Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 7*x^20 - 4*x^19 + 31*x^18 - 18*x^17 + 77*x^16 - 41*x^15 + 140*x^14 - 96*x^13 + 153*x^12 - 129*x^11 + 145*x^10 - 148*x^9 + 84*x^8 - 72*x^7 + 94*x^6 - 20*x^5 + 39*x^4 + 12*x^3 + 18*x^2 + 4*x + 1)
gp: K = bnfinit(y^22 - y^21 + 7*y^20 - 4*y^19 + 31*y^18 - 18*y^17 + 77*y^16 - 41*y^15 + 140*y^14 - 96*y^13 + 153*y^12 - 129*y^11 + 145*y^10 - 148*y^9 + 84*y^8 - 72*y^7 + 94*y^6 - 20*y^5 + 39*y^4 + 12*y^3 + 18*y^2 + 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - x^21 + 7*x^20 - 4*x^19 + 31*x^18 - 18*x^17 + 77*x^16 - 41*x^15 + 140*x^14 - 96*x^13 + 153*x^12 - 129*x^11 + 145*x^10 - 148*x^9 + 84*x^8 - 72*x^7 + 94*x^6 - 20*x^5 + 39*x^4 + 12*x^3 + 18*x^2 + 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - x^21 + 7*x^20 - 4*x^19 + 31*x^18 - 18*x^17 + 77*x^16 - 41*x^15 + 140*x^14 - 96*x^13 + 153*x^12 - 129*x^11 + 145*x^10 - 148*x^9 + 84*x^8 - 72*x^7 + 94*x^6 - 20*x^5 + 39*x^4 + 12*x^3 + 18*x^2 + 4*x + 1)
\( x^{22} - x^{21} + 7 x^{20} - 4 x^{19} + 31 x^{18} - 18 x^{17} + 77 x^{16} - 41 x^{15} + 140 x^{14} + \cdots + 1 \)
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: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-10046547996724887091059294601443\)
\(\medspace = -\,3^{11}\cdot 29^{2}\cdot 131^{2}\cdot 5399^{2}\cdot 367163^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(25.66\) | 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: | $3^{1/2}29^{1/2}131^{1/2}5399^{1/2}367163^{1/2}\approx 4753148.607259087$ | ||
| Ramified primes: |
\(3\), \(29\), \(131\), \(5399\), \(367163\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{13\!\cdots\!21}a^{21}+\frac{12\!\cdots\!28}{13\!\cdots\!21}a^{20}-\frac{50\!\cdots\!81}{13\!\cdots\!21}a^{19}+\frac{265569078960793}{13\!\cdots\!21}a^{18}+\frac{140683497308997}{13\!\cdots\!21}a^{17}+\frac{30\!\cdots\!62}{13\!\cdots\!21}a^{16}+\frac{704891312247647}{13\!\cdots\!21}a^{15}+\frac{36\!\cdots\!76}{13\!\cdots\!21}a^{14}-\frac{275290191621555}{13\!\cdots\!21}a^{13}+\frac{65\!\cdots\!54}{13\!\cdots\!21}a^{12}-\frac{23\!\cdots\!02}{13\!\cdots\!21}a^{11}-\frac{24\!\cdots\!95}{13\!\cdots\!21}a^{10}+\frac{32\!\cdots\!29}{13\!\cdots\!21}a^{9}-\frac{37\!\cdots\!10}{13\!\cdots\!21}a^{8}+\frac{57\!\cdots\!06}{13\!\cdots\!21}a^{7}-\frac{14\!\cdots\!27}{13\!\cdots\!21}a^{6}-\frac{259556439495174}{13\!\cdots\!21}a^{5}-\frac{18\!\cdots\!60}{13\!\cdots\!21}a^{4}+\frac{66\!\cdots\!72}{13\!\cdots\!21}a^{3}+\frac{28\!\cdots\!42}{13\!\cdots\!21}a^{2}-\frac{22\!\cdots\!60}{13\!\cdots\!21}a-\frac{42\!\cdots\!69}{13\!\cdots\!21}$
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: | $10$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{2685629293987735}{13677239212089721} a^{21} + \frac{2425506734097330}{13677239212089721} a^{20} - \frac{18387432369558813}{13677239212089721} a^{19} + \frac{9109022292712439}{13677239212089721} a^{18} - \frac{81671737681889821}{13677239212089721} a^{17} + \frac{42322341360437781}{13677239212089721} a^{16} - \frac{200264700221761619}{13677239212089721} a^{15} + \frac{99373435198554347}{13677239212089721} a^{14} - \frac{366524544292272972}{13677239212089721} a^{13} + \frac{247513061564379161}{13677239212089721} a^{12} - \frac{397088590715102046}{13677239212089721} a^{11} + \frac{353837120676200861}{13677239212089721} a^{10} - \frac{398897307145589961}{13677239212089721} a^{9} + \frac{417045500203026235}{13677239212089721} a^{8} - \frac{248328683492113525}{13677239212089721} a^{7} + \frac{224633125767905671}{13677239212089721} a^{6} - \frac{300744197938415810}{13677239212089721} a^{5} + \frac{63232402023086114}{13677239212089721} a^{4} - \frac{120862632834872891}{13677239212089721} a^{3} - \frac{7178304024482238}{13677239212089721} a^{2} - \frac{55662381431239864}{13677239212089721} a + \frac{1319886993338294}{13677239212089721} \)
(order $6$)
| 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: |
$\frac{12\!\cdots\!27}{13\!\cdots\!21}a^{21}-\frac{15\!\cdots\!62}{13\!\cdots\!21}a^{20}+\frac{88\!\cdots\!19}{13\!\cdots\!21}a^{19}-\frac{67\!\cdots\!21}{13\!\cdots\!21}a^{18}+\frac{39\!\cdots\!76}{13\!\cdots\!21}a^{17}-\frac{30\!\cdots\!07}{13\!\cdots\!21}a^{16}+\frac{99\!\cdots\!60}{13\!\cdots\!21}a^{15}-\frac{70\!\cdots\!26}{13\!\cdots\!21}a^{14}+\frac{18\!\cdots\!27}{13\!\cdots\!21}a^{13}-\frac{15\!\cdots\!64}{13\!\cdots\!21}a^{12}+\frac{21\!\cdots\!92}{13\!\cdots\!21}a^{11}-\frac{19\!\cdots\!29}{13\!\cdots\!21}a^{10}+\frac{21\!\cdots\!76}{13\!\cdots\!21}a^{9}-\frac{22\!\cdots\!57}{13\!\cdots\!21}a^{8}+\frac{14\!\cdots\!03}{13\!\cdots\!21}a^{7}-\frac{11\!\cdots\!69}{13\!\cdots\!21}a^{6}+\frac{13\!\cdots\!09}{13\!\cdots\!21}a^{5}-\frac{54\!\cdots\!50}{13\!\cdots\!21}a^{4}+\frac{54\!\cdots\!67}{13\!\cdots\!21}a^{3}+\frac{27\!\cdots\!33}{13\!\cdots\!21}a^{2}+\frac{21\!\cdots\!48}{13\!\cdots\!21}a-\frac{62\!\cdots\!56}{13\!\cdots\!21}$, $\frac{11\!\cdots\!45}{13\!\cdots\!21}a^{21}-\frac{13\!\cdots\!06}{13\!\cdots\!21}a^{20}+\frac{83\!\cdots\!88}{13\!\cdots\!21}a^{19}-\frac{58\!\cdots\!17}{13\!\cdots\!21}a^{18}+\frac{35\!\cdots\!25}{13\!\cdots\!21}a^{17}-\frac{25\!\cdots\!64}{13\!\cdots\!21}a^{16}+\frac{86\!\cdots\!67}{13\!\cdots\!21}a^{15}-\frac{55\!\cdots\!67}{13\!\cdots\!21}a^{14}+\frac{14\!\cdots\!43}{13\!\cdots\!21}a^{13}-\frac{11\!\cdots\!65}{13\!\cdots\!21}a^{12}+\frac{15\!\cdots\!38}{13\!\cdots\!21}a^{11}-\frac{14\!\cdots\!89}{13\!\cdots\!21}a^{10}+\frac{12\!\cdots\!01}{13\!\cdots\!21}a^{9}-\frac{14\!\cdots\!93}{13\!\cdots\!21}a^{8}+\frac{85\!\cdots\!08}{13\!\cdots\!21}a^{7}-\frac{54\!\cdots\!13}{13\!\cdots\!21}a^{6}+\frac{77\!\cdots\!00}{13\!\cdots\!21}a^{5}-\frac{331994046173336}{13\!\cdots\!21}a^{4}+\frac{35\!\cdots\!01}{13\!\cdots\!21}a^{3}+\frac{60\!\cdots\!60}{13\!\cdots\!21}a^{2}-\frac{26\!\cdots\!24}{13\!\cdots\!21}a-\frac{113983791873586}{13\!\cdots\!21}$, $\frac{25\!\cdots\!60}{13\!\cdots\!21}a^{21}-\frac{11\!\cdots\!98}{13\!\cdots\!21}a^{20}+\frac{16\!\cdots\!72}{13\!\cdots\!21}a^{19}-\frac{16\!\cdots\!65}{13\!\cdots\!21}a^{18}+\frac{75\!\cdots\!16}{13\!\cdots\!21}a^{17}-\frac{11\!\cdots\!02}{13\!\cdots\!21}a^{16}+\frac{17\!\cdots\!55}{13\!\cdots\!21}a^{15}-\frac{32\!\cdots\!63}{13\!\cdots\!21}a^{14}+\frac{32\!\cdots\!00}{13\!\cdots\!21}a^{13}-\frac{13\!\cdots\!21}{13\!\cdots\!21}a^{12}+\frac{30\!\cdots\!55}{13\!\cdots\!21}a^{11}-\frac{25\!\cdots\!41}{13\!\cdots\!21}a^{10}+\frac{29\!\cdots\!09}{13\!\cdots\!21}a^{9}-\frac{31\!\cdots\!34}{13\!\cdots\!21}a^{8}+\frac{14\!\cdots\!83}{13\!\cdots\!21}a^{7}-\frac{17\!\cdots\!85}{13\!\cdots\!21}a^{6}+\frac{27\!\cdots\!45}{13\!\cdots\!21}a^{5}+\frac{16\!\cdots\!12}{13\!\cdots\!21}a^{4}+\frac{11\!\cdots\!29}{13\!\cdots\!21}a^{3}+\frac{99\!\cdots\!58}{13\!\cdots\!21}a^{2}+\frac{59\!\cdots\!06}{13\!\cdots\!21}a-\frac{10\!\cdots\!77}{13\!\cdots\!21}$, $\frac{525956480228918}{13\!\cdots\!21}a^{21}+\frac{822823643940461}{13\!\cdots\!21}a^{20}+\frac{15\!\cdots\!41}{13\!\cdots\!21}a^{19}+\frac{83\!\cdots\!86}{13\!\cdots\!21}a^{18}+\frac{55\!\cdots\!84}{13\!\cdots\!21}a^{17}+\frac{36\!\cdots\!98}{13\!\cdots\!21}a^{16}-\frac{68\!\cdots\!31}{13\!\cdots\!21}a^{15}+\frac{98\!\cdots\!14}{13\!\cdots\!21}a^{14}-\frac{40\!\cdots\!99}{13\!\cdots\!21}a^{13}+\frac{16\!\cdots\!89}{13\!\cdots\!21}a^{12}-\frac{15\!\cdots\!84}{13\!\cdots\!21}a^{11}+\frac{20\!\cdots\!10}{13\!\cdots\!21}a^{10}-\frac{20\!\cdots\!99}{13\!\cdots\!21}a^{9}+\frac{18\!\cdots\!29}{13\!\cdots\!21}a^{8}-\frac{23\!\cdots\!00}{13\!\cdots\!21}a^{7}+\frac{15\!\cdots\!20}{13\!\cdots\!21}a^{6}-\frac{74\!\cdots\!69}{13\!\cdots\!21}a^{5}+\frac{12\!\cdots\!83}{13\!\cdots\!21}a^{4}-\frac{25\!\cdots\!38}{13\!\cdots\!21}a^{3}+\frac{60\!\cdots\!91}{13\!\cdots\!21}a^{2}+\frac{13\!\cdots\!50}{13\!\cdots\!21}a-\frac{10\!\cdots\!68}{13\!\cdots\!21}$, $\frac{21\!\cdots\!98}{13\!\cdots\!21}a^{21}+\frac{10\!\cdots\!61}{13\!\cdots\!21}a^{20}+\frac{10\!\cdots\!96}{13\!\cdots\!21}a^{19}+\frac{15\!\cdots\!44}{13\!\cdots\!21}a^{18}+\frac{41\!\cdots\!31}{13\!\cdots\!21}a^{17}+\frac{72\!\cdots\!67}{13\!\cdots\!21}a^{16}+\frac{51\!\cdots\!66}{13\!\cdots\!21}a^{15}+\frac{20\!\cdots\!03}{13\!\cdots\!21}a^{14}+\frac{19\!\cdots\!23}{13\!\cdots\!21}a^{13}+\frac{36\!\cdots\!28}{13\!\cdots\!21}a^{12}-\frac{25\!\cdots\!85}{13\!\cdots\!21}a^{11}+\frac{45\!\cdots\!64}{13\!\cdots\!21}a^{10}-\frac{42\!\cdots\!19}{13\!\cdots\!21}a^{9}+\frac{44\!\cdots\!31}{13\!\cdots\!21}a^{8}-\frac{59\!\cdots\!95}{13\!\cdots\!21}a^{7}+\frac{40\!\cdots\!43}{13\!\cdots\!21}a^{6}-\frac{19\!\cdots\!31}{13\!\cdots\!21}a^{5}+\frac{38\!\cdots\!64}{13\!\cdots\!21}a^{4}-\frac{12\!\cdots\!12}{13\!\cdots\!21}a^{3}+\frac{18\!\cdots\!67}{13\!\cdots\!21}a^{2}+\frac{41\!\cdots\!48}{13\!\cdots\!21}a+\frac{23\!\cdots\!70}{13\!\cdots\!21}$, $\frac{260122559890405}{13\!\cdots\!21}a^{21}-\frac{411972688355332}{13\!\cdots\!21}a^{20}+\frac{16\!\cdots\!01}{13\!\cdots\!21}a^{19}-\frac{15\!\cdots\!64}{13\!\cdots\!21}a^{18}+\frac{60\!\cdots\!49}{13\!\cdots\!21}a^{17}-\frac{65\!\cdots\!76}{13\!\cdots\!21}a^{16}+\frac{10\!\cdots\!88}{13\!\cdots\!21}a^{15}-\frac{94\!\cdots\!28}{13\!\cdots\!21}a^{14}+\frac{10\!\cdots\!99}{13\!\cdots\!21}a^{13}-\frac{13\!\cdots\!09}{13\!\cdots\!21}a^{12}-\frac{73\!\cdots\!46}{13\!\cdots\!21}a^{11}+\frac{94\!\cdots\!86}{13\!\cdots\!21}a^{10}-\frac{19\!\cdots\!55}{13\!\cdots\!21}a^{9}+\frac{22\!\cdots\!85}{13\!\cdots\!21}a^{8}-\frac{31\!\cdots\!51}{13\!\cdots\!21}a^{7}+\frac{48\!\cdots\!20}{13\!\cdots\!21}a^{6}-\frac{95\!\cdots\!14}{13\!\cdots\!21}a^{5}+\frac{16\!\cdots\!26}{13\!\cdots\!21}a^{4}-\frac{25\!\cdots\!82}{13\!\cdots\!21}a^{3}+\frac{73\!\cdots\!34}{13\!\cdots\!21}a^{2}+\frac{16\!\cdots\!87}{13\!\cdots\!21}a-\frac{16\!\cdots\!56}{13\!\cdots\!21}$, $\frac{25\!\cdots\!18}{13\!\cdots\!21}a^{21}+\frac{24\!\cdots\!60}{13\!\cdots\!21}a^{20}+\frac{95\!\cdots\!69}{13\!\cdots\!21}a^{19}+\frac{28\!\cdots\!63}{13\!\cdots\!21}a^{18}+\frac{36\!\cdots\!29}{13\!\cdots\!21}a^{17}+\frac{12\!\cdots\!36}{13\!\cdots\!21}a^{16}+\frac{82\!\cdots\!51}{13\!\cdots\!21}a^{15}+\frac{33\!\cdots\!96}{13\!\cdots\!21}a^{14}-\frac{78\!\cdots\!09}{13\!\cdots\!21}a^{13}+\frac{57\!\cdots\!79}{13\!\cdots\!21}a^{12}-\frac{48\!\cdots\!06}{13\!\cdots\!21}a^{11}+\frac{71\!\cdots\!34}{13\!\cdots\!21}a^{10}-\frac{66\!\cdots\!37}{13\!\cdots\!21}a^{9}+\frac{66\!\cdots\!97}{13\!\cdots\!21}a^{8}-\frac{86\!\cdots\!03}{13\!\cdots\!21}a^{7}+\frac{60\!\cdots\!95}{13\!\cdots\!21}a^{6}-\frac{27\!\cdots\!94}{13\!\cdots\!21}a^{5}+\frac{50\!\cdots\!30}{13\!\cdots\!21}a^{4}-\frac{22\!\cdots\!75}{13\!\cdots\!21}a^{3}+\frac{24\!\cdots\!82}{13\!\cdots\!21}a^{2}+\frac{54\!\cdots\!03}{13\!\cdots\!21}a+\frac{38\!\cdots\!33}{13\!\cdots\!21}$, $\frac{17\!\cdots\!11}{13\!\cdots\!21}a^{21}-\frac{18\!\cdots\!43}{13\!\cdots\!21}a^{20}+\frac{12\!\cdots\!87}{13\!\cdots\!21}a^{19}-\frac{72\!\cdots\!67}{13\!\cdots\!21}a^{18}+\frac{53\!\cdots\!32}{13\!\cdots\!21}a^{17}-\frac{32\!\cdots\!63}{13\!\cdots\!21}a^{16}+\frac{13\!\cdots\!41}{13\!\cdots\!21}a^{15}-\frac{74\!\cdots\!96}{13\!\cdots\!21}a^{14}+\frac{23\!\cdots\!74}{13\!\cdots\!21}a^{13}-\frac{17\!\cdots\!15}{13\!\cdots\!21}a^{12}+\frac{25\!\cdots\!62}{13\!\cdots\!21}a^{11}-\frac{22\!\cdots\!51}{13\!\cdots\!21}a^{10}+\frac{24\!\cdots\!73}{13\!\cdots\!21}a^{9}-\frac{25\!\cdots\!53}{13\!\cdots\!21}a^{8}+\frac{13\!\cdots\!23}{13\!\cdots\!21}a^{7}-\frac{11\!\cdots\!21}{13\!\cdots\!21}a^{6}+\frac{15\!\cdots\!60}{13\!\cdots\!21}a^{5}-\frac{30\!\cdots\!60}{13\!\cdots\!21}a^{4}+\frac{54\!\cdots\!43}{13\!\cdots\!21}a^{3}+\frac{20\!\cdots\!64}{13\!\cdots\!21}a^{2}+\frac{24\!\cdots\!98}{13\!\cdots\!21}a+\frac{54\!\cdots\!15}{13\!\cdots\!21}$, $\frac{75\!\cdots\!36}{13\!\cdots\!21}a^{21}-\frac{11\!\cdots\!92}{13\!\cdots\!21}a^{20}+\frac{56\!\cdots\!99}{13\!\cdots\!21}a^{19}-\frac{54\!\cdots\!58}{13\!\cdots\!21}a^{18}+\frac{24\!\cdots\!24}{13\!\cdots\!21}a^{17}-\frac{23\!\cdots\!61}{13\!\cdots\!21}a^{16}+\frac{63\!\cdots\!42}{13\!\cdots\!21}a^{15}-\frac{55\!\cdots\!22}{13\!\cdots\!21}a^{14}+\frac{11\!\cdots\!29}{13\!\cdots\!21}a^{13}-\frac{11\!\cdots\!22}{13\!\cdots\!21}a^{12}+\frac{14\!\cdots\!90}{13\!\cdots\!21}a^{11}-\frac{13\!\cdots\!82}{13\!\cdots\!21}a^{10}+\frac{13\!\cdots\!86}{13\!\cdots\!21}a^{9}-\frac{14\!\cdots\!84}{13\!\cdots\!21}a^{8}+\frac{10\!\cdots\!46}{13\!\cdots\!21}a^{7}-\frac{69\!\cdots\!02}{13\!\cdots\!21}a^{6}+\frac{79\!\cdots\!74}{13\!\cdots\!21}a^{5}-\frac{36\!\cdots\!47}{13\!\cdots\!21}a^{4}+\frac{30\!\cdots\!42}{13\!\cdots\!21}a^{3}+\frac{16\!\cdots\!24}{13\!\cdots\!21}a^{2}+\frac{56\!\cdots\!39}{13\!\cdots\!21}a-\frac{34\!\cdots\!66}{13\!\cdots\!21}$, $\frac{34\!\cdots\!78}{13\!\cdots\!21}a^{21}-\frac{44\!\cdots\!31}{13\!\cdots\!21}a^{20}+\frac{23\!\cdots\!98}{13\!\cdots\!21}a^{19}-\frac{15\!\cdots\!14}{13\!\cdots\!21}a^{18}+\frac{92\!\cdots\!82}{13\!\cdots\!21}a^{17}-\frac{58\!\cdots\!99}{13\!\cdots\!21}a^{16}+\frac{19\!\cdots\!37}{13\!\cdots\!21}a^{15}-\frac{66\!\cdots\!01}{13\!\cdots\!21}a^{14}+\frac{24\!\cdots\!15}{13\!\cdots\!21}a^{13}-\frac{78\!\cdots\!38}{13\!\cdots\!21}a^{12}+\frac{51\!\cdots\!41}{13\!\cdots\!21}a^{11}+\frac{13\!\cdots\!17}{13\!\cdots\!21}a^{10}-\frac{21\!\cdots\!42}{13\!\cdots\!21}a^{9}+\frac{21\!\cdots\!76}{13\!\cdots\!21}a^{8}-\frac{37\!\cdots\!41}{13\!\cdots\!21}a^{7}+\frac{38\!\cdots\!34}{13\!\cdots\!21}a^{6}-\frac{14\!\cdots\!02}{13\!\cdots\!21}a^{5}+\frac{13\!\cdots\!44}{13\!\cdots\!21}a^{4}+\frac{36\!\cdots\!56}{13\!\cdots\!21}a^{3}+\frac{64\!\cdots\!46}{13\!\cdots\!21}a^{2}+\frac{13\!\cdots\!90}{13\!\cdots\!21}a+\frac{42\!\cdots\!38}{13\!\cdots\!21}$
| 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: | \( 2461752.74517 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 2461752.74517 \cdot 2}{6\cdot\sqrt{10046547996724887091059294601443}}\cr\approx \mathstrut & 0.155988756643 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 7*x^20 - 4*x^19 + 31*x^18 - 18*x^17 + 77*x^16 - 41*x^15 + 140*x^14 - 96*x^13 + 153*x^12 - 129*x^11 + 145*x^10 - 148*x^9 + 84*x^8 - 72*x^7 + 94*x^6 - 20*x^5 + 39*x^4 + 12*x^3 + 18*x^2 + 4*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 - x^21 + 7*x^20 - 4*x^19 + 31*x^18 - 18*x^17 + 77*x^16 - 41*x^15 + 140*x^14 - 96*x^13 + 153*x^12 - 129*x^11 + 145*x^10 - 148*x^9 + 84*x^8 - 72*x^7 + 94*x^6 - 20*x^5 + 39*x^4 + 12*x^3 + 18*x^2 + 4*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 - x^21 + 7*x^20 - 4*x^19 + 31*x^18 - 18*x^17 + 77*x^16 - 41*x^15 + 140*x^14 - 96*x^13 + 153*x^12 - 129*x^11 + 145*x^10 - 148*x^9 + 84*x^8 - 72*x^7 + 94*x^6 - 20*x^5 + 39*x^4 + 12*x^3 + 18*x^2 + 4*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 - x^21 + 7*x^20 - 4*x^19 + 31*x^18 - 18*x^17 + 77*x^16 - 41*x^15 + 140*x^14 - 96*x^13 + 153*x^12 - 129*x^11 + 145*x^10 - 148*x^9 + 84*x^8 - 72*x^7 + 94*x^6 - 20*x^5 + 39*x^4 + 12*x^3 + 18*x^2 + 4*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 $C_2\times S_{11}$ are not computed |
| Character table for $C_2\times S_{11}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 11.9.7530807227563.1 |
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 | ${\href{/padicField/2.8.0.1}{8} }^{2}{,}\,{\href{/padicField/2.6.0.1}{6} }$ | R | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | $18{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{4}$ | $22$ | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $22$ | ${\href{/padicField/53.8.0.1}{8} }^{2}{,}\,{\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.10.5.2 | $x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(29\)
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.10.0.1 | $x^{10} + x^{6} + 25 x^{5} + 8 x^{4} + 17 x^{3} + 2 x^{2} + 22 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
|
\(131\)
| 131.4.2.1 | $x^{4} + 254 x^{3} + 16395 x^{2} + 33782 x + 2129540$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 131.4.0.1 | $x^{4} + 9 x^{2} + 109 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 131.4.0.1 | $x^{4} + 9 x^{2} + 109 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 131.10.0.1 | $x^{10} + 124 x^{5} + 97 x^{4} + 9 x^{3} + 126 x^{2} + 44 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
|
\(5399\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
|
\(367163\)
| Deg $4$ | $2$ | $2$ | $2$ | |||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |