdots\!69}a+\frac{56\!\cdots\!01}{87\!\cdots\!69}$, $\frac{23\!\cdots\!74}{67\!\cdots\!69}a^{19}-\frac{15\!\cdots\!87}{67\!\cdots\!69}a^{18}+\frac{58\!\cdots\!10}{67\!\cdots\!69}a^{17}-\frac{48\!\cdots\!33}{67\!\cdots\!69}a^{16}+\frac{74\!\cdots\!58}{67\!\cdots\!69}a^{15}-\frac{38\!\cdots\!02}{67\!\cdots\!69}a^{14}+\frac{17\!\cdots\!35}{67\!\cdots\!69}a^{13}+\frac{11\!\cdots\!14}{67\!\cdots\!69}a^{12}+\frac{21\!\cdots\!34}{67\!\cdots\!69}a^{11}-\frac{55\!\cdots\!80}{67\!\cdots\!69}a^{10}-\frac{80\!\cdots\!78}{67\!\cdots\!69}a^{9}+\frac{11\!\cdots\!56}{67\!\cdots\!69}a^{8}+\frac{34\!\cdots\!53}{67\!\cdots\!69}a^{7}+\frac{62\!\cdots\!66}{67\!\cdots\!69}a^{6}-\frac{48\!\cdots\!40}{67\!\cdots\!69}a^{5}-\frac{35\!\cdots\!86}{67\!\cdots\!69}a^{4}+\frac{30\!\cdots\!00}{67\!\cdots\!69}a^{3}+\frac{20\!\cdots\!59}{67\!\cdots\!69}a^{2}-\frac{72\!\cdots\!79}{67\!\cdots\!69}a-\frac{87\!\cdots\!46}{67\!\cdots\!69}$ 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:  \( 7783721.02563 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 7783721.02563 \cdot 14}{2\cdot\sqrt{333025103817911062616373171479401}}\cr\approx \mathstrut & 0.286315921835 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 + 3*x^18 - 19*x^17 + 21*x^16 - 159*x^15 - 7*x^14 + 354*x^13 + 1233*x^12 + 655*x^11 - 2385*x^10 + 2997*x^9 + 15528*x^8 + 15281*x^7 - 4769*x^6 - 14351*x^5 - 1363*x^4 + 4718*x^3 + 3265*x^2 - 9*x + 77)
 
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 + 3*x^18 - 19*x^17 + 21*x^16 - 159*x^15 - 7*x^14 + 354*x^13 + 1233*x^12 + 655*x^11 - 2385*x^10 + 2997*x^9 + 15528*x^8 + 15281*x^7 - 4769*x^6 - 14351*x^5 - 1363*x^4 + 4718*x^3 + 3265*x^2 - 9*x + 77, 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 + 3*x^18 - 19*x^17 + 21*x^16 - 159*x^15 - 7*x^14 + 354*x^13 + 1233*x^12 + 655*x^11 - 2385*x^10 + 2997*x^9 + 15528*x^8 + 15281*x^7 - 4769*x^6 - 14351*x^5 - 1363*x^4 + 4718*x^3 + 3265*x^2 - 9*x + 77);
 
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 + 3*x^18 - 19*x^17 + 21*x^16 - 159*x^15 - 7*x^14 + 354*x^13 + 1233*x^12 + 655*x^11 - 2385*x^10 + 2997*x^9 + 15528*x^8 + 15281*x^7 - 4769*x^6 - 14351*x^5 - 1363*x^4 + 4718*x^3 + 3265*x^2 - 9*x + 77);
 
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

\(\Q(\sqrt{-19}) \), 10.10.960472390437121.1

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: 10.0.18248975418305299.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.10.0.1}{10} }^{2}$ ${\href{/padicField/3.10.0.1}{10} }^{2}$ ${\href{/padicField/5.5.0.1}{5} }^{4}$ ${\href{/padicField/7.3.0.1}{3} }^{6}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.3.0.1}{3} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.10.0.1}{10} }^{2}$ ${\href{/padicField/17.5.0.1}{5} }^{4}$ R ${\href{/padicField/23.3.0.1}{3} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.6.0.1}{6} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.10.0.1}{10} }^{2}$ ${\href{/padicField/43.3.0.1}{3} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.5.0.1}{5} }^{4}$ ${\href{/padicField/53.6.0.1}{6} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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
\(19\) Copy content Toggle raw display 19.2.1.2$x^{2} + 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(293\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$