Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 10*x^18 - 20*x^17 + 18*x^16 - 7*x^15 - 14*x^14 + 86*x^13 - 128*x^12 + 44*x^11 + 106*x^10 - 209*x^9 - 61*x^8 + 380*x^7 - 323*x^6 + 172*x^5 + 19*x^4 - 52*x^3 + 19*x^2 - 7*x + 1)
gp: K = bnfinit(y^20 - 3*y^19 + 10*y^18 - 20*y^17 + 18*y^16 - 7*y^15 - 14*y^14 + 86*y^13 - 128*y^12 + 44*y^11 + 106*y^10 - 209*y^9 - 61*y^8 + 380*y^7 - 323*y^6 + 172*y^5 + 19*y^4 - 52*y^3 + 19*y^2 - 7*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 3*x^19 + 10*x^18 - 20*x^17 + 18*x^16 - 7*x^15 - 14*x^14 + 86*x^13 - 128*x^12 + 44*x^11 + 106*x^10 - 209*x^9 - 61*x^8 + 380*x^7 - 323*x^6 + 172*x^5 + 19*x^4 - 52*x^3 + 19*x^2 - 7*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 3*x^19 + 10*x^18 - 20*x^17 + 18*x^16 - 7*x^15 - 14*x^14 + 86*x^13 - 128*x^12 + 44*x^11 + 106*x^10 - 209*x^9 - 61*x^8 + 380*x^7 - 323*x^6 + 172*x^5 + 19*x^4 - 52*x^3 + 19*x^2 - 7*x + 1)
\( x^{20} - 3 x^{19} + 10 x^{18} - 20 x^{17} + 18 x^{16} - 7 x^{15} - 14 x^{14} + 86 x^{13} - 128 x^{12} + \cdots + 1 \)
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: |
\(6802209109663753569286129\)
\(\medspace = 11^{16}\cdot 23^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.44\) | 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: | $11^{4/5}23^{1/2}\approx 32.65713384043754$ | ||
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}+\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}+\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{14}+\frac{1}{4}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{11}-\frac{1}{2}a^{10}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{8}a^{18}-\frac{1}{8}a^{17}-\frac{1}{8}a^{16}-\frac{1}{8}a^{15}-\frac{1}{8}a^{14}-\frac{1}{8}a^{12}-\frac{1}{4}a^{11}-\frac{3}{8}a^{10}+\frac{3}{8}a^{8}-\frac{3}{8}a^{7}-\frac{1}{2}a^{6}+\frac{3}{8}a^{5}+\frac{1}{8}a^{4}+\frac{3}{8}a^{3}-\frac{3}{8}a+\frac{1}{8}$, $\frac{1}{37\!\cdots\!16}a^{19}+\frac{27\!\cdots\!25}{18\!\cdots\!08}a^{18}+\frac{11\!\cdots\!05}{93\!\cdots\!04}a^{17}+\frac{21\!\cdots\!23}{23\!\cdots\!51}a^{16}-\frac{56\!\cdots\!91}{18\!\cdots\!08}a^{15}+\frac{23\!\cdots\!99}{37\!\cdots\!16}a^{14}+\frac{42\!\cdots\!49}{37\!\cdots\!16}a^{13}+\frac{21\!\cdots\!55}{37\!\cdots\!16}a^{12}+\frac{20\!\cdots\!35}{37\!\cdots\!16}a^{11}+\frac{15\!\cdots\!63}{37\!\cdots\!16}a^{10}+\frac{66\!\cdots\!73}{37\!\cdots\!16}a^{9}-\frac{10\!\cdots\!89}{93\!\cdots\!04}a^{8}+\frac{31\!\cdots\!47}{37\!\cdots\!16}a^{7}+\frac{57\!\cdots\!47}{37\!\cdots\!16}a^{6}-\frac{46\!\cdots\!23}{93\!\cdots\!04}a^{5}-\frac{25\!\cdots\!13}{93\!\cdots\!04}a^{4}+\frac{72\!\cdots\!23}{37\!\cdots\!16}a^{3}-\frac{64\!\cdots\!25}{37\!\cdots\!16}a^{2}+\frac{65\!\cdots\!91}{18\!\cdots\!08}a+\frac{42\!\cdots\!87}{37\!\cdots\!16}$
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 $2$)
| 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: |
$\frac{33\!\cdots\!97}{23\!\cdots\!51}a^{19}-\frac{85\!\cdots\!06}{23\!\cdots\!51}a^{18}+\frac{58\!\cdots\!47}{46\!\cdots\!02}a^{17}-\frac{21\!\cdots\!07}{93\!\cdots\!04}a^{16}+\frac{13\!\cdots\!59}{93\!\cdots\!04}a^{15}-\frac{39\!\cdots\!65}{46\!\cdots\!02}a^{14}-\frac{22\!\cdots\!01}{93\!\cdots\!04}a^{13}+\frac{53\!\cdots\!93}{46\!\cdots\!02}a^{12}-\frac{12\!\cdots\!85}{93\!\cdots\!04}a^{11}-\frac{22\!\cdots\!45}{93\!\cdots\!04}a^{10}+\frac{80\!\cdots\!95}{46\!\cdots\!02}a^{9}-\frac{22\!\cdots\!81}{93\!\cdots\!04}a^{8}-\frac{47\!\cdots\!23}{23\!\cdots\!51}a^{7}+\frac{23\!\cdots\!41}{46\!\cdots\!02}a^{6}-\frac{25\!\cdots\!27}{93\!\cdots\!04}a^{5}+\frac{67\!\cdots\!55}{93\!\cdots\!04}a^{4}+\frac{32\!\cdots\!78}{23\!\cdots\!51}a^{3}-\frac{15\!\cdots\!59}{46\!\cdots\!02}a^{2}-\frac{21\!\cdots\!22}{23\!\cdots\!51}a+\frac{13\!\cdots\!01}{23\!\cdots\!51}$, $\frac{24\!\cdots\!07}{37\!\cdots\!16}a^{19}-\frac{16\!\cdots\!09}{18\!\cdots\!08}a^{18}+\frac{36\!\cdots\!93}{93\!\cdots\!04}a^{17}-\frac{35\!\cdots\!51}{93\!\cdots\!04}a^{16}-\frac{94\!\cdots\!83}{18\!\cdots\!08}a^{15}+\frac{20\!\cdots\!05}{37\!\cdots\!16}a^{14}-\frac{35\!\cdots\!37}{37\!\cdots\!16}a^{13}+\frac{15\!\cdots\!29}{37\!\cdots\!16}a^{12}+\frac{43\!\cdots\!21}{37\!\cdots\!16}a^{11}-\frac{25\!\cdots\!35}{37\!\cdots\!16}a^{10}+\frac{21\!\cdots\!31}{37\!\cdots\!16}a^{9}-\frac{10\!\cdots\!23}{93\!\cdots\!04}a^{8}-\frac{78\!\cdots\!83}{37\!\cdots\!16}a^{7}+\frac{30\!\cdots\!97}{37\!\cdots\!16}a^{6}+\frac{14\!\cdots\!07}{93\!\cdots\!04}a^{5}-\frac{19\!\cdots\!53}{46\!\cdots\!02}a^{4}+\frac{24\!\cdots\!85}{37\!\cdots\!16}a^{3}+\frac{15\!\cdots\!81}{37\!\cdots\!16}a^{2}-\frac{16\!\cdots\!25}{18\!\cdots\!08}a-\frac{30\!\cdots\!19}{37\!\cdots\!16}$, $\frac{16\!\cdots\!57}{37\!\cdots\!16}a^{19}-\frac{56\!\cdots\!19}{46\!\cdots\!02}a^{18}+\frac{76\!\cdots\!79}{18\!\cdots\!08}a^{17}-\frac{14\!\cdots\!17}{18\!\cdots\!08}a^{16}+\frac{27\!\cdots\!81}{46\!\cdots\!02}a^{15}-\frac{58\!\cdots\!79}{37\!\cdots\!16}a^{14}-\frac{24\!\cdots\!91}{37\!\cdots\!16}a^{13}+\frac{13\!\cdots\!41}{37\!\cdots\!16}a^{12}-\frac{17\!\cdots\!01}{37\!\cdots\!16}a^{11}+\frac{29\!\cdots\!69}{37\!\cdots\!16}a^{10}+\frac{18\!\cdots\!05}{37\!\cdots\!16}a^{9}-\frac{14\!\cdots\!99}{18\!\cdots\!08}a^{8}-\frac{17\!\cdots\!83}{37\!\cdots\!16}a^{7}+\frac{57\!\cdots\!91}{37\!\cdots\!16}a^{6}-\frac{19\!\cdots\!01}{18\!\cdots\!08}a^{5}+\frac{92\!\cdots\!41}{18\!\cdots\!08}a^{4}+\frac{84\!\cdots\!89}{37\!\cdots\!16}a^{3}-\frac{72\!\cdots\!13}{37\!\cdots\!16}a^{2}+\frac{49\!\cdots\!55}{93\!\cdots\!04}a-\frac{54\!\cdots\!95}{37\!\cdots\!16}$, $\frac{10\!\cdots\!33}{37\!\cdots\!16}a^{19}-\frac{69\!\cdots\!29}{93\!\cdots\!04}a^{18}+\frac{47\!\cdots\!43}{18\!\cdots\!08}a^{17}-\frac{86\!\cdots\!17}{18\!\cdots\!08}a^{16}+\frac{28\!\cdots\!31}{93\!\cdots\!04}a^{15}-\frac{11\!\cdots\!59}{37\!\cdots\!16}a^{14}-\frac{16\!\cdots\!19}{37\!\cdots\!16}a^{13}+\frac{85\!\cdots\!85}{37\!\cdots\!16}a^{12}-\frac{10\!\cdots\!57}{37\!\cdots\!16}a^{11}-\frac{24\!\cdots\!43}{37\!\cdots\!16}a^{10}+\frac{12\!\cdots\!13}{37\!\cdots\!16}a^{9}-\frac{87\!\cdots\!79}{18\!\cdots\!08}a^{8}-\frac{14\!\cdots\!27}{37\!\cdots\!16}a^{7}+\frac{36\!\cdots\!79}{37\!\cdots\!16}a^{6}-\frac{91\!\cdots\!17}{18\!\cdots\!08}a^{5}+\frac{38\!\cdots\!23}{18\!\cdots\!08}a^{4}+\frac{66\!\cdots\!93}{37\!\cdots\!16}a^{3}-\frac{40\!\cdots\!65}{37\!\cdots\!16}a^{2}+\frac{23\!\cdots\!21}{23\!\cdots\!51}a-\frac{29\!\cdots\!07}{37\!\cdots\!16}$, $\frac{87\!\cdots\!39}{37\!\cdots\!16}a^{19}-\frac{64\!\cdots\!95}{93\!\cdots\!04}a^{18}+\frac{43\!\cdots\!61}{18\!\cdots\!08}a^{17}-\frac{86\!\cdots\!21}{18\!\cdots\!08}a^{16}+\frac{19\!\cdots\!95}{46\!\cdots\!02}a^{15}-\frac{60\!\cdots\!93}{37\!\cdots\!16}a^{14}-\frac{12\!\cdots\!97}{37\!\cdots\!16}a^{13}+\frac{74\!\cdots\!03}{37\!\cdots\!16}a^{12}-\frac{10\!\cdots\!31}{37\!\cdots\!16}a^{11}+\frac{38\!\cdots\!03}{37\!\cdots\!16}a^{10}+\frac{92\!\cdots\!23}{37\!\cdots\!16}a^{9}-\frac{91\!\cdots\!43}{18\!\cdots\!08}a^{8}-\frac{54\!\cdots\!09}{37\!\cdots\!16}a^{7}+\frac{32\!\cdots\!17}{37\!\cdots\!16}a^{6}-\frac{14\!\cdots\!89}{18\!\cdots\!08}a^{5}+\frac{76\!\cdots\!99}{18\!\cdots\!08}a^{4}+\frac{29\!\cdots\!99}{37\!\cdots\!16}a^{3}-\frac{44\!\cdots\!15}{37\!\cdots\!16}a^{2}+\frac{27\!\cdots\!77}{46\!\cdots\!02}a-\frac{73\!\cdots\!77}{37\!\cdots\!16}$, $\frac{36\!\cdots\!77}{46\!\cdots\!02}a^{19}-\frac{23\!\cdots\!03}{18\!\cdots\!08}a^{18}+\frac{88\!\cdots\!79}{18\!\cdots\!08}a^{17}-\frac{10\!\cdots\!07}{18\!\cdots\!08}a^{16}-\frac{97\!\cdots\!59}{18\!\cdots\!08}a^{15}+\frac{16\!\cdots\!71}{18\!\cdots\!08}a^{14}-\frac{62\!\cdots\!25}{46\!\cdots\!02}a^{13}+\frac{92\!\cdots\!29}{18\!\cdots\!08}a^{12}-\frac{24\!\cdots\!29}{23\!\cdots\!51}a^{11}-\frac{15\!\cdots\!11}{18\!\cdots\!08}a^{10}+\frac{44\!\cdots\!25}{46\!\cdots\!02}a^{9}-\frac{63\!\cdots\!69}{18\!\cdots\!08}a^{8}-\frac{45\!\cdots\!01}{18\!\cdots\!08}a^{7}+\frac{39\!\cdots\!44}{23\!\cdots\!51}a^{6}+\frac{29\!\cdots\!75}{18\!\cdots\!08}a^{5}-\frac{20\!\cdots\!37}{18\!\cdots\!08}a^{4}+\frac{20\!\cdots\!55}{18\!\cdots\!08}a^{3}+\frac{20\!\cdots\!99}{93\!\cdots\!04}a^{2}-\frac{55\!\cdots\!89}{18\!\cdots\!08}a+\frac{18\!\cdots\!17}{18\!\cdots\!08}$, $\frac{76\!\cdots\!57}{37\!\cdots\!16}a^{19}-\frac{28\!\cdots\!27}{46\!\cdots\!02}a^{18}+\frac{37\!\cdots\!93}{18\!\cdots\!08}a^{17}-\frac{75\!\cdots\!77}{18\!\cdots\!08}a^{16}+\frac{32\!\cdots\!99}{93\!\cdots\!04}a^{15}-\frac{41\!\cdots\!79}{37\!\cdots\!16}a^{14}-\frac{11\!\cdots\!67}{37\!\cdots\!16}a^{13}+\frac{65\!\cdots\!85}{37\!\cdots\!16}a^{12}-\frac{95\!\cdots\!33}{37\!\cdots\!16}a^{11}+\frac{26\!\cdots\!65}{37\!\cdots\!16}a^{10}+\frac{87\!\cdots\!29}{37\!\cdots\!16}a^{9}-\frac{78\!\cdots\!61}{18\!\cdots\!08}a^{8}-\frac{59\!\cdots\!27}{37\!\cdots\!16}a^{7}+\frac{30\!\cdots\!51}{37\!\cdots\!16}a^{6}-\frac{11\!\cdots\!43}{18\!\cdots\!08}a^{5}+\frac{51\!\cdots\!71}{18\!\cdots\!08}a^{4}+\frac{22\!\cdots\!61}{37\!\cdots\!16}a^{3}-\frac{39\!\cdots\!17}{37\!\cdots\!16}a^{2}+\frac{95\!\cdots\!87}{46\!\cdots\!02}a-\frac{20\!\cdots\!91}{37\!\cdots\!16}$, $\frac{24\!\cdots\!93}{37\!\cdots\!16}a^{19}-\frac{34\!\cdots\!63}{46\!\cdots\!02}a^{18}+\frac{60\!\cdots\!49}{18\!\cdots\!08}a^{17}-\frac{36\!\cdots\!27}{18\!\cdots\!08}a^{16}-\frac{82\!\cdots\!55}{93\!\cdots\!04}a^{15}+\frac{39\!\cdots\!97}{37\!\cdots\!16}a^{14}-\frac{48\!\cdots\!55}{37\!\cdots\!16}a^{13}+\frac{14\!\cdots\!97}{37\!\cdots\!16}a^{12}+\frac{54\!\cdots\!31}{37\!\cdots\!16}a^{11}-\frac{34\!\cdots\!39}{37\!\cdots\!16}a^{10}+\frac{31\!\cdots\!77}{37\!\cdots\!16}a^{9}-\frac{12\!\cdots\!03}{18\!\cdots\!08}a^{8}-\frac{92\!\cdots\!19}{37\!\cdots\!16}a^{7}+\frac{41\!\cdots\!87}{37\!\cdots\!16}a^{6}+\frac{36\!\cdots\!73}{18\!\cdots\!08}a^{5}-\frac{26\!\cdots\!97}{18\!\cdots\!08}a^{4}+\frac{51\!\cdots\!49}{37\!\cdots\!16}a^{3}+\frac{72\!\cdots\!55}{37\!\cdots\!16}a^{2}-\frac{29\!\cdots\!53}{93\!\cdots\!04}a+\frac{31\!\cdots\!41}{37\!\cdots\!16}$, $\frac{63\!\cdots\!97}{37\!\cdots\!16}a^{19}-\frac{80\!\cdots\!09}{18\!\cdots\!08}a^{18}+\frac{35\!\cdots\!98}{23\!\cdots\!51}a^{17}-\frac{25\!\cdots\!33}{93\!\cdots\!04}a^{16}+\frac{37\!\cdots\!79}{18\!\cdots\!08}a^{15}-\frac{21\!\cdots\!41}{37\!\cdots\!16}a^{14}-\frac{90\!\cdots\!99}{37\!\cdots\!16}a^{13}+\frac{50\!\cdots\!71}{37\!\cdots\!16}a^{12}-\frac{59\!\cdots\!29}{37\!\cdots\!16}a^{11}+\frac{70\!\cdots\!27}{37\!\cdots\!16}a^{10}+\frac{64\!\cdots\!81}{37\!\cdots\!16}a^{9}-\frac{25\!\cdots\!07}{93\!\cdots\!04}a^{8}-\frac{77\!\cdots\!85}{37\!\cdots\!16}a^{7}+\frac{19\!\cdots\!51}{37\!\cdots\!16}a^{6}-\frac{31\!\cdots\!93}{93\!\cdots\!04}a^{5}+\frac{87\!\cdots\!83}{46\!\cdots\!02}a^{4}+\frac{23\!\cdots\!51}{37\!\cdots\!16}a^{3}-\frac{14\!\cdots\!61}{37\!\cdots\!16}a^{2}+\frac{43\!\cdots\!35}{18\!\cdots\!08}a-\frac{55\!\cdots\!73}{37\!\cdots\!16}$, $\frac{29\!\cdots\!57}{93\!\cdots\!04}a^{19}-\frac{22\!\cdots\!31}{23\!\cdots\!51}a^{18}+\frac{29\!\cdots\!11}{93\!\cdots\!04}a^{17}-\frac{14\!\cdots\!58}{23\!\cdots\!51}a^{16}+\frac{12\!\cdots\!75}{23\!\cdots\!51}a^{15}-\frac{42\!\cdots\!29}{23\!\cdots\!51}a^{14}-\frac{10\!\cdots\!70}{23\!\cdots\!51}a^{13}+\frac{12\!\cdots\!01}{46\!\cdots\!02}a^{12}-\frac{18\!\cdots\!41}{46\!\cdots\!02}a^{11}+\frac{28\!\cdots\!02}{23\!\cdots\!51}a^{10}+\frac{16\!\cdots\!65}{46\!\cdots\!02}a^{9}-\frac{61\!\cdots\!25}{93\!\cdots\!04}a^{8}-\frac{20\!\cdots\!43}{93\!\cdots\!04}a^{7}+\frac{11\!\cdots\!89}{93\!\cdots\!04}a^{6}-\frac{93\!\cdots\!41}{93\!\cdots\!04}a^{5}+\frac{10\!\cdots\!73}{23\!\cdots\!51}a^{4}+\frac{93\!\cdots\!23}{93\!\cdots\!04}a^{3}-\frac{15\!\cdots\!69}{93\!\cdots\!04}a^{2}+\frac{30\!\cdots\!37}{93\!\cdots\!04}a-\frac{10\!\cdots\!13}{93\!\cdots\!04}$, $\frac{86\!\cdots\!79}{37\!\cdots\!16}a^{19}-\frac{28\!\cdots\!91}{18\!\cdots\!08}a^{18}+\frac{32\!\cdots\!25}{93\!\cdots\!04}a^{17}-\frac{29\!\cdots\!73}{23\!\cdots\!51}a^{16}+\frac{38\!\cdots\!03}{18\!\cdots\!08}a^{15}-\frac{43\!\cdots\!03}{37\!\cdots\!16}a^{14}+\frac{50\!\cdots\!83}{37\!\cdots\!16}a^{13}+\frac{84\!\cdots\!93}{37\!\cdots\!16}a^{12}-\frac{41\!\cdots\!59}{37\!\cdots\!16}a^{11}+\frac{39\!\cdots\!85}{37\!\cdots\!16}a^{10}+\frac{62\!\cdots\!79}{37\!\cdots\!16}a^{9}-\frac{13\!\cdots\!37}{93\!\cdots\!04}a^{8}+\frac{70\!\cdots\!81}{37\!\cdots\!16}a^{7}+\frac{88\!\cdots\!77}{37\!\cdots\!16}a^{6}-\frac{37\!\cdots\!21}{93\!\cdots\!04}a^{5}+\frac{17\!\cdots\!71}{93\!\cdots\!04}a^{4}-\frac{38\!\cdots\!07}{37\!\cdots\!16}a^{3}-\frac{35\!\cdots\!43}{37\!\cdots\!16}a^{2}+\frac{19\!\cdots\!15}{18\!\cdots\!08}a+\frac{29\!\cdots\!21}{37\!\cdots\!16}$
| 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: | \( 25950.7198404 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 25950.7198404 \cdot 1}{2\cdot\sqrt{6802209109663753569286129}}\cr\approx \mathstrut & 0.193354085806 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 10*x^18 - 20*x^17 + 18*x^16 - 7*x^15 - 14*x^14 + 86*x^13 - 128*x^12 + 44*x^11 + 106*x^10 - 209*x^9 - 61*x^8 + 380*x^7 - 323*x^6 + 172*x^5 + 19*x^4 - 52*x^3 + 19*x^2 - 7*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 - 3*x^19 + 10*x^18 - 20*x^17 + 18*x^16 - 7*x^15 - 14*x^14 + 86*x^13 - 128*x^12 + 44*x^11 + 106*x^10 - 209*x^9 - 61*x^8 + 380*x^7 - 323*x^6 + 172*x^5 + 19*x^4 - 52*x^3 + 19*x^2 - 7*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 - 3*x^19 + 10*x^18 - 20*x^17 + 18*x^16 - 7*x^15 - 14*x^14 + 86*x^13 - 128*x^12 + 44*x^11 + 106*x^10 - 209*x^9 - 61*x^8 + 380*x^7 - 323*x^6 + 172*x^5 + 19*x^4 - 52*x^3 + 19*x^2 - 7*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 - 3*x^19 + 10*x^18 - 20*x^17 + 18*x^16 - 7*x^15 - 14*x^14 + 86*x^13 - 128*x^12 + 44*x^11 + 106*x^10 - 209*x^9 - 61*x^8 + 380*x^7 - 323*x^6 + 172*x^5 + 19*x^4 - 52*x^3 + 19*x^2 - 7*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 20T40):
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
| \(\Q(\zeta_{11})^+\), 10.4.4930254263.1, 10.4.2608104505127.1, 10.6.113395848049.1 |
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 |
| Arithmetically equvalently siblings: | data not computed |
| Minimal sibling: | 10.4.4930254263.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{4}$ | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/13.5.0.1}{5} }^{4}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.5.0.1}{5} }^{4}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |
|
\(23\)
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | 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 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 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 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |