Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 53*x^20 + 963*x^18 - 8057*x^16 + 34825*x^14 - 83894*x^12 + 119260*x^10 - 102849*x^8 + 53662*x^6 - 16302*x^4 + 2614*x^2 - 169)
gp: K = bnfinit(y^22 - 53*y^20 + 963*y^18 - 8057*y^16 + 34825*y^14 - 83894*y^12 + 119260*y^10 - 102849*y^8 + 53662*y^6 - 16302*y^4 + 2614*y^2 - 169, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 53*x^20 + 963*x^18 - 8057*x^16 + 34825*x^14 - 83894*x^12 + 119260*x^10 - 102849*x^8 + 53662*x^6 - 16302*x^4 + 2614*x^2 - 169);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 53*x^20 + 963*x^18 - 8057*x^16 + 34825*x^14 - 83894*x^12 + 119260*x^10 - 102849*x^8 + 53662*x^6 - 16302*x^4 + 2614*x^2 - 169)
\( x^{22} - 53 x^{20} + 963 x^{18} - 8057 x^{16} + 34825 x^{14} - 83894 x^{12} + 119260 x^{10} - 102849 x^{8} + \cdots - 169 \)
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: | $[22, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(4129233136056857981979443884256982828952059904\)
\(\medspace = 2^{22}\cdot 74843^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(118.43\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(74843\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{121998283921461}a^{20}-\frac{14409201320112}{40666094640487}a^{18}+\frac{18926811921010}{40666094640487}a^{16}-\frac{57921341971871}{121998283921461}a^{14}-\frac{13951320244354}{40666094640487}a^{12}-\frac{26427443611160}{121998283921461}a^{10}+\frac{15212170142341}{40666094640487}a^{8}-\frac{485342475304}{40666094640487}a^{6}-\frac{49631067509534}{121998283921461}a^{4}+\frac{25203520741757}{121998283921461}a^{2}-\frac{60448737560050}{121998283921461}$, $\frac{1}{15\!\cdots\!93}a^{21}-\frac{95741390601086}{528659230326331}a^{19}+\frac{181591190482958}{528659230326331}a^{17}-\frac{545914477657715}{15\!\cdots\!93}a^{15}+\frac{189379152958081}{528659230326331}a^{13}-\frac{270424011454082}{15\!\cdots\!93}a^{11}-\frac{66120019138633}{528659230326331}a^{9}-\frac{41151437115791}{528659230326331}a^{7}-\frac{293627635352456}{15\!\cdots\!93}a^{5}-\frac{54368167906693}{121998283921461}a^{3}+\frac{427544398125794}{15\!\cdots\!93}a$
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: | $21$ | 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: |
$\frac{26252115157607}{40666094640487}a^{20}-\frac{13\!\cdots\!99}{40666094640487}a^{18}+\frac{24\!\cdots\!84}{40666094640487}a^{16}-\frac{19\!\cdots\!12}{40666094640487}a^{14}+\frac{76\!\cdots\!84}{40666094640487}a^{12}-\frac{16\!\cdots\!49}{40666094640487}a^{10}+\frac{18\!\cdots\!40}{40666094640487}a^{8}-\frac{12\!\cdots\!11}{40666094640487}a^{6}+\frac{43\!\cdots\!90}{40666094640487}a^{4}-\frac{76\!\cdots\!38}{40666094640487}a^{2}+\frac{52\!\cdots\!89}{40666094640487}$, $a-1$, $\frac{54843352816052}{40666094640487}a^{20}-\frac{28\!\cdots\!55}{40666094640487}a^{18}+\frac{51\!\cdots\!21}{40666094640487}a^{16}-\frac{41\!\cdots\!64}{40666094640487}a^{14}+\frac{17\!\cdots\!08}{40666094640487}a^{12}-\frac{38\!\cdots\!93}{40666094640487}a^{10}+\frac{48\!\cdots\!56}{40666094640487}a^{8}-\frac{35\!\cdots\!15}{40666094640487}a^{6}+\frac{14\!\cdots\!91}{40666094640487}a^{4}-\frac{30\!\cdots\!78}{40666094640487}a^{2}+\frac{24\!\cdots\!53}{40666094640487}$, $a+1$, $\frac{13373660424214}{121998283921461}a^{20}-\frac{229308313706686}{40666094640487}a^{18}+\frac{39\!\cdots\!50}{40666094640487}a^{16}-\frac{88\!\cdots\!87}{121998283921461}a^{14}+\frac{10\!\cdots\!86}{40666094640487}a^{12}-\frac{56\!\cdots\!52}{121998283921461}a^{10}+\frac{16\!\cdots\!84}{40666094640487}a^{8}-\frac{69\!\cdots\!51}{40666094640487}a^{6}+\frac{26\!\cdots\!75}{121998283921461}a^{4}+\frac{48\!\cdots\!73}{121998283921461}a^{2}-\frac{11\!\cdots\!72}{121998283921461}$, $\frac{377932943590739}{15\!\cdots\!93}a^{21}-\frac{66\!\cdots\!74}{528659230326331}a^{19}+\frac{11\!\cdots\!21}{528659230326331}a^{17}-\frac{29\!\cdots\!34}{15\!\cdots\!93}a^{15}+\frac{41\!\cdots\!66}{528659230326331}a^{13}-\frac{28\!\cdots\!02}{15\!\cdots\!93}a^{11}+\frac{12\!\cdots\!65}{528659230326331}a^{9}-\frac{90\!\cdots\!62}{528659230326331}a^{7}+\frac{11\!\cdots\!24}{15\!\cdots\!93}a^{5}-\frac{17\!\cdots\!98}{121998283921461}a^{3}+\frac{18\!\cdots\!07}{15\!\cdots\!93}a$, $\frac{241738522895680}{528659230326331}a^{21}+\frac{54843352816052}{40666094640487}a^{20}-\frac{12\!\cdots\!33}{528659230326331}a^{19}-\frac{28\!\cdots\!55}{40666094640487}a^{18}+\frac{22\!\cdots\!03}{528659230326331}a^{17}+\frac{51\!\cdots\!21}{40666094640487}a^{16}-\frac{18\!\cdots\!44}{528659230326331}a^{15}-\frac{41\!\cdots\!64}{40666094640487}a^{14}+\frac{77\!\cdots\!86}{528659230326331}a^{13}+\frac{17\!\cdots\!08}{40666094640487}a^{12}-\frac{17\!\cdots\!34}{528659230326331}a^{11}-\frac{38\!\cdots\!93}{40666094640487}a^{10}+\frac{22\!\cdots\!43}{528659230326331}a^{9}+\frac{48\!\cdots\!56}{40666094640487}a^{8}-\frac{17\!\cdots\!69}{528659230326331}a^{7}-\frac{35\!\cdots\!15}{40666094640487}a^{6}+\frac{75\!\cdots\!37}{528659230326331}a^{5}+\frac{14\!\cdots\!91}{40666094640487}a^{4}-\frac{12\!\cdots\!22}{40666094640487}a^{3}-\frac{30\!\cdots\!78}{40666094640487}a^{2}+\frac{14\!\cdots\!13}{528659230326331}a+\frac{24\!\cdots\!66}{40666094640487}$, $\frac{241738522895680}{528659230326331}a^{21}-\frac{54843352816052}{40666094640487}a^{20}-\frac{12\!\cdots\!33}{528659230326331}a^{19}+\frac{28\!\cdots\!55}{40666094640487}a^{18}+\frac{22\!\cdots\!03}{528659230326331}a^{17}-\frac{51\!\cdots\!21}{40666094640487}a^{16}-\frac{18\!\cdots\!44}{528659230326331}a^{15}+\frac{41\!\cdots\!64}{40666094640487}a^{14}+\frac{77\!\cdots\!86}{528659230326331}a^{13}-\frac{17\!\cdots\!08}{40666094640487}a^{12}-\frac{17\!\cdots\!34}{528659230326331}a^{11}+\frac{38\!\cdots\!93}{40666094640487}a^{10}+\frac{22\!\cdots\!43}{528659230326331}a^{9}-\frac{48\!\cdots\!56}{40666094640487}a^{8}-\frac{17\!\cdots\!69}{528659230326331}a^{7}+\frac{35\!\cdots\!15}{40666094640487}a^{6}+\frac{75\!\cdots\!37}{528659230326331}a^{5}-\frac{14\!\cdots\!91}{40666094640487}a^{4}-\frac{12\!\cdots\!22}{40666094640487}a^{3}+\frac{30\!\cdots\!78}{40666094640487}a^{2}+\frac{14\!\cdots\!13}{528659230326331}a-\frac{24\!\cdots\!66}{40666094640487}$, $\frac{16\!\cdots\!13}{15\!\cdots\!93}a^{21}-\frac{29\!\cdots\!43}{528659230326331}a^{19}+\frac{53\!\cdots\!34}{528659230326331}a^{17}-\frac{13\!\cdots\!29}{15\!\cdots\!93}a^{15}+\frac{18\!\cdots\!80}{528659230326331}a^{13}-\frac{13\!\cdots\!42}{15\!\cdots\!93}a^{11}+\frac{59\!\cdots\!59}{528659230326331}a^{9}-\frac{46\!\cdots\!58}{528659230326331}a^{7}+\frac{61\!\cdots\!75}{15\!\cdots\!93}a^{5}-\frac{10\!\cdots\!59}{121998283921461}a^{3}+\frac{12\!\cdots\!57}{15\!\cdots\!93}a$, $\frac{24\!\cdots\!68}{15\!\cdots\!93}a^{21}-\frac{88109110406689}{121998283921461}a^{20}-\frac{43\!\cdots\!01}{528659230326331}a^{19}+\frac{15\!\cdots\!02}{40666094640487}a^{18}+\frac{77\!\cdots\!83}{528659230326331}a^{17}-\frac{27\!\cdots\!10}{40666094640487}a^{16}-\frac{18\!\cdots\!30}{15\!\cdots\!93}a^{15}+\frac{68\!\cdots\!45}{121998283921461}a^{14}+\frac{26\!\cdots\!23}{528659230326331}a^{13}-\frac{95\!\cdots\!19}{40666094640487}a^{12}-\frac{17\!\cdots\!95}{15\!\cdots\!93}a^{11}+\frac{64\!\cdots\!56}{121998283921461}a^{10}+\frac{76\!\cdots\!00}{528659230326331}a^{9}-\frac{28\!\cdots\!27}{40666094640487}a^{8}-\frac{56\!\cdots\!68}{528659230326331}a^{7}+\frac{20\!\cdots\!39}{40666094640487}a^{6}+\frac{69\!\cdots\!38}{15\!\cdots\!93}a^{5}-\frac{26\!\cdots\!72}{121998283921461}a^{4}-\frac{10\!\cdots\!76}{121998283921461}a^{3}+\frac{54\!\cdots\!22}{121998283921461}a^{2}+\frac{11\!\cdots\!07}{15\!\cdots\!93}a-\frac{43\!\cdots\!34}{121998283921461}$, $\frac{24\!\cdots\!68}{15\!\cdots\!93}a^{21}+\frac{88109110406689}{121998283921461}a^{20}-\frac{43\!\cdots\!01}{528659230326331}a^{19}-\frac{15\!\cdots\!02}{40666094640487}a^{18}+\frac{77\!\cdots\!83}{528659230326331}a^{17}+\frac{27\!\cdots\!10}{40666094640487}a^{16}-\frac{18\!\cdots\!30}{15\!\cdots\!93}a^{15}-\frac{68\!\cdots\!45}{121998283921461}a^{14}+\frac{26\!\cdots\!23}{528659230326331}a^{13}+\frac{95\!\cdots\!19}{40666094640487}a^{12}-\frac{17\!\cdots\!95}{15\!\cdots\!93}a^{11}-\frac{64\!\cdots\!56}{121998283921461}a^{10}+\frac{76\!\cdots\!00}{528659230326331}a^{9}+\frac{28\!\cdots\!27}{40666094640487}a^{8}-\frac{56\!\cdots\!68}{528659230326331}a^{7}-\frac{20\!\cdots\!39}{40666094640487}a^{6}+\frac{69\!\cdots\!38}{15\!\cdots\!93}a^{5}+\frac{26\!\cdots\!72}{121998283921461}a^{4}-\frac{10\!\cdots\!76}{121998283921461}a^{3}-\frac{54\!\cdots\!22}{121998283921461}a^{2}+\frac{11\!\cdots\!07}{15\!\cdots\!93}a+\frac{43\!\cdots\!34}{121998283921461}$, $\frac{186308607656578}{528659230326331}a^{21}+\frac{38049082286312}{121998283921461}a^{20}-\frac{97\!\cdots\!17}{528659230326331}a^{19}-\frac{658737685099696}{40666094640487}a^{18}+\frac{17\!\cdots\!50}{528659230326331}a^{17}+\frac{11\!\cdots\!60}{40666094640487}a^{16}-\frac{14\!\cdots\!49}{528659230326331}a^{15}-\frac{26\!\cdots\!22}{121998283921461}a^{14}+\frac{58\!\cdots\!33}{528659230326331}a^{13}+\frac{34\!\cdots\!13}{40666094640487}a^{12}-\frac{12\!\cdots\!18}{528659230326331}a^{11}-\frac{20\!\cdots\!84}{121998283921461}a^{10}+\frac{16\!\cdots\!49}{528659230326331}a^{9}+\frac{75\!\cdots\!56}{40666094640487}a^{8}-\frac{11\!\cdots\!83}{528659230326331}a^{7}-\frac{45\!\cdots\!78}{40666094640487}a^{6}+\frac{45\!\cdots\!21}{528659230326331}a^{5}+\frac{42\!\cdots\!77}{121998283921461}a^{4}-\frac{68\!\cdots\!09}{40666094640487}a^{3}-\frac{62\!\cdots\!70}{121998283921461}a^{2}+\frac{65\!\cdots\!37}{528659230326331}a+\frac{32\!\cdots\!02}{121998283921461}$, $\frac{186308607656578}{528659230326331}a^{21}-\frac{38049082286312}{121998283921461}a^{20}-\frac{97\!\cdots\!17}{528659230326331}a^{19}+\frac{658737685099696}{40666094640487}a^{18}+\frac{17\!\cdots\!50}{528659230326331}a^{17}-\frac{11\!\cdots\!60}{40666094640487}a^{16}-\frac{14\!\cdots\!49}{528659230326331}a^{15}+\frac{26\!\cdots\!22}{121998283921461}a^{14}+\frac{58\!\cdots\!33}{528659230326331}a^{13}-\frac{34\!\cdots\!13}{40666094640487}a^{12}-\frac{12\!\cdots\!18}{528659230326331}a^{11}+\frac{20\!\cdots\!84}{121998283921461}a^{10}+\frac{16\!\cdots\!49}{528659230326331}a^{9}-\frac{75\!\cdots\!56}{40666094640487}a^{8}-\frac{11\!\cdots\!83}{528659230326331}a^{7}+\frac{45\!\cdots\!78}{40666094640487}a^{6}+\frac{45\!\cdots\!21}{528659230326331}a^{5}-\frac{42\!\cdots\!77}{121998283921461}a^{4}-\frac{68\!\cdots\!09}{40666094640487}a^{3}+\frac{62\!\cdots\!70}{121998283921461}a^{2}+\frac{65\!\cdots\!37}{528659230326331}a-\frac{32\!\cdots\!02}{121998283921461}$, $\frac{23\!\cdots\!03}{15\!\cdots\!93}a^{21}-\frac{167494209167152}{121998283921461}a^{20}-\frac{41\!\cdots\!40}{528659230326331}a^{19}+\frac{29\!\cdots\!04}{40666094640487}a^{18}+\frac{74\!\cdots\!46}{528659230326331}a^{17}-\frac{52\!\cdots\!37}{40666094640487}a^{16}-\frac{18\!\cdots\!66}{15\!\cdots\!93}a^{15}+\frac{12\!\cdots\!07}{121998283921461}a^{14}+\frac{24\!\cdots\!69}{528659230326331}a^{13}-\frac{17\!\cdots\!96}{40666094640487}a^{12}-\frac{16\!\cdots\!05}{15\!\cdots\!93}a^{11}+\frac{11\!\cdots\!60}{121998283921461}a^{10}+\frac{68\!\cdots\!85}{528659230326331}a^{9}-\frac{49\!\cdots\!96}{40666094640487}a^{8}-\frac{49\!\cdots\!39}{528659230326331}a^{7}+\frac{36\!\cdots\!88}{40666094640487}a^{6}+\frac{58\!\cdots\!53}{15\!\cdots\!93}a^{5}-\frac{43\!\cdots\!53}{121998283921461}a^{4}-\frac{87\!\cdots\!99}{121998283921461}a^{3}+\frac{85\!\cdots\!50}{121998283921461}a^{2}+\frac{84\!\cdots\!20}{15\!\cdots\!93}a-\frac{65\!\cdots\!18}{121998283921461}$, $\frac{125691868743335}{121998283921461}a^{21}-\frac{9691742806196}{121998283921461}a^{20}-\frac{21\!\cdots\!53}{40666094640487}a^{19}+\frac{177293544561118}{40666094640487}a^{18}+\frac{38\!\cdots\!39}{40666094640487}a^{17}-\frac{34\!\cdots\!69}{40666094640487}a^{16}-\frac{93\!\cdots\!39}{121998283921461}a^{15}+\frac{94\!\cdots\!80}{121998283921461}a^{14}+\frac{12\!\cdots\!13}{40666094640487}a^{13}-\frac{15\!\cdots\!41}{40666094640487}a^{12}-\frac{80\!\cdots\!56}{121998283921461}a^{11}+\frac{12\!\cdots\!56}{121998283921461}a^{10}+\frac{32\!\cdots\!33}{40666094640487}a^{9}-\frac{67\!\cdots\!14}{40666094640487}a^{8}-\frac{22\!\cdots\!94}{40666094640487}a^{7}+\frac{60\!\cdots\!70}{40666094640487}a^{6}+\frac{26\!\cdots\!04}{121998283921461}a^{5}-\frac{89\!\cdots\!44}{121998283921461}a^{4}-\frac{56\!\cdots\!86}{121998283921461}a^{3}+\frac{22\!\cdots\!19}{121998283921461}a^{2}+\frac{47\!\cdots\!22}{121998283921461}a-\frac{20\!\cdots\!02}{121998283921461}$, $\frac{10\!\cdots\!77}{15\!\cdots\!93}a^{21}+\frac{19184663667007}{40666094640487}a^{20}-\frac{17\!\cdots\!75}{528659230326331}a^{19}-\frac{998868606518093}{40666094640487}a^{18}+\frac{31\!\cdots\!99}{528659230326331}a^{17}+\frac{17\!\cdots\!99}{40666094640487}a^{16}-\frac{77\!\cdots\!72}{15\!\cdots\!93}a^{15}-\frac{13\!\cdots\!83}{40666094640487}a^{14}+\frac{10\!\cdots\!33}{528659230326331}a^{13}+\frac{53\!\cdots\!30}{40666094640487}a^{12}-\frac{70\!\cdots\!95}{15\!\cdots\!93}a^{11}-\frac{11\!\cdots\!93}{40666094640487}a^{10}+\frac{29\!\cdots\!31}{528659230326331}a^{9}+\frac{12\!\cdots\!40}{40666094640487}a^{8}-\frac{21\!\cdots\!04}{528659230326331}a^{7}-\frac{81\!\cdots\!44}{40666094640487}a^{6}+\frac{27\!\cdots\!13}{15\!\cdots\!93}a^{5}+\frac{29\!\cdots\!13}{40666094640487}a^{4}-\frac{44\!\cdots\!39}{121998283921461}a^{3}-\frac{56\!\cdots\!41}{40666094640487}a^{2}+\frac{48\!\cdots\!77}{15\!\cdots\!93}a+\frac{44\!\cdots\!90}{40666094640487}$, $\frac{73747933546279}{15\!\cdots\!93}a^{21}+\frac{32610094018690}{40666094640487}a^{20}-\frac{13\!\cdots\!88}{528659230326331}a^{19}-\frac{17\!\cdots\!56}{40666094640487}a^{18}+\frac{28\!\cdots\!22}{528659230326331}a^{17}+\frac{30\!\cdots\!60}{40666094640487}a^{16}-\frac{84\!\cdots\!75}{15\!\cdots\!93}a^{15}-\frac{23\!\cdots\!57}{40666094640487}a^{14}+\frac{15\!\cdots\!85}{528659230326331}a^{13}+\frac{94\!\cdots\!25}{40666094640487}a^{12}-\frac{13\!\cdots\!07}{15\!\cdots\!93}a^{11}-\frac{19\!\cdots\!60}{40666094640487}a^{10}+\frac{76\!\cdots\!02}{528659230326331}a^{9}+\frac{23\!\cdots\!31}{40666094640487}a^{8}-\frac{73\!\cdots\!73}{528659230326331}a^{7}-\frac{14\!\cdots\!67}{40666094640487}a^{6}+\frac{11\!\cdots\!78}{15\!\cdots\!93}a^{5}+\frac{50\!\cdots\!16}{40666094640487}a^{4}-\frac{22\!\cdots\!94}{121998283921461}a^{3}-\frac{82\!\cdots\!77}{40666094640487}a^{2}+\frac{27\!\cdots\!82}{15\!\cdots\!93}a+\frac{49\!\cdots\!82}{40666094640487}$, $\frac{326157035729995}{121998283921461}a^{21}+\frac{335480982333268}{121998283921461}a^{20}-\frac{56\!\cdots\!88}{40666094640487}a^{19}-\frac{58\!\cdots\!66}{40666094640487}a^{18}+\frac{10\!\cdots\!65}{40666094640487}a^{17}+\frac{10\!\cdots\!97}{40666094640487}a^{16}-\frac{23\!\cdots\!42}{121998283921461}a^{15}-\frac{24\!\cdots\!89}{121998283921461}a^{14}+\frac{31\!\cdots\!01}{40666094640487}a^{13}+\frac{31\!\cdots\!32}{40666094640487}a^{12}-\frac{20\!\cdots\!95}{121998283921461}a^{11}-\frac{19\!\cdots\!92}{121998283921461}a^{10}+\frac{79\!\cdots\!74}{40666094640487}a^{9}+\frac{76\!\cdots\!37}{40666094640487}a^{8}-\frac{54\!\cdots\!02}{40666094640487}a^{7}-\frac{49\!\cdots\!05}{40666094640487}a^{6}+\frac{63\!\cdots\!96}{121998283921461}a^{5}+\frac{51\!\cdots\!18}{121998283921461}a^{4}-\frac{12\!\cdots\!47}{121998283921461}a^{3}-\frac{88\!\cdots\!52}{121998283921461}a^{2}+\frac{10\!\cdots\!01}{121998283921461}a+\frac{56\!\cdots\!60}{121998283921461}$, $\frac{13\!\cdots\!66}{15\!\cdots\!93}a^{21}-\frac{905938801223849}{121998283921461}a^{20}-\frac{23\!\cdots\!56}{528659230326331}a^{19}+\frac{15\!\cdots\!63}{40666094640487}a^{18}+\frac{40\!\cdots\!92}{528659230326331}a^{17}-\frac{27\!\cdots\!62}{40666094640487}a^{16}-\frac{96\!\cdots\!31}{15\!\cdots\!93}a^{15}+\frac{65\!\cdots\!42}{121998283921461}a^{14}+\frac{12\!\cdots\!50}{528659230326331}a^{13}-\frac{85\!\cdots\!41}{40666094640487}a^{12}-\frac{78\!\cdots\!49}{15\!\cdots\!93}a^{11}+\frac{52\!\cdots\!82}{121998283921461}a^{10}+\frac{30\!\cdots\!14}{528659230326331}a^{9}-\frac{20\!\cdots\!50}{40666094640487}a^{8}-\frac{19\!\cdots\!82}{528659230326331}a^{7}+\frac{12\!\cdots\!23}{40666094640487}a^{6}+\frac{20\!\cdots\!56}{15\!\cdots\!93}a^{5}-\frac{13\!\cdots\!35}{121998283921461}a^{4}-\frac{29\!\cdots\!30}{121998283921461}a^{3}+\frac{24\!\cdots\!00}{121998283921461}a^{2}+\frac{26\!\cdots\!49}{15\!\cdots\!93}a-\frac{16\!\cdots\!60}{121998283921461}$, $\frac{30\!\cdots\!10}{15\!\cdots\!93}a^{21}-\frac{158809342722301}{121998283921461}a^{20}-\frac{53\!\cdots\!49}{528659230326331}a^{19}+\frac{27\!\cdots\!08}{40666094640487}a^{18}+\frac{94\!\cdots\!98}{528659230326331}a^{17}-\frac{48\!\cdots\!40}{40666094640487}a^{16}-\frac{22\!\cdots\!70}{15\!\cdots\!93}a^{15}+\frac{11\!\cdots\!14}{121998283921461}a^{14}+\frac{29\!\cdots\!38}{528659230326331}a^{13}-\frac{14\!\cdots\!92}{40666094640487}a^{12}-\frac{18\!\cdots\!03}{15\!\cdots\!93}a^{11}+\frac{91\!\cdots\!95}{121998283921461}a^{10}+\frac{69\!\cdots\!10}{528659230326331}a^{9}-\frac{34\!\cdots\!87}{40666094640487}a^{8}-\frac{44\!\cdots\!03}{528659230326331}a^{7}+\frac{21\!\cdots\!34}{40666094640487}a^{6}+\frac{47\!\cdots\!73}{15\!\cdots\!93}a^{5}-\frac{22\!\cdots\!84}{121998283921461}a^{4}-\frac{66\!\cdots\!05}{121998283921461}a^{3}+\frac{39\!\cdots\!35}{121998283921461}a^{2}+\frac{62\!\cdots\!92}{15\!\cdots\!93}a-\frac{27\!\cdots\!36}{121998283921461}$, $\frac{394391806379006}{15\!\cdots\!93}a^{21}+\frac{21598340067245}{121998283921461}a^{20}-\frac{67\!\cdots\!50}{528659230326331}a^{19}-\frac{375727891802368}{40666094640487}a^{18}+\frac{11\!\cdots\!65}{528659230326331}a^{17}+\frac{66\!\cdots\!11}{40666094640487}a^{16}-\frac{24\!\cdots\!04}{15\!\cdots\!93}a^{15}-\frac{15\!\cdots\!94}{121998283921461}a^{14}+\frac{28\!\cdots\!09}{528659230326331}a^{13}+\frac{20\!\cdots\!33}{40666094640487}a^{12}-\frac{12\!\cdots\!87}{15\!\cdots\!93}a^{11}-\frac{13\!\cdots\!75}{121998283921461}a^{10}+\frac{23\!\cdots\!78}{528659230326331}a^{9}+\frac{54\!\cdots\!77}{40666094640487}a^{8}+\frac{86\!\cdots\!52}{528659230326331}a^{7}-\frac{39\!\cdots\!94}{40666094640487}a^{6}-\frac{44\!\cdots\!03}{15\!\cdots\!93}a^{5}+\frac{49\!\cdots\!43}{121998283921461}a^{4}+\frac{12\!\cdots\!77}{121998283921461}a^{3}-\frac{10\!\cdots\!46}{121998283921461}a^{2}-\frac{18\!\cdots\!93}{15\!\cdots\!93}a+\frac{90\!\cdots\!15}{121998283921461}$
| 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: | \( 10853217975800000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{22}\cdot(2\pi)^{0}\cdot 10853217975800000 \cdot 1}{2\cdot\sqrt{4129233136056857981979443884256982828952059904}}\cr\approx \mathstrut & 0.354204221163405 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 53*x^20 + 963*x^18 - 8057*x^16 + 34825*x^14 - 83894*x^12 + 119260*x^10 - 102849*x^8 + 53662*x^6 - 16302*x^4 + 2614*x^2 - 169)
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 - 53*x^20 + 963*x^18 - 8057*x^16 + 34825*x^14 - 83894*x^12 + 119260*x^10 - 102849*x^8 + 53662*x^6 - 16302*x^4 + 2614*x^2 - 169, 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 - 53*x^20 + 963*x^18 - 8057*x^16 + 34825*x^14 - 83894*x^12 + 119260*x^10 - 102849*x^8 + 53662*x^6 - 16302*x^4 + 2614*x^2 - 169);
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 - 53*x^20 + 963*x^18 - 8057*x^16 + 34825*x^14 - 83894*x^12 + 119260*x^10 - 102849*x^8 + 53662*x^6 - 16302*x^4 + 2614*x^2 - 169);
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
$C_2^{10}.\PSL(2,11)$ (as 22T39):
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 675840 |
| The 56 conjugacy class representatives for $C_2^{10}.\PSL(2,11)$ |
| Character table for $C_2^{10}.\PSL(2,11)$ |
Intermediate fields
| 11.11.31376518243389673201.2 |
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 |
| 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 | R | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.11.0.1}{11} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.11.0.1}{11} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.11.0.1}{11} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $22$ | $2$ | $11$ | $22$ | |||
|
\(74843\)
| $\Q_{74843}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{74843}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $8$ | $2$ | $4$ | $4$ |