Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 86*x^18 + 155*x^17 + 3456*x^16 - 5762*x^15 - 85725*x^14 + 132481*x^13 + 1467570*x^12 - 2030017*x^11 - 18368583*x^10 + 21318004*x^9 + 172316543*x^8 - 153736813*x^7 - 1206080098*x^6 + 717065378*x^5 + 6066158054*x^4 - 1732999488*x^3 - 19912550031*x^2 + 988623515*x + 33019779529)
gp: K = bnfinit(y^20 - 2*y^19 - 86*y^18 + 155*y^17 + 3456*y^16 - 5762*y^15 - 85725*y^14 + 132481*y^13 + 1467570*y^12 - 2030017*y^11 - 18368583*y^10 + 21318004*y^9 + 172316543*y^8 - 153736813*y^7 - 1206080098*y^6 + 717065378*y^5 + 6066158054*y^4 - 1732999488*y^3 - 19912550031*y^2 + 988623515*y + 33019779529, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 - 86*x^18 + 155*x^17 + 3456*x^16 - 5762*x^15 - 85725*x^14 + 132481*x^13 + 1467570*x^12 - 2030017*x^11 - 18368583*x^10 + 21318004*x^9 + 172316543*x^8 - 153736813*x^7 - 1206080098*x^6 + 717065378*x^5 + 6066158054*x^4 - 1732999488*x^3 - 19912550031*x^2 + 988623515*x + 33019779529);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 - 86*x^18 + 155*x^17 + 3456*x^16 - 5762*x^15 - 85725*x^14 + 132481*x^13 + 1467570*x^12 - 2030017*x^11 - 18368583*x^10 + 21318004*x^9 + 172316543*x^8 - 153736813*x^7 - 1206080098*x^6 + 717065378*x^5 + 6066158054*x^4 - 1732999488*x^3 - 19912550031*x^2 + 988623515*x + 33019779529)
\( x^{20} - 2 x^{19} - 86 x^{18} + 155 x^{17} + 3456 x^{16} - 5762 x^{15} - 85725 x^{14} + \cdots + 33019779529 \)
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: |
\(1600573236263365245152401689453125\)
\(\medspace = 5^{10}\cdot 7^{15}\cdot 11^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(45.73\) | 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: | $5^{1/2}7^{3/4}11^{9/10}\approx 83.2840715937989$ | ||
| Ramified primes: |
\(5\), \(7\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{77}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $10$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{79}a^{18}-\frac{32}{79}a^{17}-\frac{17}{79}a^{16}+\frac{26}{79}a^{15}-\frac{31}{79}a^{14}-\frac{32}{79}a^{13}-\frac{27}{79}a^{12}+\frac{11}{79}a^{11}+\frac{14}{79}a^{10}+\frac{16}{79}a^{9}+\frac{25}{79}a^{8}+\frac{16}{79}a^{7}+\frac{2}{79}a^{6}+\frac{14}{79}a^{5}+\frac{35}{79}a^{4}-\frac{20}{79}a^{3}-\frac{27}{79}a^{2}+\frac{35}{79}a-\frac{8}{79}$, $\frac{1}{32\!\cdots\!77}a^{19}-\frac{27\!\cdots\!08}{32\!\cdots\!77}a^{18}+\frac{28\!\cdots\!78}{32\!\cdots\!77}a^{17}+\frac{69\!\cdots\!89}{32\!\cdots\!77}a^{16}+\frac{50\!\cdots\!93}{32\!\cdots\!77}a^{15}+\frac{14\!\cdots\!28}{32\!\cdots\!77}a^{14}-\frac{11\!\cdots\!56}{32\!\cdots\!77}a^{13}+\frac{12\!\cdots\!96}{32\!\cdots\!77}a^{12}+\frac{98\!\cdots\!76}{32\!\cdots\!77}a^{11}-\frac{88\!\cdots\!15}{32\!\cdots\!77}a^{10}-\frac{42\!\cdots\!27}{32\!\cdots\!77}a^{9}-\frac{65\!\cdots\!08}{32\!\cdots\!77}a^{8}+\frac{19\!\cdots\!99}{32\!\cdots\!77}a^{7}+\frac{60\!\cdots\!84}{86\!\cdots\!49}a^{6}+\frac{79\!\cdots\!87}{32\!\cdots\!77}a^{5}-\frac{59\!\cdots\!71}{32\!\cdots\!77}a^{4}-\frac{12\!\cdots\!47}{32\!\cdots\!77}a^{3}+\frac{85\!\cdots\!01}{32\!\cdots\!77}a^{2}-\frac{11\!\cdots\!54}{32\!\cdots\!77}a+\frac{86\!\cdots\!66}{32\!\cdots\!77}$
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
$C_{2}$, which has order $2$ (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 $2$)
| 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: |
$\frac{28\!\cdots\!44}{32\!\cdots\!77}a^{19}-\frac{28\!\cdots\!29}{32\!\cdots\!77}a^{18}-\frac{14\!\cdots\!89}{32\!\cdots\!77}a^{17}+\frac{20\!\cdots\!56}{32\!\cdots\!77}a^{16}+\frac{31\!\cdots\!74}{32\!\cdots\!77}a^{15}-\frac{68\!\cdots\!52}{32\!\cdots\!77}a^{14}-\frac{24\!\cdots\!98}{32\!\cdots\!77}a^{13}+\frac{14\!\cdots\!23}{32\!\cdots\!77}a^{12}-\frac{26\!\cdots\!26}{32\!\cdots\!77}a^{11}-\frac{20\!\cdots\!83}{32\!\cdots\!77}a^{10}+\frac{82\!\cdots\!80}{32\!\cdots\!77}a^{9}+\frac{21\!\cdots\!16}{32\!\cdots\!77}a^{8}-\frac{95\!\cdots\!54}{32\!\cdots\!77}a^{7}-\frac{45\!\cdots\!32}{86\!\cdots\!49}a^{6}+\frac{62\!\cdots\!71}{32\!\cdots\!77}a^{5}+\frac{95\!\cdots\!08}{32\!\cdots\!77}a^{4}-\frac{18\!\cdots\!50}{32\!\cdots\!77}a^{3}-\frac{35\!\cdots\!29}{32\!\cdots\!77}a^{2}+\frac{12\!\cdots\!26}{32\!\cdots\!77}a+\frac{70\!\cdots\!48}{32\!\cdots\!77}$, $\frac{19\!\cdots\!30}{32\!\cdots\!77}a^{19}-\frac{82\!\cdots\!38}{32\!\cdots\!77}a^{18}-\frac{12\!\cdots\!68}{32\!\cdots\!77}a^{17}+\frac{63\!\cdots\!22}{32\!\cdots\!77}a^{16}+\frac{38\!\cdots\!94}{32\!\cdots\!77}a^{15}-\frac{22\!\cdots\!00}{32\!\cdots\!77}a^{14}-\frac{67\!\cdots\!04}{32\!\cdots\!77}a^{13}+\frac{50\!\cdots\!37}{32\!\cdots\!77}a^{12}+\frac{72\!\cdots\!08}{32\!\cdots\!77}a^{11}-\frac{75\!\cdots\!95}{32\!\cdots\!77}a^{10}-\frac{48\!\cdots\!46}{32\!\cdots\!77}a^{9}+\frac{82\!\cdots\!39}{32\!\cdots\!77}a^{8}+\frac{19\!\cdots\!52}{32\!\cdots\!77}a^{7}-\frac{17\!\cdots\!42}{86\!\cdots\!49}a^{6}-\frac{74\!\cdots\!97}{32\!\cdots\!77}a^{5}+\frac{40\!\cdots\!93}{32\!\cdots\!77}a^{4}-\frac{24\!\cdots\!19}{32\!\cdots\!77}a^{3}-\frac{16\!\cdots\!62}{32\!\cdots\!77}a^{2}-\frac{35\!\cdots\!50}{32\!\cdots\!77}a+\frac{41\!\cdots\!52}{32\!\cdots\!77}$, $\frac{85\!\cdots\!10}{32\!\cdots\!77}a^{19}+\frac{11\!\cdots\!14}{32\!\cdots\!77}a^{18}-\frac{87\!\cdots\!38}{32\!\cdots\!77}a^{17}-\frac{89\!\cdots\!66}{32\!\cdots\!77}a^{16}+\frac{38\!\cdots\!00}{32\!\cdots\!77}a^{15}+\frac{33\!\cdots\!65}{32\!\cdots\!77}a^{14}-\frac{10\!\cdots\!31}{32\!\cdots\!77}a^{13}-\frac{75\!\cdots\!88}{32\!\cdots\!77}a^{12}+\frac{18\!\cdots\!87}{32\!\cdots\!77}a^{11}+\frac{11\!\cdots\!34}{32\!\cdots\!77}a^{10}-\frac{22\!\cdots\!07}{32\!\cdots\!77}a^{9}-\frac{13\!\cdots\!38}{32\!\cdots\!77}a^{8}+\frac{20\!\cdots\!59}{32\!\cdots\!77}a^{7}+\frac{30\!\cdots\!24}{86\!\cdots\!49}a^{6}-\frac{12\!\cdots\!37}{32\!\cdots\!77}a^{5}-\frac{67\!