Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 9*x^18 - 9*x^17 + 11*x^16 - 21*x^15 - 16*x^14 + 141*x^13 - 361*x^12 + 484*x^11 - 461*x^10 + 363*x^9 - 445*x^8 + 410*x^7 - 133*x^6 + 133*x^5 - 143*x^4 + 39*x^3 - 8*x^2 - x + 1)
gp: K = bnfinit(y^20 - 5*y^19 + 9*y^18 - 9*y^17 + 11*y^16 - 21*y^15 - 16*y^14 + 141*y^13 - 361*y^12 + 484*y^11 - 461*y^10 + 363*y^9 - 445*y^8 + 410*y^7 - 133*y^6 + 133*y^5 - 143*y^4 + 39*y^3 - 8*y^2 - y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 5*x^19 + 9*x^18 - 9*x^17 + 11*x^16 - 21*x^15 - 16*x^14 + 141*x^13 - 361*x^12 + 484*x^11 - 461*x^10 + 363*x^9 - 445*x^8 + 410*x^7 - 133*x^6 + 133*x^5 - 143*x^4 + 39*x^3 - 8*x^2 - x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 5*x^19 + 9*x^18 - 9*x^17 + 11*x^16 - 21*x^15 - 16*x^14 + 141*x^13 - 361*x^12 + 484*x^11 - 461*x^10 + 363*x^9 - 445*x^8 + 410*x^7 - 133*x^6 + 133*x^5 - 143*x^4 + 39*x^3 - 8*x^2 - x + 1)
\( x^{20} - 5 x^{19} + 9 x^{18} - 9 x^{17} + 11 x^{16} - 21 x^{15} - 16 x^{14} + 141 x^{13} - 361 x^{12} + \cdots + 1 \)
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: |
\(823067302269314181883621609\)
\(\medspace = 11^{18}\cdot 23^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(22.17\) | 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: | $11^{9/10}23^{1/2}\approx 41.50661671665305$ | ||
| Ramified primes: |
\(11\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{13}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{14}-\frac{1}{3}a^{10}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{17}-\frac{1}{3}a^{13}-\frac{1}{3}a^{11}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{18}-\frac{1}{3}a^{14}-\frac{1}{3}a^{12}-\frac{1}{3}a^{10}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{14\!\cdots\!29}a^{19}-\frac{12\!\cdots\!89}{14\!\cdots\!29}a^{18}-\frac{62\!\cdots\!80}{47\!\cdots\!43}a^{17}+\frac{63\!\cdots\!25}{47\!\cdots\!43}a^{16}-\frac{79\!\cdots\!64}{47\!\cdots\!43}a^{15}+\frac{59\!\cdots\!46}{14\!\cdots\!29}a^{14}-\frac{30\!\cdots\!28}{20\!\cdots\!41}a^{13}-\frac{47\!\cdots\!01}{14\!\cdots\!29}a^{12}+\frac{19\!\cdots\!59}{14\!\cdots\!29}a^{11}+\frac{10\!\cdots\!62}{47\!\cdots\!43}a^{10}+\frac{29\!\cdots\!95}{14\!\cdots\!29}a^{9}-\frac{16\!\cdots\!60}{47\!\cdots\!43}a^{8}-\frac{40\!\cdots\!01}{14\!\cdots\!29}a^{7}+\frac{39\!\cdots\!24}{47\!\cdots\!43}a^{6}+\frac{29\!\cdots\!01}{14\!\cdots\!29}a^{5}-\frac{49\!\cdots\!13}{14\!\cdots\!29}a^{4}+\frac{41\!\cdots\!40}{14\!\cdots\!29}a^{3}-\frac{59\!\cdots\!40}{47\!\cdots\!43}a^{2}+\frac{11\!\cdots\!30}{47\!\cdots\!43}a+\frac{15\!\cdots\!55}{14\!\cdots\!29}$
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 $2$
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 $2$)
| 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: |
$\frac{30\!\cdots\!68}{14\!\cdots\!29}a^{19}-\frac{15\!\cdots\!09}{14\!\cdots\!29}a^{18}+\frac{26\!\cdots\!50}{14\!\cdots\!29}a^{17}-\frac{21\!\cdots\!05}{14\!\cdots\!29}a^{16}+\frac{27\!\cdots\!68}{14\!\cdots\!29}a^{15}-\frac{20\!\cdots\!56}{47\!\cdots\!43}a^{14}-\frac{25\!\cdots\!79}{62\!\cdots\!23}a^{13}+\frac{45\!\cdots\!21}{14\!\cdots\!29}a^{12}-\frac{35\!\cdots\!19}{47\!\cdots\!43}a^{11}+\frac{43\!\cdots\!24}{47\!\cdots\!43}a^{10}-\frac{35\!\cdots\!01}{47\!\cdots\!43}a^{9}+\frac{28\!\cdots\!26}{47\!\cdots\!43}a^{8}-\frac{11\!\cdots\!15}{14\!\cdots\!29}a^{7}+\frac{38\!\cdots\!58}{47\!\cdots\!43}a^{6}-\frac{16\!\cdots\!04}{14\!\cdots\!29}a^{5}+\frac{36\!\cdots\!06}{14\!\cdots\!29}a^{4}-\frac{18\!\cdots\!14}{47\!\cdots\!43}a^{3}-\frac{74\!\cdots\!25}{14\!\cdots\!29}a^{2}-\frac{79\!\cdots\!59}{47\!\cdots\!43}a+\frac{47\!\cdots\!67}{47\!\cdots\!43}$, $\frac{14\!\cdots\!32}{14\!\cdots\!29}a^{19}-\frac{29\!\cdots\!31}{47\!\cdots\!43}a^{18}+\frac{19\!\cdots\!73}{14\!\cdots\!29}a^{17}-\frac{21\!\cdots\!86}{14\!\cdots\!29}a^{16}+\frac{77\!\cdots\!98}{47\!\cdots\!43}a^{15}-\frac{42\!\cdots\!90}{14\!\cdots\!29}a^{14}+\frac{48\!\cdots\!43}{62\!\cdots\!23}a^{13}+\frac{83\!\cdots\!30}{47\!\cdots\!43}a^{12}-\frac{70\!\cdots\!17}{14\!\cdots\!29}a^{11}+\frac{37\!\cdots\!59}{47\!\cdots\!43}a^{10}-\frac{36\!\cdots\!43}{47\!\cdots\!43}a^{9}+\frac{30\!\cdots\!57}{47\!\cdots\!43}a^{8}-\frac{89\!\cdots\!67}{14\!\cdots\!29}a^{7}+\frac{11\!\cdots\!79}{14\!\cdots\!29}a^{6}-\frac{43\!\cdots\!36}{14\!\cdots\!29}a^{5}+\frac{26\!\cdots\!86}{14\!\cdots\!29}a^{4}-\frac{40\!\cdots\!19}{14\!\cdots\!29}a^{3}+\frac{11\!\cdots\!26}{14\!\cdots\!29}a^{2}-\frac{50\!\cdots\!84}{14\!\cdots\!29}a-\frac{21\!\cdots\!41}{47\!\cdots\!43}$, $\frac{38\!\cdots\!22}{62\!\cdots\!23}a^{19}-\frac{23\!\cdots\!52}{62\!\cdots\!23}a^{18}-\frac{57\!\cdots\!22}{62\!\cdots\!23}a^{17}+\frac{14\!\cdots\!20}{62\!\cdots\!23}a^{16}-\frac{11\!\cdots\!72}{62\!\cdots\!23}a^{15}+\frac{24\!\cdots\!61}{20\!\cdots\!41}a^{14}-\frac{13\!\cdots\!33}{20\!\cdots\!41}a^{13}+\frac{29\!\cdots\!50}{62\!\cdots\!23}a^{12}+\frac{44\!\cdots\!11}{20\!\cdots\!41}a^{11}-\frac{16\!\cdots\!31}{20\!\cdots\!41}a^{10}+\frac{73\!\cdots\!35}{62\!\cdots\!23}a^{9}-\frac{19\!\cdots\!78}{20\!\cdots\!41}a^{8}+\frac{25\!\cdots\!81}{62\!\cdots\!23}a^{7}-\frac{35\!\cdots\!13}{62\!\cdots\!23}a^{6}+\frac{56\!\cdots\!27}{62\!\cdots\!23}a^{5}+\frac{10\!\cdots\!09}{20\!\cdots\!41}a^{4}-\frac{49\!\cdots\!30}{20\!\cdots\!41}a^{3}-\frac{77\!\cdots\!11}{20\!\cdots\!41}a^{2}+\frac{33\!\cdots\!07}{62\!\cdots\!23}a+\frac{69\!\cdots\!15}{62\!\cdots\!23}$, $\frac{11\!\cdots\!68}{47\!\cdots\!43}a^{19}-\frac{54\!\cdots\!71}{47\!\cdots\!43}a^{18}+\frac{27\!\cdots\!38}{14\!\cdots\!29}a^{17}-\frac{25\!\cdots\!95}{14\!\cdots\!29}a^{16}+\frac{11\!\cdots\!60}{47\!\cdots\!43}a^{15}-\frac{66\!\cdots\!95}{14\!\cdots\!29}a^{14}-\frac{29\!\cdots\!05}{62\!\cdots\!23}a^{13}+\frac{15\!\cdots\!03}{47\!\cdots\!43}a^{12}-\frac{11\!\cdots\!72}{14\!\cdots\!29}a^{11}+\frac{14\!\cdots\!55}{14\!\cdots\!29}a^{10}-\frac{13\!\cdots\!43}{14\!\cdots\!29}a^{9}+\frac{35\!\cdots\!05}{47\!\cdots\!43}a^{8}-\frac{14\!\cdots\!80}{14\!\cdots\!29}a^{7}+\frac{11\!\cdots\!84}{14\!\cdots\!29}a^{6}-\frac{33\!\cdots\!17}{14\!\cdots\!29}a^{5}+\frac{50\!\cdots\!35}{14\!\cdots\!29}a^{4}-\frac{42\!\cdots\!97}{14\!\cdots\!29}a^{3}+\frac{95\!\cdots\!09}{14\!\cdots\!29}a^{2}-\frac{39\!\cdots\!91}{14\!\cdots\!29}a+\frac{19\!\cdots\!35}{14\!\cdots\!29}$, $\frac{39\!\cdots\!16}{20\!\cdots\!41}a^{19}-\frac{16\!\cdots\!81}{20\!\cdots\!41}a^{18}+\frac{64\!\cdots\!11}{62\!\cdots\!23}a^{17}-\frac{11\!\cdots\!49}{20\!\cdots\!41}a^{16}+\frac{62\!\cdots\!34}{62\!\cdots\!23}a^{15}-\frac{54\!\cdots\!73}{20\!\cdots\!41}a^{14}-\frac{36\!\cdots\!04}{62\!\cdots\!23}a^{13}+\frac{50\!\cdots\!06}{20\!\cdots\!41}a^{12}-\frac{30\!\cdots\!02}{62\!\cdots\!23}a^{11}+\frac{92\!\cdots\!48}{20\!\cdots\!41}a^{10}-\frac{56\!\cdots\!19}{20\!\cdots\!41}a^{9}+\frac{84\!\cdots\!47}{62\!\cdots\!23}a^{8}-\frac{26\!\cdots\!24}{62\!\cdots\!23}a^{7}+\frac{45\!\cdots\!89}{20\!\cdots\!41}a^{6}+\frac{49\!\cdots\!62}{20\!\cdots\!41}a^{5}+\frac{29\!\cdots\!52}{20\!\cdots\!41}a^{4}-\frac{62\!\cdots\!28}{62\!\cdots\!23}a^{3}-\frac{21\!\cdots\!93}{20\!\cdots\!41}a^{2}+\frac{93\!\cdots\!36}{62\!\cdots\!23}a-\frac{25\!\cdots\!21}{62\!\cdots\!23}$, $\frac{51\!\cdots\!23}{47\!\cdots\!43}a^{19}+\frac{15\!\cdots\!62}{14\!\cdots\!29}a^{18}-\frac{61\!\cdots\!07}{14\!\cdots\!29}a^{17}+\frac{64\!\cdots\!80}{14\!\cdots\!29}a^{16}-\frac{12\!\cdots\!20}{47\!\cdots\!43}a^{15}+\frac{29\!\cdots\!43}{47\!\cdots\!43}a^{14}-\frac{87\!\cdots\!71}{62\!\cdots\!23}a^{13}-\frac{53\!\cdots\!91}{14\!\cdots\!29}a^{12}+\frac{17\!\cdots\!40}{14\!\cdots\!29}a^{11}-\frac{35\!\cdots\!45}{14\!\cdots\!29}a^{10}+\frac{89\!\cdots\!02}{47\!\cdots\!43}a^{9}-\frac{22\!\cdots\!45}{14\!\cdots\!29}a^{8}+\frac{10\!\cdots\!06}{14\!\cdots\!29}a^{7}-\frac{39\!\cdots\!40}{14\!\cdots\!29}a^{6}+\frac{47\!\cdots\!38}{14\!\cdots\!29}a^{5}+\frac{10\!\cdots\!18}{14\!\cdots\!29}a^{4}+\frac{19\!\cdots\!77}{14\!\cdots\!29}a^{3}-\frac{75\!\cdots\!67}{14\!\cdots\!29}a^{2}-\frac{17\!\cdots\!69}{14\!\cdots\!29}a+\frac{30\!\cdots\!68}{14\!\cdots\!29}$, $\frac{23\!\cdots\!44}{20\!\cdots\!41}a^{19}-\frac{28\!\cdots\!92}{62\!\cdots\!23}a^{18}+\frac{27\!\cdots\!99}{62\!\cdots\!23}a^{17}+\frac{14\!\cdots\!47}{62\!\cdots\!23}a^{16}+\frac{17\!\cdots\!38}{62\!\cdots\!23}a^{15}-\frac{80\!\cdots\!86}{62\!\cdots\!23}a^{14}-\frac{85\!\cdots\!74}{20\!\cdots\!41}a^{13}+\frac{88\!\cdots\!97}{62\!\cdots\!23}a^{12}-\frac{15\!\cdots\!80}{62\!\cdots\!23}a^{11}+\frac{27\!\cdots\!55}{20\!\cdots\!41}a^{10}+\frac{42\!\cdots\!54}{20\!\cdots\!41}a^{9}-\frac{11\!\cdots\!97}{20\!\cdots\!41}a^{8}-\frac{35\!\cdots\!92}{20\!\cdots\!41}a^{7}+\frac{15\!\cdots\!65}{62\!\cdots\!23}a^{6}+\frac{60\!\cdots\!28}{20\!\cdots\!41}a^{5}+\frac{42\!\cdots\!40}{62\!\cdots\!23}a^{4}-\frac{59\!\cdots\!01}{62\!\cdots\!23}a^{3}-\frac{26\!\cdots\!79}{20\!\cdots\!41}a^{2}+\frac{11\!\cdots\!09}{62\!\cdots\!23}a+\frac{30\!\cdots\!96}{20\!\cdots\!41}$, $\frac{15\!\cdots\!77}{47\!\cdots\!43}a^{19}-\frac{20\!\cdots\!51}{14\!\cdots\!29}a^{18}+\frac{88\!\cdots\!56}{47\!\cdots\!43}a^{17}-\frac{14\!\cdots\!06}{14\!\cdots\!29}a^{16}+\frac{24\!\cdots\!94}{14\!\cdots\!29}a^{15}-\frac{66\!\cdots\!71}{14\!\cdots\!29}a^{14}-\frac{58\!\cdots\!92}{62\!\cdots\!23}a^{13}+\frac{60\!\cdots\!60}{14\!\cdots\!29}a^{12}-\frac{41\!\cdots\!09}{47\!\cdots\!43}a^{11}+\frac{38\!\cdots\!83}{47\!\cdots\!43}a^{10}-\frac{22\!\cdots\!00}{47\!\cdots\!43}a^{9}+\frac{36\!\cdots\!77}{14\!\cdots\!29}a^{8}-\frac{10\!\cdots\!26}{14\!\cdots\!29}a^{7}+\frac{65\!\cdots\!86}{14\!\cdots\!29}a^{6}+\frac{20\!\cdots\!47}{47\!\cdots\!43}a^{5}+\frac{33\!\cdots\!53}{14\!\cdots\!29}a^{4}-\frac{12\!\cdots\!35}{47\!\cdots\!43}a^{3}-\frac{97\!\cdots\!97}{47\!\cdots\!43}a^{2}+\frac{34\!\cdots\!52}{14\!\cdots\!29}a+\frac{41\!\cdots\!52}{14\!\cdots\!29}$, $\frac{40\!\cdots\!28}{14\!\cdots\!29}a^{19}-\frac{19\!\cdots\!97}{14\!\cdots\!29}a^{18}+\frac{30\!\cdots\!70}{14\!\cdots\!29}a^{17}-\frac{22\!\cdots\!75}{14\!\cdots\!29}a^{16}+\frac{96\!\cdots\!84}{47\!\cdots\!43}a^{15}-\frac{70\!\cdots\!35}{14\!\cdots\!29}a^{14}-\frac{39\!\cdots\!61}{62\!\cdots\!23}a^{13}+\frac{57\!\cdots\!42}{14\!\cdots\!29}a^{12}-\frac{12\!\cdots\!94}{14\!\cdots\!29}a^{11}+\frac{14\!\cdots\!78}{14\!\cdots\!29}a^{10}-\frac{10\!\cdots\!37}{14\!\cdots\!29}a^{9}+\frac{72\!\cdots\!77}{14\!\cdots\!29}a^{8}-\frac{40\!\cdots\!16}{47\!\cdots\!43}a^{7}+\frac{11\!\cdots\!61}{14\!\cdots\!29}a^{6}+\frac{17\!\cdots\!09}{14\!\cdots\!29}a^{5}+\frac{10\!\cdots\!91}{47\!\cdots\!43}a^{4}-\frac{16\!\cdots\!44}{47\!\cdots\!43}a^{3}-\frac{12\!\cdots\!99}{14\!\cdots\!29}a^{2}-\frac{52\!\cdots\!70}{14\!\cdots\!29}a+\frac{99\!\cdots\!57}{47\!\cdots\!43}$, $\frac{13\!\cdots\!24}{47\!\cdots\!43}a^{19}-\frac{65\!\cdots\!04}{47\!\cdots\!43}a^{18}+\frac{11\!\cdots\!76}{47\!\cdots\!43}a^{17}-\frac{10\!\cdots\!77}{47\!\cdots\!43}a^{16}+\frac{40\!\cdots\!46}{14\!\cdots\!29}a^{15}-\frac{26\!\cdots\!92}{47\!\cdots\!43}a^{14}-\frac{30\!\cdots\!21}{62\!\cdots\!23}a^{13}+\frac{18\!\cdots\!80}{47\!\cdots\!43}a^{12}-\frac{46\!\cdots\!22}{47\!\cdots\!43}a^{11}+\frac{59\!\cdots\!52}{47\!\cdots\!43}a^{10}-\frac{16\!\cdots\!32}{14\!\cdots\!29}a^{9}+\frac{43\!\cdots\!98}{47\!\cdots\!43}a^{8}-\frac{16\!\cdots\!49}{14\!\cdots\!29}a^{7}+\frac{15\!\cdots\!81}{14\!\cdots\!29}a^{6}-\frac{12\!\cdots\!94}{47\!\cdots\!43}a^{5}+\frac{48\!\cdots\!67}{14\!\cdots\!29}a^{4}-\frac{19\!\cdots\!18}{47\!\cdots\!43}a^{3}+\frac{10\!\cdots\!77}{14\!\cdots\!29}a^{2}-\frac{65\!\cdots\!48}{14\!\cdots\!29}a-\frac{84\!\cdots\!43}{14\!\cdots\!29}$, $\frac{11\!\cdots\!45}{47\!\cdots\!43}a^{19}-\frac{17\!\cdots\!75}{14\!\cdots\!29}a^{18}+\frac{33\!\cdots\!45}{14\!\cdots\!29}a^{17}-\frac{10\!\cdots\!25}{47\!\cdots\!43}a^{16}+\frac{12\!\cdots\!80}{47\!\cdots\!43}a^{15}-\frac{75\!\cdots\!24}{14\!\cdots\!29}a^{14}-\frac{21\!\cdots\!34}{62\!\cdots\!23}a^{13}+\frac{51\!\cdots\!00}{14\!\cdots\!29}a^{12}-\frac{13\!\cdots\!12}{14\!\cdots\!29}a^{11}+\frac{17\!\cdots\!00}{14\!\cdots\!29}a^{10}-\frac{16\!\cdots\!49}{14\!\cdots\!29}a^{9}+\frac{12\!\cdots\!60}{14\!\cdots\!29}a^{8}-\frac{15\!\cdots\!86}{14\!\cdots\!29}a^{7}+\frac{15\!\cdots\!24}{14\!\cdots\!29}a^{6}-\frac{38\!\cdots\!55}{14\!\cdots\!29}a^{5}+\frac{13\!\cdots\!39}{47\!\cdots\!43}a^{4}-\frac{62\!\cdots\!74}{14\!\cdots\!29}a^{3}+\frac{32\!\cdots\!92}{47\!\cdots\!43}a^{2}-\frac{74\!\cdots\!74}{47\!\cdots\!43}a+\frac{16\!\cdots\!67}{14\!\cdots\!29}$
| 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: | \( 173434.395473 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 173434.395473 \cdot 2}{2\cdot\sqrt{823067302269314181883621609}}\cr\approx \mathstrut & 0.234950560770 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 9*x^18 - 9*x^17 + 11*x^16 - 21*x^15 - 16*x^14 + 141*x^13 - 361*x^12 + 484*x^11 - 461*x^10 + 363*x^9 - 445*x^8 + 410*x^7 - 133*x^6 + 133*x^5 - 143*x^4 + 39*x^3 - 8*x^2 - 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^19 + 9*x^18 - 9*x^17 + 11*x^16 - 21*x^15 - 16*x^14 + 141*x^13 - 361*x^12 + 484*x^11 - 461*x^10 + 363*x^9 - 445*x^8 + 410*x^7 - 133*x^6 + 133*x^5 - 143*x^4 + 39*x^3 - 8*x^2 - 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^19 + 9*x^18 - 9*x^17 + 11*x^16 - 21*x^15 - 16*x^14 + 141*x^13 - 361*x^12 + 484*x^11 - 461*x^10 + 363*x^9 - 445*x^8 + 410*x^7 - 133*x^6 + 133*x^5 - 143*x^4 + 39*x^3 - 8*x^2 - 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^19 + 9*x^18 - 9*x^17 + 11*x^16 - 21*x^15 - 16*x^14 + 141*x^13 - 361*x^12 + 484*x^11 - 461*x^10 + 363*x^9 - 445*x^8 + 410*x^7 - 133*x^6 + 133*x^5 - 143*x^4 + 39*x^3 - 8*x^2 - 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
$C_2\wr C_5$ (as 20T46):
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 160 |
| The 16 conjugacy class representatives for $C_2\wr C_5$ |
| Character table for $C_2\wr C_5$ |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.6.28689149556397.1, 10.6.54232796893.1, 10.2.113395848049.1 |
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 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.6.54232796893.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.5.0.1}{5} }^{4}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.10.9.1 | $x^{10} + 110$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
| 11.10.9.1 | $x^{10} + 110$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |
|
\(23\)
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |