Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 11*x^18 - 21*x^16 + 13*x^15 + 55*x^14 - 217*x^13 + 434*x^12 - 360*x^11 - 205*x^10 + 805*x^9 - 845*x^8 - 70*x^7 + 2180*x^6 - 4240*x^5 + 4450*x^4 - 2890*x^3 + 1155*x^2 - 250*x + 25)
gp: K = bnfinit(y^20 - 6*y^19 + 11*y^18 - 21*y^16 + 13*y^15 + 55*y^14 - 217*y^13 + 434*y^12 - 360*y^11 - 205*y^10 + 805*y^9 - 845*y^8 - 70*y^7 + 2180*y^6 - 4240*y^5 + 4450*y^4 - 2890*y^3 + 1155*y^2 - 250*y + 25, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 6*x^19 + 11*x^18 - 21*x^16 + 13*x^15 + 55*x^14 - 217*x^13 + 434*x^12 - 360*x^11 - 205*x^10 + 805*x^9 - 845*x^8 - 70*x^7 + 2180*x^6 - 4240*x^5 + 4450*x^4 - 2890*x^3 + 1155*x^2 - 250*x + 25);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 6*x^19 + 11*x^18 - 21*x^16 + 13*x^15 + 55*x^14 - 217*x^13 + 434*x^12 - 360*x^11 - 205*x^10 + 805*x^9 - 845*x^8 - 70*x^7 + 2180*x^6 - 4240*x^5 + 4450*x^4 - 2890*x^3 + 1155*x^2 - 250*x + 25)
\( x^{20} - 6 x^{19} + 11 x^{18} - 21 x^{16} + 13 x^{15} + 55 x^{14} - 217 x^{13} + 434 x^{12} - 360 x^{11} + \cdots + 25 \)
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: |
\(61839639189243316650390625\)
\(\medspace = 5^{22}\cdot 11^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.48\) | 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: | $5^{17/10}11^{1/2}\approx 51.16174513964836$ | ||
| Ramified primes: |
\(5\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $10$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{35}a^{15}-\frac{2}{35}a^{14}+\frac{3}{35}a^{13}-\frac{11}{35}a^{12}-\frac{2}{5}a^{11}+\frac{13}{35}a^{10}+\frac{3}{7}a^{7}+\frac{2}{7}a^{6}+\frac{3}{7}a^{5}-\frac{1}{7}a^{4}-\frac{1}{7}a^{3}-\frac{1}{7}a^{2}-\frac{2}{7}a+\frac{1}{7}$, $\frac{1}{35}a^{16}-\frac{1}{35}a^{14}-\frac{1}{7}a^{13}-\frac{1}{35}a^{12}-\frac{3}{7}a^{11}-\frac{9}{35}a^{10}+\frac{3}{7}a^{8}+\frac{1}{7}a^{7}-\frac{2}{7}a^{5}-\frac{3}{7}a^{4}-\frac{3}{7}a^{3}+\frac{3}{7}a^{2}-\frac{3}{7}a+\frac{2}{7}$, $\frac{1}{35}a^{17}-\frac{1}{5}a^{14}+\frac{2}{35}a^{13}+\frac{9}{35}a^{12}+\frac{12}{35}a^{11}+\frac{13}{35}a^{10}+\frac{3}{7}a^{9}+\frac{1}{7}a^{8}+\frac{3}{7}a^{7}+\frac{3}{7}a^{4}+\frac{2}{7}a^{3}+\frac{3}{7}a^{2}+\frac{1}{7}$, $\frac{1}{35}a^{18}-\frac{12}{35}a^{14}-\frac{1}{7}a^{13}+\frac{1}{7}a^{12}-\frac{3}{7}a^{11}+\frac{1}{35}a^{10}+\frac{1}{7}a^{9}+\frac{3}{7}a^{8}+\frac{3}{7}a^{5}+\frac{2}{7}a^{4}+\frac{3}{7}a^{3}+\frac{1}{7}a$, $\frac{1}{12\!\cdots\!95}a^{19}-\frac{32\!\cdots\!32}{24\!\cdots\!99}a^{18}+\frac{16\!\cdots\!37}{12\!\cdots\!95}a^{17}-\frac{13\!\cdots\!24}{12\!\cdots\!95}a^{16}-\frac{26\!\cdots\!25}{24\!\cdots\!99}a^{15}+\frac{22\!\cdots\!36}{12\!\cdots\!95}a^{14}-\frac{13\!\cdots\!72}{34\!\cdots\!57}a^{13}-\frac{20\!\cdots\!59}{34\!\cdots\!57}a^{12}+\frac{84\!\cdots\!27}{12\!\cdots\!95}a^{11}-\frac{48\!\cdots\!87}{12\!\cdots\!95}a^{10}-\frac{96\!\cdots\!87}{24\!\cdots\!99}a^{9}+\frac{41\!\cdots\!04}{41\!\cdots\!61}a^{8}-\frac{10\!\cdots\!01}{24\!\cdots\!99}a^{7}-\frac{10\!\cdots\!51}{24\!\cdots\!99}a^{6}-\frac{79\!\cdots\!77}{24\!\cdots\!99}a^{5}-\frac{70\!\cdots\!92}{24\!\cdots\!99}a^{4}-\frac{97\!\cdots\!08}{24\!\cdots\!99}a^{3}+\frac{87\!\cdots\!28}{34\!\cdots\!57}a^{2}-\frac{10\!\cdots\!54}{24\!\cdots\!99}a+\frac{84\!\cdots\!23}{24\!\cdots\!99}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $5$ |
Class group and class number
$C_{2}$, which has order $2$
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: |
$\frac{5118495079342}{83209661083619}a^{19}-\frac{19354817203284}{59435472202585}a^{18}+\frac{26191158045098}{59435472202585}a^{17}+\frac{140146938222694}{416048305418095}a^{16}-\frac{445226215516113}{416048305418095}a^{15}-\frac{2415385290559}{416048305418095}a^{14}+\frac{14\!\cdots\!23}{416048305418095}a^{13}-\frac{45\!\cdots\!02}{416048305418095}a^{12}+\frac{78\!\cdots\!59}{416048305418095}a^{11}-\frac{677460684357262}{83209661083619}a^{10}-\frac{16\!\cdots\!82}{83209661083619}a^{9}+\frac{29\!\cdots\!30}{83209661083619}a^{8}-\frac{301710533637717}{11887094440517}a^{7}-\frac{19\!\cdots\!18}{83209661083619}a^{6}+\frac{98\!\cdots\!05}{83209661083619}a^{5}-\frac{14\!\cdots\!21}{83209661083619}a^{4}+\frac{11\!\cdots\!67}{83209661083619}a^{3}-\frac{57\!\cdots\!38}{83209661083619}a^{2}+\frac{201036802755843}{11887094440517}a-\frac{23297836206048}{83209661083619}$, $\frac{97\!\cdots\!44}{12\!\cdots\!95}a^{19}-\frac{54\!\cdots\!01}{12\!\cdots\!95}a^{18}+\frac{86\!\cdots\!49}{12\!\cdots\!95}a^{17}+\frac{27\!\cdots\!24}{12\!\cdots\!95}a^{16}-\frac{37\!\cdots\!63}{24\!\cdots\!99}a^{15}+\frac{65\!\cdots\!54}{12\!\cdots\!95}a^{14}+\frac{54\!\cdots\!57}{12\!\cdots\!95}a^{13}-\frac{19\!\cdots\!31}{12\!\cdots\!95}a^{12}+\frac{71\!\cdots\!54}{24\!\cdots\!99}a^{11}-\frac{23\!\cdots\!81}{12\!\cdots\!95}a^{10}-\frac{50\!\cdots\!32}{24\!\cdots\!99}a^{9}+\frac{23\!\cdots\!54}{41\!\cdots\!61}a^{8}-\frac{17\!\cdots\!40}{34\!\cdots\!57}a^{7}-\frac{49\!\cdots\!79}{24\!\cdots\!99}a^{6}+\frac{57\!\cdots\!39}{34\!\cdots\!57}a^{5}-\frac{68\!\cdots\!58}{24\!\cdots\!99}a^{4}+\frac{65\!\cdots\!52}{24\!\cdots\!99}a^{3}-\frac{37\!\cdots\!93}{24\!\cdots\!99}a^{2}+\frac{11\!\cdots\!65}{24\!\cdots\!99}a-\frac{15\!\cdots\!33}{24\!\cdots\!99}$, $\frac{12\!\cdots\!01}{24\!\cdots\!99}a^{19}-\frac{66\!\cdots\!13}{24\!\cdots\!99}a^{18}+\frac{92\!\cdots\!35}{24\!\cdots\!99}a^{17}+\frac{66\!\cdots\!66}{24\!\cdots\!99}a^{16}-\frac{32\!\cdots\!49}{34\!\cdots\!57}a^{15}+\frac{87\!\cdots\!65}{24\!\cdots\!99}a^{14}+\frac{71\!\cdots\!00}{24\!\cdots\!99}a^{13}-\frac{22\!\cdots\!16}{24\!\cdots\!99}a^{12}+\frac{38\!\cdots\!49}{24\!\cdots\!99}a^{11}-\frac{17\!\cdots\!20}{24\!\cdots\!99}a^{10}-\frac{41\!\cdots\!23}{24\!\cdots\!99}a^{9}+\frac{12\!\cdots\!84}{41\!\cdots\!61}a^{8}-\frac{51\!\cdots\!89}{24\!\cdots\!99}a^{7}-\frac{51\!\cdots\!92}{24\!\cdots\!99}a^{6}+\frac{24\!\cdots\!78}{24\!\cdots\!99}a^{5}-\frac{36\!\cdots\!37}{24\!\cdots\!99}a^{4}+\frac{28\!\cdots\!39}{24\!\cdots\!99}a^{3}-\frac{12\!\cdots\!89}{24\!\cdots\!99}a^{2}+\frac{17\!\cdots\!10}{24\!\cdots\!99}a+\frac{46\!\cdots\!39}{24\!\cdots\!99}$, $\frac{10\!\cdots\!62}{12\!\cdots\!95}a^{19}-\frac{61\!\cdots\!38}{12\!\cdots\!95}a^{18}+\frac{14\!\cdots\!58}{17\!\cdots\!85}a^{17}+\frac{59\!\cdots\!71}{24\!\cdots\!99}a^{16}-\frac{22\!\cdots\!88}{12\!\cdots\!95}a^{15}+\frac{76\!\cdots\!37}{12\!\cdots\!95}a^{14}+\frac{12\!\cdots\!71}{24\!\cdots\!99}a^{13}-\frac{21\!\cdots\!52}{12\!\cdots\!95}a^{12}+\frac{80\!\cdots\!20}{24\!\cdots\!99}a^{11}-\frac{38\!\cdots\!01}{17\!\cdots\!85}a^{10}-\frac{62\!\cdots\!96}{24\!\cdots\!99}a^{9}+\frac{26\!\cdots\!83}{41\!\cdots\!61}a^{8}-\frac{13\!\cdots\!79}{24\!\cdots\!99}a^{7}-\frac{57\!\cdots\!71}{24\!\cdots\!99}a^{6}+\frac{45\!\cdots\!57}{24\!\cdots\!99}a^{5}-\frac{78\!\cdots\!94}{24\!\cdots\!99}a^{4}+\frac{71\!\cdots\!58}{24\!\cdots\!99}a^{3}-\frac{38\!\cdots\!66}{24\!\cdots\!99}a^{2}+\frac{11\!\cdots\!75}{24\!\cdots\!99}a-\frac{17\!\cdots\!27}{34\!\cdots\!57}$, $\frac{39\!\cdots\!69}{12\!\cdots\!95}a^{19}-\frac{22\!\cdots\!56}{12\!\cdots\!95}a^{18}+\frac{10\!\cdots\!56}{34\!\cdots\!57}a^{17}+\frac{11\!\cdots\!91}{12\!\cdots\!95}a^{16}-\frac{79\!\cdots\!47}{12\!\cdots\!95}a^{15}+\frac{37\!\cdots\!57}{17\!\cdots\!85}a^{14}+\frac{22\!\cdots\!83}{12\!\cdots\!95}a^{13}-\frac{78\!\cdots\!57}{12\!\cdots\!95}a^{12}+\frac{29\!\cdots\!62}{24\!\cdots\!99}a^{11}-\frac{97\!\cdots\!12}{12\!\cdots\!95}a^{10}-\frac{31\!\cdots\!93}{34\!\cdots\!57}a^{9}+\frac{95\!\cdots\!53}{41\!\cdots\!61}a^{8}-\frac{49\!\cdots\!69}{24\!\cdots\!99}a^{7}-\frac{28\!\cdots\!77}{34\!\cdots\!57}a^{6}+\frac{16\!\cdots\!07}{24\!\cdots\!99}a^{5}-\frac{28\!\cdots\!91}{24\!\cdots\!99}a^{4}+\frac{26\!\cdots\!86}{24\!\cdots\!99}a^{3}-\frac{14\!\cdots\!52}{24\!\cdots\!99}a^{2}+\frac{44\!\cdots\!50}{24\!\cdots\!99}a-\frac{84\!\cdots\!01}{34\!\cdots\!57}$, $\frac{68\!\cdots\!21}{12\!\cdots\!95}a^{19}-\frac{36\!\cdots\!03}{12\!\cdots\!95}a^{18}+\frac{13\!\cdots\!62}{24\!\cdots\!99}a^{17}-\frac{33\!\cdots\!14}{12\!\cdots\!95}a^{16}-\frac{86\!\cdots\!98}{12\!\cdots\!95}a^{15}+\frac{23\!\cdots\!01}{17\!\cdots\!85}a^{14}+\frac{36\!\cdots\!81}{17\!\cdots\!85}a^{13}-\frac{28\!\cdots\!79}{24\!\cdots\!99}a^{12}+\frac{57\!\cdots\!86}{24\!\cdots\!99}a^{11}-\frac{85\!\cdots\!76}{34\!\cdots\!57}a^{10}+\frac{97\!\cdots\!48}{34\!\cdots\!57}a^{9}+\frac{18\!\cdots\!52}{41\!\cdots\!61}a^{8}-\frac{15\!\cdots\!37}{24\!\cdots\!99}a^{7}+\frac{33\!\cdots\!92}{34\!\cdots\!57}a^{6}+\frac{25\!\cdots\!24}{24\!\cdots\!99}a^{5}-\frac{57\!\cdots\!31}{24\!\cdots\!99}a^{4}+\frac{77\!\cdots\!07}{24\!\cdots\!99}a^{3}-\frac{55\!\cdots\!60}{24\!\cdots\!99}a^{2}+\frac{20\!\cdots\!42}{24\!\cdots\!99}a-\frac{25\!\cdots\!58}{24\!\cdots\!99}$, $\frac{12\!\cdots\!33}{24\!\cdots\!99}a^{19}-\frac{36\!\cdots\!46}{12\!\cdots\!95}a^{18}+\frac{60\!\cdots\!46}{12\!\cdots\!95}a^{17}+\frac{17\!\cdots\!73}{12\!\cdots\!95}a^{16}-\frac{12\!\cdots\!57}{12\!\cdots\!95}a^{15}+\frac{41\!\cdots\!46}{12\!\cdots\!95}a^{14}+\frac{36\!\cdots\!56}{12\!\cdots\!95}a^{13}-\frac{12\!\cdots\!72}{12\!\cdots\!95}a^{12}+\frac{48\!\cdots\!55}{24\!\cdots\!99}a^{11}-\frac{16\!\cdots\!91}{12\!\cdots\!95}a^{10}-\frac{35\!\cdots\!69}{24\!\cdots\!99}a^{9}+\frac{15\!\cdots\!04}{41\!\cdots\!61}a^{8}-\frac{81\!\cdots\!90}{24\!\cdots\!99}a^{7}-\frac{29\!\cdots\!98}{24\!\cdots\!99}a^{6}+\frac{26\!\cdots\!22}{24\!\cdots\!99}a^{5}-\frac{46\!\cdots\!23}{24\!\cdots\!99}a^{4}+\frac{43\!\cdots\!74}{24\!\cdots\!99}a^{3}-\frac{25\!\cdots\!43}{24\!\cdots\!99}a^{2}+\frac{87\!\cdots\!67}{24\!\cdots\!99}a-\frac{13\!\cdots\!02}{24\!\cdots\!99}$, $\frac{69\!\cdots\!12}{12\!\cdots\!95}a^{19}-\frac{38\!\cdots\!79}{12\!\cdots\!95}a^{18}+\frac{58\!\cdots\!88}{12\!\cdots\!95}a^{17}+\frac{56\!\cdots\!23}{24\!\cdots\!99}a^{16}-\frac{13\!\cdots\!57}{12\!\cdots\!95}a^{15}+\frac{25\!\cdots\!53}{12\!\cdots\!95}a^{14}+\frac{39\!\cdots\!42}{12\!\cdots\!95}a^{13}-\frac{26\!\cdots\!78}{24\!\cdots\!99}a^{12}+\frac{48\!\cdots\!81}{24\!\cdots\!99}a^{11}-\frac{13\!\cdots\!74}{12\!\cdots\!95}a^{10}-\frac{42\!\cdots\!34}{24\!\cdots\!99}a^{9}+\frac{15\!\cdots\!28}{41\!\cdots\!61}a^{8}-\frac{74\!\cdots\!90}{24\!\cdots\!99}a^{7}-\frac{44\!\cdots\!54}{24\!\cdots\!99}a^{6}+\frac{28\!\cdots\!17}{24\!\cdots\!99}a^{5}-\frac{45\!\cdots\!57}{24\!\cdots\!99}a^{4}+\frac{57\!\cdots\!13}{34\!\cdots\!57}a^{3}-\frac{21\!\cdots\!77}{24\!\cdots\!99}a^{2}+\frac{64\!\cdots\!77}{24\!\cdots\!99}a-\frac{76\!\cdots\!43}{24\!\cdots\!99}$, $\frac{37\!\cdots\!37}{12\!\cdots\!95}a^{19}-\frac{19\!\cdots\!24}{12\!\cdots\!95}a^{18}+\frac{23\!\cdots\!13}{12\!\cdots\!95}a^{17}+\frac{27\!\cdots\!41}{12\!\cdots\!95}a^{16}-\frac{64\!\cdots\!29}{12\!\cdots\!95}a^{15}-\frac{19\!\cdots\!29}{12\!\cdots\!95}a^{14}+\frac{21\!\cdots\!31}{12\!\cdots\!95}a^{13}-\frac{62\!\cdots\!19}{12\!\cdots\!95}a^{12}+\frac{10\!\cdots\!23}{12\!\cdots\!95}a^{11}-\frac{28\!\cdots\!44}{12\!\cdots\!95}a^{10}-\frac{26\!\cdots\!29}{24\!\cdots\!99}a^{9}+\frac{63\!\cdots\!44}{41\!\cdots\!61}a^{8}-\frac{20\!\cdots\!77}{24\!\cdots\!99}a^{7}-\frac{33\!\cdots\!85}{24\!\cdots\!99}a^{6}+\frac{13\!\cdots\!28}{24\!\cdots\!99}a^{5}-\frac{18\!\cdots\!50}{24\!\cdots\!99}a^{4}+\frac{12\!\cdots\!69}{24\!\cdots\!99}a^{3}-\frac{55\!\cdots\!63}{24\!\cdots\!99}a^{2}+\frac{19\!\cdots\!89}{24\!\cdots\!99}a-\frac{49\!\cdots\!89}{34\!\cdots\!57}$
| 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: | \( 50369.577816307596 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 50369.577816307596 \cdot 2}{2\cdot\sqrt{61839639189243316650390625}}\cr\approx \mathstrut & 0.614233530926613 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 11*x^18 - 21*x^16 + 13*x^15 + 55*x^14 - 217*x^13 + 434*x^12 - 360*x^11 - 205*x^10 + 805*x^9 - 845*x^8 - 70*x^7 + 2180*x^6 - 4240*x^5 + 4450*x^4 - 2890*x^3 + 1155*x^2 - 250*x + 25)
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 + 11*x^18 - 21*x^16 + 13*x^15 + 55*x^14 - 217*x^13 + 434*x^12 - 360*x^11 - 205*x^10 + 805*x^9 - 845*x^8 - 70*x^7 + 2180*x^6 - 4240*x^5 + 4450*x^4 - 2890*x^3 + 1155*x^2 - 250*x + 25, 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 + 11*x^18 - 21*x^16 + 13*x^15 + 55*x^14 - 217*x^13 + 434*x^12 - 360*x^11 - 205*x^10 + 805*x^9 - 845*x^8 - 70*x^7 + 2180*x^6 - 4240*x^5 + 4450*x^4 - 2890*x^3 + 1155*x^2 - 250*x + 25);
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 + 11*x^18 - 21*x^16 + 13*x^15 + 55*x^14 - 217*x^13 + 434*x^12 - 360*x^11 - 205*x^10 + 805*x^9 - 845*x^8 - 70*x^7 + 2180*x^6 - 4240*x^5 + 4450*x^4 - 2890*x^3 + 1155*x^2 - 250*x + 25);
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
$C_5\times D_{10}$ (as 20T24):
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 100 |
| The 40 conjugacy class representatives for $C_5\times D_{10}$ |
| Character table for $C_5\times D_{10}$ |
Intermediate fields
| \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}, \sqrt{-11})\), 10.0.62910546875.1 |
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 sibling: | deg 20 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{10}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{10}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{10}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.10.17.1 | $x^{10} + 20 x^{8} + 5$ | $10$ | $1$ | $17$ | $C_{10}$ | $[2]_{2}$ | |
|
\(11\)
| 11.10.5.2 | $x^{10} + 55 x^{8} + 20 x^{7} + 1210 x^{6} - 202 x^{5} + 13410 x^{4} - 14080 x^{3} + 75585 x^{2} - 68970 x + 171252$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 11.10.5.2 | $x^{10} + 55 x^{8} + 20 x^{7} + 1210 x^{6} - 202 x^{5} + 13410 x^{4} - 14080 x^{3} + 75585 x^{2} - 68970 x + 171252$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.11.2t1.a.a | $1$ | $ 11 $ | \(\Q(\sqrt{-11}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.55.2t1.a.a | $1$ | $ 5 \cdot 11 $ | \(\Q(\sqrt{-55}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.275.10t1.a.c | $1$ | $ 5^{2} \cdot 11 $ | 10.0.24574432373046875.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| 1.25.10t1.a.c | $1$ | $ 5^{2}$ | \(\Q(\zeta_{25})^+\) | $C_{10}$ (as 10T1) | $0$ | $1$ | |
| 1.275.10t1.b.d | $1$ | $ 5^{2} \cdot 11 $ | 10.0.122872161865234375.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| 1.25.5t1.a.d | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ | |
| 1.25.10t1.a.b | $1$ | $ 5^{2}$ | \(\Q(\zeta_{25})^+\) | $C_{10}$ (as 10T1) | $0$ | $1$ | |
| 1.25.5t1.a.a | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ | |
| 1.275.10t1.b.c | $1$ | $ 5^{2} \cdot 11 $ | 10.0.122872161865234375.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| 1.25.10t1.a.d | $1$ | $ 5^{2}$ | \(\Q(\zeta_{25})^+\) | $C_{10}$ (as 10T1) | $0$ | $1$ | |
| 1.275.10t1.a.d | $1$ | $ 5^{2} \cdot 11 $ | 10.0.24574432373046875.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| 1.275.10t1.b.b | $1$ | $ 5^{2} \cdot 11 $ | 10.0.122872161865234375.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| 1.275.10t1.b.a | $1$ | $ 5^{2} \cdot 11 $ | 10.0.122872161865234375.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| 1.275.10t1.a.a | $1$ | $ 5^{2} \cdot 11 $ | 10.0.24574432373046875.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| 1.25.10t1.a.a | $1$ | $ 5^{2}$ | \(\Q(\zeta_{25})^+\) | $C_{10}$ (as 10T1) | $0$ | $1$ | |
| 1.25.5t1.a.b | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ | |
| 1.25.5t1.a.c | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ | |
| 1.275.10t1.a.b | $1$ | $ 5^{2} \cdot 11 $ | 10.0.24574432373046875.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| 2.6875.20t24.a.b | $2$ | $ 5^{4} \cdot 11 $ | 20.0.61839639189243316650390625.1 | $C_5\times D_{10}$ (as 20T24) | $0$ | $0$ | |
| 2.6875.20t24.a.d | $2$ | $ 5^{4} \cdot 11 $ | 20.0.61839639189243316650390625.1 | $C_5\times D_{10}$ (as 20T24) | $0$ | $0$ | |
| 2.6875.5t2.a.a | $2$ | $ 5^{4} \cdot 11 $ | 5.1.47265625.1 | $D_{5}$ (as 5T2) | $1$ | $0$ | |
| * | 2.275.10t6.a.d | $2$ | $ 5^{2} \cdot 11 $ | 10.0.62910546875.1 | $D_5\times C_5$ (as 10T6) | $0$ | $0$ |
| 2.6875.10t6.a.a | $2$ | $ 5^{4} \cdot 11 $ | 10.0.62910546875.1 | $D_5\times C_5$ (as 10T6) | $0$ | $0$ | |
| 2.6875.10t6.a.d | $2$ | $ 5^{4} \cdot 11 $ | 10.0.62910546875.1 | $D_5\times C_5$ (as 10T6) | $0$ | $0$ | |
| * | 2.1375.20t24.a.a | $2$ | $ 5^{3} \cdot 11 $ | 20.0.61839639189243316650390625.1 | $C_5\times D_{10}$ (as 20T24) | $0$ | $0$ |
| 2.6875.10t3.a.a | $2$ | $ 5^{4} \cdot 11 $ | 10.0.122872161865234375.5 | $D_{10}$ (as 10T3) | $1$ | $0$ | |
| * | 2.275.10t6.a.a | $2$ | $ 5^{2} \cdot 11 $ | 10.0.62910546875.1 | $D_5\times C_5$ (as 10T6) | $0$ | $0$ |
| 2.6875.10t6.a.b | $2$ | $ 5^{4} \cdot 11 $ | 10.0.62910546875.1 | $D_5\times C_5$ (as 10T6) | $0$ | $0$ | |
| 2.6875.20t24.a.c | $2$ | $ 5^{4} \cdot 11 $ | 20.0.61839639189243316650390625.1 | $C_5\times D_{10}$ (as 20T24) | $0$ | $0$ | |
| * | 2.1375.20t24.a.c | $2$ | $ 5^{3} \cdot 11 $ | 20.0.61839639189243316650390625.1 | $C_5\times D_{10}$ (as 20T24) | $0$ | $0$ |
| * | 2.1375.20t24.a.b | $2$ | $ 5^{3} \cdot 11 $ | 20.0.61839639189243316650390625.1 | $C_5\times D_{10}$ (as 20T24) | $0$ | $0$ |
| 2.6875.10t3.a.b | $2$ | $ 5^{4} \cdot 11 $ | 10.0.122872161865234375.5 | $D_{10}$ (as 10T3) | $1$ | $0$ | |
| * | 2.1375.20t24.a.d | $2$ | $ 5^{3} \cdot 11 $ | 20.0.61839639189243316650390625.1 | $C_5\times D_{10}$ (as 20T24) | $0$ | $0$ |
| * | 2.275.10t6.a.c | $2$ | $ 5^{2} \cdot 11 $ | 10.0.62910546875.1 | $D_5\times C_5$ (as 10T6) | $0$ | $0$ |
| * | 2.275.10t6.a.b | $2$ | $ 5^{2} \cdot 11 $ | 10.0.62910546875.1 | $D_5\times C_5$ (as 10T6) | $0$ | $0$ |
| 2.6875.10t6.a.c | $2$ | $ 5^{4} \cdot 11 $ | 10.0.62910546875.1 | $D_5\times C_5$ (as 10T6) | $0$ | $0$ | |
| 2.6875.20t24.a.a | $2$ | $ 5^{4} \cdot 11 $ | 20.0.61839639189243316650390625.1 | $C_5\times D_{10}$ (as 20T24) | $0$ | $0$ | |
| 2.6875.5t2.a.b | $2$ | $ 5^{4} \cdot 11 $ | 5.1.47265625.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.