\cdots\!73}{32\!\cdots\!77}a^{4}+\frac{44\!\cdots\!49}{32\!\cdots\!77}a^{3}+\frac{26\!\cdots\!26}{32\!\cdots\!77}a^{2}-\frac{60\!\cdots\!02}{32\!\cdots\!77}a-\frac{56\!\cdots\!39}{32\!\cdots\!77}$, $\frac{65\!\cdots\!22}{32\!\cdots\!77}a^{19}-\frac{35\!\cdots\!73}{32\!\cdots\!77}a^{18}-\frac{44\!\cdots\!31}{32\!\cdots\!77}a^{17}+\frac{24\!\cdots\!37}{32\!\cdots\!77}a^{16}+\frac{14\!\cdots\!64}{32\!\cdots\!77}a^{15}-\frac{80\!\cdots\!15}{32\!\cdots\!77}a^{14}-\frac{27\!\cdots\!99}{32\!\cdots\!77}a^{13}+\frac{16\!\cdots\!87}{32\!\cdots\!77}a^{12}+\frac{37\!\cdots\!07}{32\!\cdots\!77}a^{11}-\frac{22\!\cdots\!20}{32\!\cdots\!77}a^{10}-\frac{38\!\cdots\!34}{32\!\cdots\!77}a^{9}+\frac{21\!\cdots\!38}{32\!\cdots\!77}a^{8}+\frac{30\!\cdots\!09}{32\!\cdots\!77}a^{7}-\frac{40\!\cdots\!63}{86\!\cdots\!49}a^{6}-\frac{18\!\cdots\!59}{32\!\cdots\!77}a^{5}+\frac{74\!\cdots\!59}{32\!\cdots\!77}a^{4}+\frac{80\!\cdots\!59}{32\!\cdots\!77}a^{3}-\frac{23\!\cdots\!08}{32\!\cdots\!77}a^{2}-\frac{19\!\cdots\!10}{32\!\cdots\!77}a+\frac{40\!\cdots\!10}{32\!\cdots\!77}$, $\frac{56\!\cdots\!09}{32\!\cdots\!77}a^{19}-\frac{87\!\cdots\!50}{32\!\cdots\!77}a^{18}-\frac{48\!\cdots\!94}{32\!\cdots\!77}a^{17}+\frac{68\!\cdots\!33}{32\!\cdots\!77}a^{16}+\frac{18\!\cdots\!98}{32\!\cdots\!77}a^{15}-\frac{26\!\cdots\!50}{32\!\cdots\!77}a^{14}-\frac{42\!\cdots\!25}{32\!\cdots\!77}a^{13}+\frac{63\!\cdots\!67}{32\!\cdots\!77}a^{12}+\frac{62\!\cdots\!16}{32\!\cdots\!77}a^{11}-\frac{10\!\cdots\!53}{32\!\cdots\!77}a^{10}-\frac{64\!\cdots\!92}{32\!\cdots\!77}a^{9}+\frac{11\!\cdots\!22}{32\!\cdots\!77}a^{8}+\frac{45\!\cdots\!59}{32\!\cdots\!77}a^{7}-\frac{25\!\cdots\!40}{86\!\cdots\!49}a^{6}-\frac{22\!\cdots\!32}{32\!\cdots\!77}a^{5}+\frac{54\!\cdots\!10}{32\!\cdots\!77}a^{4}+\frac{65\!\cdots\!00}{32\!\cdots\!77}a^{3}-\frac{20\!\cdots\!02}{32\!\cdots\!77}a^{2}-\frac{91\!\cdots\!88}{32\!\cdots\!77}a+\frac{36\!\cdots\!59}{32\!\cdots\!77}$, $\frac{82\!\cdots\!67}{32\!\cdots\!77}a^{19}-\frac{14\!\cdots\!56}{32\!\cdots\!77}a^{18}-\frac{73\!\cdots\!93}{32\!\cdots\!77}a^{17}+\frac{12\!\cdots\!63}{32\!\cdots\!77}a^{16}+\frac{29\!\cdots\!68}{32\!\cdots\!77}a^{15}-\frac{47\!\cdots\!24}{32\!\cdots\!77}a^{14}-\frac{72\!\cdots\!69}{32\!\cdots\!77}a^{13}+\frac{10\!\cdots\!38}{32\!\cdots\!77}a^{12}+\frac{11\!\cdots\!39}{32\!\cdots\!77}a^{11}-\frac{16\!\cdots\!27}{32\!\cdots\!77}a^{10}-\frac{13\!\cdots\!71}{32\!\cdots\!77}a^{9}+\frac{15\!\cdots\!12}{32\!\cdots\!77}a^{8}+\frac{11\!\cdots\!02}{32\!\cdots\!77}a^{7}-\frac{22\!\cdots\!01}{86\!\cdots\!49}a^{6}-\frac{68\!\cdots\!91}{32\!\cdots\!77}a^{5}+\frac{16\!\cdots\!70}{32\!\cdots\!77}a^{4}+\frac{26\!\cdots\!50}{32\!\cdots\!77}a^{3}+\frac{12\!\cdots\!18}{32\!\cdots\!77}a^{2}-\frac{55\!\cdots\!00}{32\!\cdots\!77}a-\frac{77\!\cdots\!50}{32\!\cdots\!77}$, $\frac{64\!\cdots\!44}{32\!\cdots\!77}a^{19}-\frac{11\!\cdots\!87}{32\!\cdots\!77}a^{18}-\frac{82\!\cdots\!90}{32\!\cdots\!77}a^{17}+\frac{81\!\cdots\!86}{32\!\cdots\!77}a^{16}-\frac{96\!\cdots\!70}{32\!\cdots\!77}a^{15}-\frac{27\!\cdots\!50}{32\!\cdots\!77}a^{14}+\frac{47\!\cdots\!17}{32\!\cdots\!77}a^{13}+\frac{58\!\cdots\!30}{32\!\cdots\!77}a^{12}-\frac{11\!\cdots\!60}{32\!\cdots\!77}a^{11}-\frac{85\!\cdots\!31}{32\!\cdots\!77}a^{10}+\frac{15\!\cdots\!98}{32\!\cdots\!77}a^{9}+\frac{93\!\cdots\!17}{32\!\cdots\!77}a^{8}-\frac{14\!\cdots\!73}{32\!\cdots\!77}a^{7}-\frac{20\!\cdots\!70}{86\!\cdots\!49}a^{6}+\frac{95\!\cdots\!73}{32\!\cdots\!77}a^{5}+\frac{46\!\cdots\!13}{32\!\cdots\!77}a^{4}-\frac{37\!\cdots\!94}{32\!\cdots\!77}a^{3}-\frac{19\!\cdots\!82}{32\!\cdots\!77}a^{2}+\frac{63\!\cdots\!10}{32\!\cdots\!77}a+\frac{43\!\cdots\!67}{32\!\cdots\!77}$, $\frac{43\!\cdots\!40}{32\!\cdots\!77}a^{19}-\frac{27\!\cdots\!31}{32\!\cdots\!77}a^{18}-\frac{26\!\cdots\!46}{32\!\cdots\!77}a^{17}+\frac{19\!\cdots\!64}{32\!\cdots\!77}a^{16}+\frac{78\!\cdots\!96}{32\!\cdots\!77}a^{15}-\frac{64\!\cdots\!01}{32\!\cdots\!77}a^{14}-\frac{13\!\cdots\!08}{32\!\cdots\!77}a^{13}+\frac{13\!\cdots\!85}{32\!\cdots\!77}a^{12}+\frac{16\!\cdots\!83}{32\!\cdots\!77}a^{11}-\frac{18\!\cdots\!45}{32\!\cdots\!77}a^{10}-\frac{14\!\cdots\!19}{32\!\cdots\!77}a^{9}+\frac{18\!\cdots\!03}{32\!\cdots\!77}a^{8}+\frac{10\!\cdots\!06}{32\!\cdots\!77}a^{7}-\frac{37\!\cdots\!75}{86\!\cdots\!49}a^{6}-\frac{62\!\cdots\!00}{32\!\cdots\!77}a^{5}+\frac{72\!\cdots\!48}{32\!\cdots\!77}a^{4}+\frac{31\!\cdots\!25}{32\!\cdots\!77}a^{3}-\frac{24\!\cdots\!61}{32\!\cdots\!77}a^{2}-\frac{11\!\cdots\!53}{41\!\cdots\!63}a+\frac{45\!\cdots\!30}{32\!\cdots\!77}$, $\frac{66\!\cdots\!03}{32\!\cdots\!77}a^{19}-\frac{35\!\cdots\!30}{32\!\cdots\!77}a^{18}-\frac{45\!\cdots\!04}{32\!\cdots\!77}a^{17}+\frac{25\!\cdots\!24}{32\!\cdots\!77}a^{16}+\frac{14\!\cdots\!87}{32\!\cdots\!77}a^{15}-\frac{86\!\cdots\!02}{32\!\cdots\!77}a^{14}-\frac{27\!\cdots\!80}{32\!\cdots\!77}a^{13}+\frac{18\!\cdots\!13}{32\!\cdots\!77}a^{12}+\frac{36\!\cdots\!35}{32\!\cdots\!77}a^{11}-\frac{25\!\cdots\!91}{32\!\cdots\!77}a^{10}-\frac{45\!\cdots\!88}{41\!\cdots\!63}a^{9}+\frac{26\!\cdots\!49}{32\!\cdots\!77}a^{8}+\frac{27\!\cdots\!66}{32\!\cdots\!77}a^{7}-\frac{51\!\cdots\!64}{86\!\cdots\!49}a^{6}-\frac{16\!\cdots\!15}{32\!\cdots\!77}a^{5}+\frac{99\!\cdots\!63}{32\!\cdots\!77}a^{4}+\frac{70\!\cdots\!83}{32\!\cdots\!77}a^{3}-\frac{34\!\cdots\!12}{32\!\cdots\!77}a^{2}-\frac{18\!\cdots\!01}{32\!\cdots\!77}a+\frac{66\!\cdots\!97}{32\!\cdots\!77}$
| 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: | \( 76205249.56986351 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 76205249.56986351 \cdot 2}{2\cdot\sqrt{1600573236263365245152401689453125}}\cr\approx \mathstrut & 0.182660986150062 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 86*x^18 + 155*x^17 + 3456*x^16 - 5762*x^15 - 85725*x^14 + 132481*x^13 + 1467570*x^12 - 2030017*x^11 - 18368583*x^10 + 21318004*x^9 + 172316543*x^8 - 153736813*x^7 - 1206080098*x^6 + 717065378*x^5 + 6066158054*x^4 - 1732999488*x^3 - 19912550031*x^2 + 988623515*x + 33019779529)
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 - 2*x^19 - 86*x^18 + 155*x^17 + 3456*x^16 - 5762*x^15 - 85725*x^14 + 132481*x^13 + 1467570*x^12 - 2030017*x^11 - 18368583*x^10 + 21318004*x^9 + 172316543*x^8 - 153736813*x^7 - 1206080098*x^6 + 717065378*x^5 + 6066158054*x^4 - 1732999488*x^3 - 19912550031*x^2 + 988623515*x + 33019779529, 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 - 2*x^19 - 86*x^18 + 155*x^17 + 3456*x^16 - 5762*x^15 - 85725*x^14 + 132481*x^13 + 1467570*x^12 - 2030017*x^11 - 18368583*x^10 + 21318004*x^9 + 172316543*x^8 - 153736813*x^7 - 1206080098*x^6 + 717065378*x^5 + 6066158054*x^4 - 1732999488*x^3 - 19912550031*x^2 + 988623515*x + 33019779529);
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 - 2*x^19 - 86*x^18 + 155*x^17 + 3456*x^16 - 5762*x^15 - 85725*x^14 + 132481*x^13 + 1467570*x^12 - 2030017*x^11 - 18368583*x^10 + 21318004*x^9 + 172316543*x^8 - 153736813*x^7 - 1206080098*x^6 + 717065378*x^5 + 6066158054*x^4 - 1732999488*x^3 - 19912550031*x^2 + 988623515*x + 33019779529);
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
$C_{10}\wr C_2$ (as 20T53):
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 solvable group of order 200 |
| The 65 conjugacy class representatives for $C_{10}\wr C_2$ |
| Character table for $C_{10}\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-7}) \), 4.0.94325.1, 10.0.246071287.1 |
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 20 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 20.0.109321305666509476480595703125.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.5.0.1}{5} }^{2}$ | $20$ | R | R | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{4}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }^{2}$ | $20$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{5}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}$ | $20$ | ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{10}$ | $20$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(5\)
| Deg $20$ | $2$ | $10$ | $10$ | |||
|
\(7\)
| 7.20.15.1 | $x^{20} + 39 x^{16} + 16 x^{15} - 344 x^{12} - 7232 x^{11} + 96 x^{10} + 4344 x^{8} + 131648 x^{7} + 38272 x^{6} + 256 x^{5} + 59536 x^{4} - 212224 x^{3} + 84096 x^{2} - 8704 x + 9328$ | $4$ | $5$ | $15$ | 20T12 | $[\ ]_{4}^{10}$ |
|
\(11\)
| 11.5.4.3 | $x^{5} + 33$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.5.4.3 | $x^{5} + 33$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.10.5.2 | $x^{10} + 55 x^{8} + 20 x^{7} + 1210 x^{6} - 202 x^{5} + 13410 x^{4} - 14080 x^{3} + 75585 x^{2} - 68970 x + 171252$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |