Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 8*x^19 - 4*x^16 + 116*x^15 - 47*x^14 - 228*x^13 + 179*x^12 + 177*x^11 - 108*x^10 + 58*x^9 + 78*x^8 + 4*x^7 + 37*x^6 - 13*x^5 - 40*x^4 - x^3 + 10*x^2 + x - 1)
gp: K = bnfinit(y^22 - 8*y^19 - 4*y^16 + 116*y^15 - 47*y^14 - 228*y^13 + 179*y^12 + 177*y^11 - 108*y^10 + 58*y^9 + 78*y^8 + 4*y^7 + 37*y^6 - 13*y^5 - 40*y^4 - y^3 + 10*y^2 + y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 8*x^19 - 4*x^16 + 116*x^15 - 47*x^14 - 228*x^13 + 179*x^12 + 177*x^11 - 108*x^10 + 58*x^9 + 78*x^8 + 4*x^7 + 37*x^6 - 13*x^5 - 40*x^4 - x^3 + 10*x^2 + x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 8*x^19 - 4*x^16 + 116*x^15 - 47*x^14 - 228*x^13 + 179*x^12 + 177*x^11 - 108*x^10 + 58*x^9 + 78*x^8 + 4*x^7 + 37*x^6 - 13*x^5 - 40*x^4 - x^3 + 10*x^2 + x - 1)
\( x^{22} - 8 x^{19} - 4 x^{16} + 116 x^{15} - 47 x^{14} - 228 x^{13} + 179 x^{12} + 177 x^{11} - 108 x^{10} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[6, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(30428929951449912919970703125\)
\(\medspace = 5^{11}\cdot 971^{2}\cdot 25709231^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.71\) | 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: | $5^{1/2}971^{1/2}25709231^{1/2}\approx 353296.357899427$ | ||
| Ramified primes: |
\(5\), \(971\), \(25709231\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{33\!\cdots\!34}a^{21}+\frac{16\!\cdots\!70}{16\!\cdots\!67}a^{20}+\frac{30\!\cdots\!49}{33\!\cdots\!34}a^{19}-\frac{13\!\cdots\!39}{33\!\cdots\!34}a^{18}-\frac{12\!\cdots\!09}{33\!\cdots\!34}a^{17}+\frac{36\!\cdots\!11}{16\!\cdots\!67}a^{16}+\frac{34\!\cdots\!77}{16\!\cdots\!67}a^{15}-\frac{33\!\cdots\!69}{33\!\cdots\!34}a^{14}+\frac{12\!\cdots\!89}{33\!\cdots\!34}a^{13}+\frac{16\!\cdots\!35}{33\!\cdots\!34}a^{12}+\frac{13\!\cdots\!41}{33\!\cdots\!34}a^{11}+\frac{67\!\cdots\!51}{33\!\cdots\!34}a^{10}+\frac{27\!\cdots\!55}{16\!\cdots\!67}a^{9}-\frac{10\!\cdots\!44}{16\!\cdots\!67}a^{8}+\frac{98\!\cdots\!99}{33\!\cdots\!34}a^{7}+\frac{80\!\cdots\!11}{16\!\cdots\!67}a^{6}+\frac{82\!\cdots\!25}{16\!\cdots\!67}a^{5}+\frac{97\!\cdots\!24}{16\!\cdots\!67}a^{4}-\frac{42\!\cdots\!73}{16\!\cdots\!67}a^{3}+\frac{68\!\cdots\!81}{33\!\cdots\!34}a^{2}-\frac{20\!\cdots\!28}{16\!\cdots\!67}a+\frac{26\!\cdots\!06}{16\!\cdots\!67}$
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 $1$ (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: | $13$ | 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: |
$\frac{18\!\cdots\!55}{33\!\cdots\!34}a^{21}+\frac{16\!\cdots\!65}{33\!\cdots\!34}a^{20}-\frac{16\!\cdots\!67}{33\!\cdots\!34}a^{19}-\frac{74\!\cdots\!03}{16\!\cdots\!67}a^{18}-\frac{75\!\cdots\!84}{16\!\cdots\!67}a^{17}+\frac{14\!\cdots\!83}{33\!\cdots\!34}a^{16}-\frac{44\!\cdots\!56}{16\!\cdots\!67}a^{15}+\frac{21\!\cdots\!53}{33\!\cdots\!34}a^{14}-\frac{34\!\cdots\!86}{16\!\cdots\!67}a^{13}-\frac{22\!\cdots\!11}{16\!\cdots\!67}a^{12}+\frac{16\!\cdots\!06}{16\!\cdots\!67}a^{11}+\frac{18\!\cdots\!42}{16\!\cdots\!67}a^{10}-\frac{21\!\cdots\!01}{33\!\cdots\!34}a^{9}+\frac{51\!\cdots\!04}{16\!\cdots\!67}a^{8}+\frac{16\!\cdots\!61}{33\!\cdots\!34}a^{7}+\frac{49\!\cdots\!91}{33\!\cdots\!34}a^{6}+\frac{36\!\cdots\!22}{16\!\cdots\!67}a^{5}-\frac{10\!\cdots\!57}{16\!\cdots\!67}a^{4}-\frac{40\!\cdots\!65}{16\!\cdots\!67}a^{3}-\frac{34\!\cdots\!73}{33\!\cdots\!34}a^{2}+\frac{20\!\cdots\!39}{33\!\cdots\!34}a+\frac{11\!\cdots\!84}{16\!\cdots\!67}$, $\frac{32\!\cdots\!15}{33\!\cdots\!34}a^{21}-\frac{11\!\cdots\!65}{33\!\cdots\!34}a^{20}+\frac{32\!\cdots\!53}{16\!\cdots\!67}a^{19}-\frac{13\!\cdots\!01}{16\!\cdots\!67}a^{18}+\frac{96\!\cdots\!69}{33\!\cdots\!34}a^{17}-\frac{26\!\cdots\!95}{16\!\cdots\!67}a^{16}-\frac{64\!\cdots\!45}{33\!\cdots\!34}a^{15}+\frac{38\!\cdots\!09}{33\!\cdots\!34}a^{14}-\frac{14\!\cdots\!23}{16\!\cdots\!67}a^{13}-\frac{60\!\cdots\!89}{33\!\cdots\!34}a^{12}+\frac{72\!\cdots\!01}{33\!\cdots\!34}a^{11}+\frac{27\!\cdots\!21}{33\!\cdots\!34}a^{10}-\frac{14\!\cdots\!12}{16\!\cdots\!67}a^{9}+\frac{25\!\cdots\!63}{33\!\cdots\!34}a^{8}+\frac{44\!\cdots\!87}{33\!\cdots\!34}a^{7}+\frac{31\!\cdots\!21}{33\!\cdots\!34}a^{6}+\frac{68\!\cdots\!67}{33\!\cdots\!34}a^{5}-\frac{57\!\cdots\!18}{16\!\cdots\!67}a^{4}-\frac{55\!\cdots\!31}{16\!\cdots\!67}a^{3}-\frac{81\!\cdots\!11}{33\!\cdots\!34}a^{2}+\frac{18\!\cdots\!33}{33\!\cdots\!34}a+\frac{72\!\cdots\!53}{33\!\cdots\!34}$, $\frac{84\!\cdots\!11}{16\!\cdots\!67}a^{21}+\frac{84\!\cdots\!35}{33\!\cdots\!34}a^{20}+\frac{15\!\cdots\!50}{16\!\cdots\!67}a^{19}-\frac{13\!\cdots\!99}{33\!\cdots\!34}a^{18}-\frac{66\!\cdots\!31}{33\!\cdots\!34}a^{17}-\frac{24\!\cdots\!11}{33\!\cdots\!34}a^{16}-\frac{38\!\cdots\!94}{16\!\cdots\!67}a^{15}+\frac{95\!\cdots\!12}{16\!\cdots\!67}a^{14}+\frac{17\!\cdots\!85}{33\!\cdots\!34}a^{13}-\frac{38\!\cdots\!05}{33\!\cdots\!34}a^{12}+\frac{10\!\cdots\!23}{33\!\cdots\!34}a^{11}+\frac{37\!\cdots\!79}{33\!\cdots\!34}a^{10}-\frac{54\!\cdots\!19}{33\!\cdots\!34}a^{9}+\frac{34\!\cdots\!90}{16\!\cdots\!67}a^{8}+\frac{88\!\cdots\!55}{16\!\cdots\!67}a^{7}+\frac{88\!\cdots\!63}{33\!\cdots\!34}a^{6}+\frac{46\!\cdots\!97}{16\!\cdots\!67}a^{5}+\frac{12\!\cdots\!75}{16\!\cdots\!67}a^{4}-\frac{30\!\cdots\!50}{16\!\cdots\!67}a^{3}-\frac{16\!\cdots\!17}{16\!\cdots\!67}a^{2}+\frac{52\!\cdots\!63}{33\!\cdots\!34}a+\frac{24\!\cdots\!29}{16\!\cdots\!67}$, $\frac{62\!\cdots\!97}{33\!\cdots\!34}a^{21}+\frac{42\!\cdots\!83}{33\!\cdots\!34}a^{20}+\frac{15\!\cdots\!48}{16\!\cdots\!67}a^{19}-\frac{24\!\cdots\!87}{16\!\cdots\!67}a^{18}-\frac{34\!\cdots\!77}{33\!\cdots\!34}a^{17}-\frac{11\!\cdots\!31}{16\!\cdots\!67}a^{16}-\frac{30\!\cdots\!39}{33\!\cdots\!34}a^{15}+\frac{70\!\cdots\!99}{33\!\cdots\!34}a^{14}+\frac{98\!\cdots\!82}{16\!\cdots\!67}a^{13}-\frac{15\!\cdots\!89}{33\!\cdots\!34}a^{12}+\frac{20\!\cdots\!91}{33\!\cdots\!34}a^{11}+\frac{17\!\cdots\!07}{33\!\cdots\!34}a^{10}+\frac{62\!\cdots\!24}{16\!\cdots\!67}a^{9}+\frac{88\!\cdots\!27}{33\!\cdots\!34}a^{8}+\frac{77\!\cdots\!69}{33\!\cdots\!34}a^{7}+\frac{31\!\cdots\!03}{33\!\cdots\!34}a^{6}+\frac{31\!\cdots\!55}{33\!\cdots\!34}a^{5}+\frac{56\!\cdots\!00}{16\!\cdots\!67}a^{4}-\frac{14\!\cdots\!03}{16\!\cdots\!67}a^{3}-\frac{15\!\cdots\!77}{33\!\cdots\!34}a^{2}+\frac{33\!\cdots\!13}{33\!\cdots\!34}a+\frac{27\!\cdots\!59}{33\!\cdots\!34}$, $\frac{15\!\cdots\!29}{33\!\cdots\!34}a^{21}+\frac{96\!\cdots\!31}{33\!\cdots\!34}a^{20}+\frac{24\!\cdots\!75}{33\!\cdots\!34}a^{19}-\frac{61\!\cdots\!43}{16\!\cdots\!67}a^{18}-\frac{38\!\cdots\!88}{16\!\cdots\!67}a^{17}-\frac{19\!\cdots\!71}{33\!\cdots\!34}a^{16}-\frac{39\!\cdots\!14}{16\!\cdots\!67}a^{15}+\frac{17\!\cdots\!95}{33\!\cdots\!34}a^{14}+\frac{18\!\cdots\!86}{16\!\cdots\!67}a^{13}-\frac{18\!\cdots\!76}{16\!\cdots\!67}a^{12}+\frac{36\!\cdots\!00}{16\!\cdots\!67}a^{11}+\frac{19\!\cdots\!52}{16\!\cdots\!67}a^{10}+\frac{11\!\cdots\!03}{33\!\cdots\!34}a^{9}+\frac{27\!\cdots\!62}{16\!\cdots\!67}a^{8}+\frac{18\!\cdots\!35}{33\!\cdots\!34}a^{7}+\frac{86\!\cdots\!37}{33\!\cdots\!34}a^{6}+\frac{47\!\cdots\!78}{16\!\cdots\!67}a^{5}+\frac{16\!\cdots\!45}{16\!\cdots\!67}a^{4}-\frac{29\!\cdots\!11}{16\!\cdots\!67}a^{3}-\frac{31\!\cdots\!59}{33\!\cdots\!34}a^{2}+\frac{49\!\cdots\!45}{33\!\cdots\!34}a+\frac{22\!\cdots\!04}{16\!\cdots\!67}$, $\frac{18\!\cdots\!11}{33\!\cdots\!34}a^{21}+\frac{49\!\cdots\!41}{16\!\cdots\!67}a^{20}+\frac{37\!\cdots\!91}{33\!\cdots\!34}a^{19}-\frac{14\!\cdots\!29}{33\!\cdots\!34}a^{18}-\frac{78\!\cdots\!07}{33\!\cdots\!34}a^{17}-\frac{14\!\cdots\!68}{16\!\cdots\!67}a^{16}-\frac{44\!\cdots\!43}{16\!\cdots\!67}a^{15}+\frac{21\!\cdots\!13}{33\!\cdots\!34}a^{14}+\frac{24\!\cdots\!99}{33\!\cdots\!34}a^{13}-\frac{43\!\cdots\!09}{33\!\cdots\!34}a^{12}+\frac{11\!\cdots\!39}{33\!\cdots\!34}a^{11}+\frac{42\!\cdots\!95}{33\!\cdots\!34}a^{10}-\frac{41\!\cdots\!54}{16\!\cdots\!67}a^{9}+\frac{41\!\cdots\!11}{16\!\cdots\!67}a^{8}+\frac{20\!\cdots\!07}{33\!\cdots\!34}a^{7}+\frac{52\!\cdots\!73}{16\!\cdots\!67}a^{6}+\frac{55\!\cdots\!42}{16\!\cdots\!67}a^{5}+\frac{16\!\cdots\!60}{16\!\cdots\!67}a^{4}-\frac{32\!\cdots\!36}{16\!\cdots\!67}a^{3}-\frac{35\!\cdots\!75}{33\!\cdots\!34}a^{2}+\frac{27\!\cdots\!76}{16\!\cdots\!67}a+\frac{24\!\cdots\!40}{16\!\cdots\!67}$, $\frac{16\!\cdots\!65}{33\!\cdots\!34}a^{21}-\frac{16\!\cdots\!67}{33\!\cdots\!34}a^{20}+\frac{80\!\cdots\!17}{16\!\cdots\!67}a^{19}-\frac{75\!\cdots\!84}{16\!\cdots\!67}a^{18}+\frac{14\!\cdots\!83}{33\!\cdots\!34}a^{17}-\frac{70\!\cdots\!46}{16\!\cdots\!67}a^{16}+\frac{70\!\cdots\!73}{33\!\cdots\!34}a^{15}+\frac{18\!\cdots\!13}{33\!\cdots\!34}a^{14}-\frac{13\!\cdots\!41}{16\!\cdots\!67}a^{13}-\frac{12\!\cdots\!33}{33\!\cdots\!34}a^{12}+\frac{42\!\cdots\!49}{33\!\cdots\!34}a^{11}-\frac{12\!\cdots\!61}{33\!\cdots\!34}a^{10}-\frac{30\!\cdots\!91}{16\!\cdots\!67}a^{9}+\frac{15\!\cdots\!71}{33\!\cdots\!34}a^{8}-\frac{25\!\cdots\!29}{33\!\cdots\!34}a^{7}+\frac{32\!\cdots\!09}{33\!\cdots\!34}a^{6}+\frac{31\!\cdots\!01}{33\!\cdots\!34}a^{5}-\frac{26\!\cdots\!65}{16\!\cdots\!67}a^{4}-\frac{77\!\cdots\!09}{16\!\cdots\!67}a^{3}+\frac{13\!\cdots\!89}{33\!\cdots\!34}a^{2}+\frac{34\!\cdots\!13}{33\!\cdots\!34}a+\frac{52\!\cdots\!89}{33\!\cdots\!34}$, $\frac{65\!\cdots\!29}{33\!\cdots\!34}a^{21}+\frac{19\!\cdots\!91}{16\!\cdots\!67}a^{20}+\frac{31\!\cdots\!89}{16\!\cdots\!67}a^{19}-\frac{52\!\cdots\!25}{33\!\cdots\!34}a^{18}-\frac{15\!\cdots\!51}{16\!\cdots\!67}a^{17}-\frac{50\!\cdots\!87}{33\!\cdots\!34}a^{16}-\frac{29\!\cdots\!39}{33\!\cdots\!34}a^{15}+\frac{74\!\cdots\!39}{33\!\cdots\!34}a^{14}+\frac{14\!\cdots\!95}{33\!\cdots\!34}a^{13}-\frac{80\!\cdots\!18}{16\!\cdots\!67}a^{12}+\frac{14\!\cdots\!03}{16\!\cdots\!67}a^{11}+\frac{85\!\cdots\!04}{16\!\cdots\!67}a^{10}+\frac{21\!\cdots\!65}{33\!\cdots\!34}a^{9}+\frac{11\!\cdots\!97}{33\!\cdots\!34}a^{8}+\frac{76\!\cdots\!91}{33\!\cdots\!34}a^{7}+\frac{16\!\cdots\!58}{16\!\cdots\!67}a^{6}+\frac{32\!\cdots\!41}{33\!\cdots\!34}a^{5}+\frac{51\!\cdots\!44}{16\!\cdots\!67}a^{4}-\frac{14\!\cdots\!39}{16\!\cdots\!67}a^{3}-\frac{15\!\cdots\!51}{33\!\cdots\!34}a^{2}+\frac{14\!\cdots\!51}{16\!\cdots\!67}a+\frac{29\!\cdots\!49}{33\!\cdots\!34}$, $\frac{14\!\cdots\!95}{33\!\cdots\!34}a^{21}+\frac{33\!\cdots\!16}{16\!\cdots\!67}a^{20}+\frac{16\!\cdots\!85}{33\!\cdots\!34}a^{19}-\frac{11\!\cdots\!67}{33\!\cdots\!34}a^{18}-\frac{53\!\cdots\!87}{33\!\cdots\!34}a^{17}-\frac{62\!\cdots\!26}{16\!\cdots\!67}a^{16}-\frac{35\!\cdots\!70}{16\!\cdots\!67}a^{15}+\frac{16\!\cdots\!33}{33\!\cdots\!34}a^{14}+\frac{70\!\cdots\!63}{33\!\cdots\!34}a^{13}-\frac{34\!\cdots\!57}{33\!\cdots\!34}a^{12}+\frac{11\!\cdots\!77}{33\!\cdots\!34}a^{11}+\frac{33\!\cdots\!95}{33\!\cdots\!34}a^{10}-\frac{18\!\cdots\!91}{16\!\cdots\!67}a^{9}+\frac{32\!\cdots\!06}{16\!\cdots\!67}a^{8}+\frac{15\!\cdots\!91}{33\!\cdots\!34}a^{7}+\frac{30\!\cdots\!86}{16\!\cdots\!67}a^{6}+\frac{39\!\cdots\!50}{16\!\cdots\!67}a^{5}+\frac{65\!\cdots\!23}{16\!\cdots\!67}a^{4}-\frac{28\!\cdots\!28}{16\!\cdots\!67}a^{3}-\frac{24\!\cdots\!87}{33\!\cdots\!34}a^{2}+\frac{29\!\cdots\!28}{16\!\cdots\!67}a+\frac{18\!\cdots\!09}{16\!\cdots\!67}$, $\frac{39\!\cdots\!54}{16\!\cdots\!67}a^{21}+\frac{32\!\cdots\!44}{16\!\cdots\!67}a^{20}+\frac{20\!\cdots\!41}{33\!\cdots\!34}a^{19}-\frac{31\!\cdots\!05}{16\!\cdots\!67}a^{18}-\frac{44\!\cdots\!97}{33\!\cdots\!34}a^{17}-\frac{17\!\cdots\!85}{33\!\cdots\!34}a^{16}-\frac{30\!\cdots\!11}{33\!\cdots\!34}a^{15}+\frac{45\!\cdots\!39}{16\!\cdots\!67}a^{14}-\frac{15\!\cdots\!23}{16\!\cdots\!67}a^{13}-\frac{16\!\cdots\!77}{33\!\cdots\!34}a^{12}+\frac{11\!\cdots\!71}{33\!\cdots\!34}a^{11}+\frac{11\!\cdots\!41}{33\!\cdots\!34}a^{10}-\frac{38\!\cdots\!79}{33\!\cdots\!34}a^{9}+\frac{62\!\cdots\!63}{33\!\cdots\!34}a^{8}+\frac{26\!\cdots\!37}{16\!\cdots\!67}a^{7}+\frac{13\!\cdots\!22}{16\!\cdots\!67}a^{6}+\frac{40\!\cdots\!89}{33\!\cdots\!34}a^{5}-\frac{13\!\cdots\!62}{16\!\cdots\!67}a^{4}-\frac{11\!\cdots\!68}{16\!\cdots\!67}a^{3}-\frac{30\!\cdots\!63}{16\!\cdots\!67}a^{2}+\frac{11\!\cdots\!45}{16\!\cdots\!67}a+\frac{69\!\cdots\!87}{33\!\cdots\!34}$, $\frac{24\!\cdots\!68}{16\!\cdots\!67}a^{21}+\frac{25\!\cdots\!75}{33\!\cdots\!34}a^{20}+\frac{41\!\cdots\!88}{16\!\cdots\!67}a^{19}-\frac{38\!\cdots\!71}{33\!\cdots\!34}a^{18}-\frac{20\!\cdots\!55}{33\!\cdots\!34}a^{17}-\frac{64\!\cdots\!91}{33\!\cdots\!34}a^{16}-\frac{12\!\cdots\!57}{16\!\cdots\!67}a^{15}+\frac{27\!\cdots\!96}{16\!\cdots\!67}a^{14}+\frac{58\!\cdots\!07}{33\!\cdots\!34}a^{13}-\frac{11\!\cdots\!25}{33\!\cdots\!34}a^{12}+\frac{32\!\cdots\!07}{33\!\cdots\!34}a^{11}+\frac{11\!\cdots\!27}{33\!\cdots\!34}a^{10}-\frac{46\!\cdots\!11}{33\!\cdots\!34}a^{9}+\frac{11\!\cdots\!67}{16\!\cdots\!67}a^{8}+\frac{26\!\cdots\!06}{16\!\cdots\!67}a^{7}+\frac{24\!\cdots\!15}{33\!\cdots\!34}a^{6}+\frac{14\!\cdots\!00}{16\!\cdots\!67}a^{5}+\frac{37\!\cdots\!15}{16\!\cdots\!67}a^{4}-\frac{88\!\cdots\!09}{16\!\cdots\!67}a^{3}-\frac{43\!\cdots\!45}{16\!\cdots\!67}a^{2}+\frac{14\!\cdots\!33}{33\!\cdots\!34}a+\frac{62\!\cdots\!59}{16\!\cdots\!67}$, $\frac{82\!\cdots\!09}{33\!\cdots\!34}a^{21}+\frac{37\!\cdots\!37}{33\!\cdots\!34}a^{20}+\frac{15\!\cdots\!55}{33\!\cdots\!34}a^{19}-\frac{32\!\cdots\!74}{16\!\cdots\!67}a^{18}-\frac{14\!\cdots\!26}{16\!\cdots\!67}a^{17}-\frac{12\!\cdots\!13}{33\!\cdots\!34}a^{16}-\frac{19\!\cdots\!01}{16\!\cdots\!67}a^{15}+\frac{94\!\cdots\!53}{33\!\cdots\!34}a^{14}+\frac{19\!\cdots\!92}{16\!\cdots\!67}a^{13}-\frac{94\!\cdots\!76}{16\!\cdots\!67}a^{12}+\frac{31\!\cdots\!82}{16\!\cdots\!67}a^{11}+\frac{89\!\cdots\!13}{16\!\cdots\!67}a^{10}-\frac{11\!\cdots\!55}{33\!\cdots\!34}a^{9}+\frac{20\!\cdots\!03}{16\!\cdots\!67}a^{8}+\frac{84\!\cdots\!03}{33\!\cdots\!34}a^{7}+\frac{40\!\cdots\!73}{33\!\cdots\!34}a^{6}+\frac{23\!\cdots\!00}{16\!\cdots\!67}a^{5}+\frac{53\!\cdots\!49}{16\!\cdots\!67}a^{4}-\frac{14\!\cdots\!80}{16\!\cdots\!67}a^{3}-\frac{13\!\cdots\!69}{33\!\cdots\!34}a^{2}+\frac{24\!\cdots\!05}{33\!\cdots\!34}a+\frac{99\!\cdots\!10}{16\!\cdots\!67}$, $\frac{64\!\cdots\!89}{33\!\cdots\!34}a^{21}+\frac{13\!\cdots\!27}{16\!\cdots\!67}a^{20}+\frac{13\!\cdots\!21}{33\!\cdots\!34}a^{19}-\frac{50\!\cdots\!55}{33\!\cdots\!34}a^{18}-\frac{21\!\cdots\!15}{33\!\cdots\!34}a^{17}-\frac{51\!\cdots\!09}{16\!\cdots\!67}a^{16}-\frac{15\!\cdots\!33}{16\!\cdots\!67}a^{15}+\frac{72\!\cdots\!03}{33\!\cdots\!34}a^{14}+\frac{66\!\cdots\!79}{33\!\cdots\!34}a^{13}-\frac{14\!\cdots\!87}{33\!\cdots\!34}a^{12}+\frac{53\!\cdots\!87}{33\!\cdots\!34}a^{11}+\frac{13\!\cdots\!47}{33\!\cdots\!34}a^{10}-\frac{53\!\cdots\!74}{16\!\cdots\!67}a^{9}+\frac{17\!\cdots\!44}{16\!\cdots\!67}a^{8}+\frac{63\!\cdots\!11}{33\!\cdots\!34}a^{7}+\frac{15\!\cdots\!92}{16\!\cdots\!67}a^{6}+\frac{19\!\cdots\!35}{16\!\cdots\!67}a^{5}+\frac{39\!\cdots\!09}{16\!\cdots\!67}a^{4}-\frac{10\!\cdots\!31}{16\!\cdots\!67}a^{3}-\frac{97\!\cdots\!61}{33\!\cdots\!34}a^{2}+\frac{89\!\cdots\!64}{16\!\cdots\!67}a+\frac{67\!\cdots\!29}{16\!\cdots\!67}$
| 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: | \( 376545.621481 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{8}\cdot 376545.621481 \cdot 1}{2\cdot\sqrt{30428929951449912919970703125}}\cr\approx \mathstrut & 0.167788884552 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 8*x^19 - 4*x^16 + 116*x^15 - 47*x^14 - 228*x^13 + 179*x^12 + 177*x^11 - 108*x^10 + 58*x^9 + 78*x^8 + 4*x^7 + 37*x^6 - 13*x^5 - 40*x^4 - x^3 + 10*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^22 - 8*x^19 - 4*x^16 + 116*x^15 - 47*x^14 - 228*x^13 + 179*x^12 + 177*x^11 - 108*x^10 + 58*x^9 + 78*x^8 + 4*x^7 + 37*x^6 - 13*x^5 - 40*x^4 - x^3 + 10*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^22 - 8*x^19 - 4*x^16 + 116*x^15 - 47*x^14 - 228*x^13 + 179*x^12 + 177*x^11 - 108*x^10 + 58*x^9 + 78*x^8 + 4*x^7 + 37*x^6 - 13*x^5 - 40*x^4 - x^3 + 10*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^22 - 8*x^19 - 4*x^16 + 116*x^15 - 47*x^14 - 228*x^13 + 179*x^12 + 177*x^11 - 108*x^10 + 58*x^9 + 78*x^8 + 4*x^7 + 37*x^6 - 13*x^5 - 40*x^4 - x^3 + 10*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
$C_2\times S_{11}$ (as 22T47):
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 79833600 |
| The 112 conjugacy class representatives for $C_2\times S_{11}$ |
| Character table for $C_2\times S_{11}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 11.3.24963663301.1 |
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 22 sibling: | data not computed |
| Degree 44 sibling: | data not computed |
| 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 | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.6.0.1}{6} }^{2}$ | $22$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}{,}\,{\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.6.0.1}{6} }$ | $18{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.7.0.1}{7} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | $18{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.10.0.1}{10} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.7.0.1}{7} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.22.11.1 | $x^{22} + 220 x^{21} + 22055 x^{20} + 1331000 x^{19} + 53791375 x^{18} + 1531447500 x^{17} + 31435820625 x^{16} + 467679300000 x^{15} + 4991151206250 x^{14} + 37171668875000 x^{13} + 183624733943756 x^{12} + 553513923250726 x^{11} + 918123669784090 x^{10} + 929291725767350 x^{9} + 623894056087500 x^{8} + 292303912609500 x^{7} + 98324330218125 x^{6} + 25190924781000 x^{5} + 17099014728125 x^{4} + 90189081743750 x^{3} + 391939091809384 x^{2} + 906877245981448 x + 669277565422109$ | $2$ | $11$ | $11$ | 22T1 | $[\ ]_{2}^{11}$ |
|
\(971\)
| $\Q_{971}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{971}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{971}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{971}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
|
\(25709231\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |