↑

Properties

Label 22.22.199...249.1
Degree $22$
Signature $[22, 0]$
Discriminant $2.000\times 10^{123}$
Root discriminant \(402\,335.24\)
Ramified primes $421,3913599589,115692385433$
Class number not computed
Class group not computed
Galois group $C_2^{10}.S_{11}$ (as 22T51)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 - 2356992835*x^20 + 23569928735*x^19 + 2204501155911423557*x^18 - 19840511074945783155*x^17 - 1082872885988662125300812370*x^16 + 8662983537627551079745723914*x^15 + 316068217715312920637575328756082112*x^14 - 2212477675609403926959065035382244332*x^13 - 58767377684409992170556055760995021176639986*x^12 + 352604294868670130111344193424712514895639548*x^11 + 7226966902317690915617765163171122089181106876101229*x^10 - 36134837743794543604950652866191954074296615184490417*x^9 - 595718839192069496369987165264666540405203201509932504884044*x^8 + 2382875573577308326894544241652333406345733476067116734044261*x^7 + 32584603183355485909634946704681447022832056003466832943811521734636*x^6 - 97753817890131117015806386829370798310090825227154792176518881828628*x^5 - 1135082737337117340979212736589910913095311476546586257204562141505863404960*x^4 + 2270165637597267278865195276060941498200493335290757444902509490795622424977*x^3 + 22798723985900078548327022529561358312453616137357242512589877765454307023391451962*x^2 - 22798725120982913639264040796618268980326961172862719464781569239076972125427474194*x - 200957389535301956135132770569928284254466122629162705805201402317826353092603069971791081)
 
gp: K = bnfinit(y^22 - 11*y^21 - 2356992835*y^20 + 23569928735*y^19 + 2204501155911423557*y^18 - 19840511074945783155*y^17 - 1082872885988662125300812370*y^16 + 8662983537627551079745723914*y^15 + 316068217715312920637575328756082112*y^14 - 2212477675609403926959065035382244332*y^13 - 58767377684409992170556055760995021176639986*y^12 + 352604294868670130111344193424712514895639548*y^11 + 7226966902317690915617765163171122089181106876101229*y^10 - 36134837743794543604950652866191954074296615184490417*y^9 - 595718839192069496369987165264666540405203201509932504884044*y^8 + 2382875573577308326894544241652333406345733476067116734044261*y^7 + 32584603183355485909634946704681447022832056003466832943811521734636*y^6 - 97753817890131117015806386829370798310090825227154792176518881828628*y^5 - 1135082737337117340979212736589910913095311476546586257204562141505863404960*y^4 + 2270165637597267278865195276060941498200493335290757444902509490795622424977*y^3 + 22798723985900078548327022529561358312453616137357242512589877765454307023391451962*y^2 - 22798725120982913639264040796618268980326961172862719464781569239076972125427474194*y - 200957389535301956135132770569928284254466122629162705805201402317826353092603069971791081, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 11*x^21 - 2356992835*x^20 + 23569928735*x^19 + 2204501155911423557*x^18 - 19840511074945783155*x^17 - 1082872885988662125300812370*x^16 + 8662983537627551079745723914*x^15 + 316068217715312920637575328756082112*x^14 - 2212477675609403926959065035382244332*x^13 - 58767377684409992170556055760995021176639986*x^12 + 352604294868670130111344193424712514895639548*x^11 + 7226966902317690915617765163171122089181106876101229*x^10 - 36134837743794543604950652866191954074296615184490417*x^9 - 595718839192069496369987165264666540405203201509932504884044*x^8 + 2382875573577308326894544241652333406345733476067116734044261*x^7 + 32584603183355485909634946704681447022832056003466832943811521734636*x^6 - 97753817890131117015806386829370798310090825227154792176518881828628*x^5 - 1135082737337117340979212736589910913095311476546586257204562141505863404960*x^4 + 2270165637597267278865195276060941498200493335290757444902509490795622424977*x^3 + 22798723985900078548327022529561358312453616137357242512589877765454307023391451962*x^2 - 22798725120982913639264040796618268980326961172862719464781569239076972125427474194*x - 200957389535301956135132770569928284254466122629162705805201402317826353092603069971791081);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 11*x^21 - 2356992835*x^20 + 23569928735*x^19 + 2204501155911423557*x^18 - 19840511074945783155*x^17 - 1082872885988662125300812370*x^16 + 8662983537627551079745723914*x^15 + 316068217715312920637575328756082112*x^14 - 2212477675609403926959065035382244332*x^13 - 58767377684409992170556055760995021176639986*x^12 + 352604294868670130111344193424712514895639548*x^11 + 7226966902317690915617765163171122089181106876101229*x^10 - 36134837743794543604950652866191954074296615184490417*x^9 - 595718839192069496369987165264666540405203201509932504884044*x^8 + 2382875573577308326894544241652333406345733476067116734044261*x^7 + 32584603183355485909634946704681447022832056003466832943811521734636*x^6 - 97753817890131117015806386829370798310090825227154792176518881828628*x^5 - 1135082737337117340979212736589910913095311476546586257204562141505863404960*x^4 + 2270165637597267278865195276060941498200493335290757444902509490795622424977*x^3 + 22798723985900078548327022529561358312453616137357242512589877765454307023391451962*x^2 - 22798725120982913639264040796618268980326961172862719464781569239076972125427474194*x - 200957389535301956135132770569928284254466122629162705805201402317826353092603069971791081)
 

\( x^{22} - 11 x^{21} - 2356992835 x^{20} + 23569928735 x^{19} + \cdots - 20\!\cdots\!81 \) Copy content Toggle raw display

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:   \(199\!\cdots\!249\) \(\medspace = 421^{2}\cdot 3913599589^{10}\cdot 115692385433^{2}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(402\,335.24\)
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:  $421^{1/2}3913599589^{1/2}115692385433^{1/2}\approx 436597888160.3858$
Ramified primes:   \(421\), \(3913599589\), \(115692385433\) Copy content Toggle raw display
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3913599589}a^{4}-\frac{2}{3913599589}a^{3}-\frac{1378592994}{3913599589}a^{2}+\frac{1378592995}{3913599589}a+\frac{1856751520}{3913599589}$, $\frac{1}{3913599589}a^{5}-\frac{1378592998}{3913599589}a^{3}-\frac{1378592993}{3913599589}a^{2}+\frac{700337921}{3913599589}a-\frac{200096549}{3913599589}$, $\frac{1}{15\!\cdots\!21}a^{6}-\frac{3}{15\!\cdots\!21}a^{5}+\frac{1556606700}{15\!\cdots\!21}a^{4}-\frac{3113213395}{15\!\cdots\!21}a^{3}-\frac{70\!\cdots\!29}{15\!\cdots\!21}a^{2}+\frac{70\!\cdots\!26}{15\!\cdots\!21}a-\frac{29\!\cdots\!72}{15\!\cdots\!21}$, $\frac{1}{15\!\cdots\!21}a^{7}+\frac{1556606691}{15\!\cdots\!21}a^{5}+\frac{1556606705}{15\!\cdots\!21}a^{4}-\frac{70\!\cdots\!14}{15\!\cdots\!21}a^{3}+\frac{12\!\cdots\!60}{15\!\cdots\!21}a^{2}+\frac{27\!\cdots\!85}{15\!\cdots\!21}a+\frac{64\!\cdots\!05}{15\!\cdots\!21}$, $\frac{1}{59\!\cdots\!69}a^{8}-\frac{4}{59\!\cdots\!69}a^{7}+\frac{578206806}{59\!\cdots\!69}a^{6}-\frac{1734620404}{59\!\cdots\!69}a^{5}+\frac{67\!\cdots\!87}{59\!\cdots\!69}a^{4}-\frac{13\!\cdots\!72}{59\!\cdots\!69}a^{3}-\frac{29\!\cdots\!38}{59\!\cdots\!69}a^{2}+\frac{29\!\cdots\!24}{59\!\cdots\!69}a+\frac{19\!\cdots\!67}{59\!\cdots\!69}$, $\frac{1}{59\!\cdots\!69}a^{9}+\frac{578206790}{59\!\cdots\!69}a^{7}+\frac{578206820}{59\!\cdots\!69}a^{6}+\frac{67\!\cdots\!71}{59\!\cdots\!69}a^{5}-\frac{17\!\cdots\!45}{59\!\cdots\!69}a^{4}-\frac{29\!\cdots\!84}{59\!\cdots\!69}a^{3}+\frac{12\!\cdots\!46}{59\!\cdots\!69}a^{2}+\frac{98\!\cdots\!68}{59\!\cdots\!69}a-\frac{11\!\cdots\!21}{59\!\cdots\!69}$, $\frac{1}{23\!\cdots\!41}a^{10}-\frac{5}{23\!\cdots\!41}a^{9}-\frac{400193087}{23\!\cdots\!41}a^{8}+\frac{1600772378}{23\!\cdots\!41}a^{7}+\frac{62\!\cdots\!09}{23\!\cdots\!41}a^{6}-\frac{18\!\cdots\!71}{23\!\cdots\!41}a^{5}-\frac{95\!\cdots\!05}{23\!\cdots\!41}a^{4}+\frac{19\!\cdots\!46}{23\!\cdots\!41}a^{3}+\frac{41\!\cdots\!38}{23\!\cdots\!41}a^{2}-\frac{41\!\cdots\!04}{23\!\cdots\!41}a-\frac{67\!\cdots\!57}{23\!\cdots\!41}$, $\frac{1}{23\!\cdots\!41}a^{11}-\frac{400193112}{23\!\cdots\!41}a^{9}-\frac{400193057}{23\!\cdots\!41}a^{8}+\frac{62\!\cdots\!99}{23\!\cdots\!41}a^{7}-\frac{29\!\cdots\!47}{23\!\cdots\!41}a^{6}-\frac{95\!\cdots\!97}{23\!\cdots\!41}a^{5}+\frac{73\!\cdots\!90}{23\!\cdots\!41}a^{4}+\frac{41\!\cdots\!25}{23\!\cdots\!41}a^{3}-\frac{42\!\cdots\!32}{23\!\cdots\!41}a^{2}-\frac{66\!\cdots\!27}{23\!\cdots\!41}a+\frac{52\!\cdots\!48}{23\!\cdots\!41}$, $\frac{1}{91\!\cdots\!49}a^{12}-\frac{6}{91\!\cdots\!49}a^{11}-\frac{1378592979}{91\!\cdots\!49}a^{10}+\frac{6892964950}{91\!\cdots\!49}a^{9}+\frac{65\!\cdots\!70}{91\!\cdots\!49}a^{8}-\frac{26\!\cdots\!46}{91\!\cdots\!49}a^{7}-\frac{15\!\cdots\!07}{91\!\cdots\!49}a^{6}+\frac{46\!\cdots\!38}{91\!\cdots\!49}a^{5}+\frac{51\!\cdots\!77}{91\!\cdots\!49}a^{4}-\frac{10\!\cdots\!04}{91\!\cdots\!49}a^{3}-\frac{14\!\cdots\!60}{91\!\cdots\!49}a^{2}+\frac{14\!\cdots\!66}{91\!\cdots\!49}a+\frac{27\!\cdots\!11}{91\!\cdots\!49}$, $\frac{1}{91\!\cdots\!49}a^{13}-\frac{1378593015}{91\!\cdots\!49}a^{11}-\frac{1378592924}{91\!\cdots\!49}a^{10}+\frac{65\!\cdots\!70}{91\!\cdots\!49}a^{9}-\frac{21\!\cdots\!47}{91\!\cdots\!49}a^{8}-\frac{15\!\cdots\!99}{91\!\cdots\!49}a^{7}+\frac{41\!\cdots\!39}{91\!\cdots\!49}a^{6}+\frac{51\!\cdots\!82}{91\!\cdots\!49}a^{5}-\frac{40\!\cdots\!10}{91\!\cdots\!49}a^{4}-\frac{14\!\cdots\!45}{91\!\cdots\!49}a^{3}+\frac{15\!\cdots\!06}{91\!\cdots\!49}a^{2}+\frac{26\!\cdots\!53}{91\!\cdots\!49}a-\frac{20\!\cdots\!78}{91\!\cdots\!49}$, $\frac{1}{35\!\cdots\!61}a^{14}-\frac{7}{35\!\cdots\!61}a^{13}+\frac{1556606719}{35\!\cdots\!61}a^{12}-\frac{9339640223}{35\!\cdots\!61}a^{11}+\frac{25\!\cdots\!52}{35\!\cdots\!61}a^{10}-\frac{12\!\cdots\!16}{35\!\cdots\!61}a^{9}+\frac{37\!\cdots\!79}{35\!\cdots\!61}a^{8}-\frac{14\!\cdots\!87}{35\!\cdots\!61}a^{7}+\frac{51\!\cdots\!95}{35\!\cdots\!61}a^{6}-\frac{15\!\cdots\!85}{35\!\cdots\!61}a^{5}+\frac{67\!\cdots\!28}{35\!\cdots\!61}a^{4}-\frac{13\!\cdots\!42}{35\!\cdots\!61}a^{3}+\frac{85\!\cdots\!34}{35\!\cdots\!61}a^{2}-\frac{85\!\cdots\!48}{35\!\cdots\!61}a+\frac{19\!\cdots\!99}{35\!\cdots\!61}$, $\frac{1}{35\!\cdots\!61}a^{15}+\frac{1556606670}{35\!\cdots\!61}a^{13}+\frac{1556606810}{35\!\cdots\!61}a^{12}+\frac{25\!\cdots\!91}{35\!\cdots\!61}a^{11}+\frac{51\!\cdots\!48}{35\!\cdots\!61}a^{10}+\frac{37\!\cdots\!67}{35\!\cdots\!61}a^{9}+\frac{11\!\cdots\!66}{35\!\cdots\!61}a^{8}+\frac{51\!\cdots\!86}{35\!\cdots\!61}a^{7}+\frac{20\!\cdots\!80}{35\!\cdots\!61}a^{6}+\frac{67\!\cdots\!33}{35\!\cdots\!61}a^{5}+\frac{33\!\cdots\!54}{35\!\cdots\!61}a^{4}+\frac{85\!\cdots\!40}{35\!\cdots\!61}a^{3}+\frac{51\!\cdots\!90}{35\!\cdots\!61}a^{2}-\frac{58\!\cdots\!37}{35\!\cdots\!61}a+\frac{13\!\cdots\!93}{35\!\cdots\!61}$, $\frac{1}{14\!\cdots\!29}a^{16}-\frac{8}{14\!\cdots\!29}a^{15}+\frac{578206829}{14\!\cdots\!29}a^{14}-\frac{4047447663}{14\!\cdots\!29}a^{13}+\frac{10\!\cdots\!32}{14\!\cdots\!29}a^{12}-\frac{61\!\cdots\!37}{14\!\cdots\!29}a^{11}+\frac{12\!\cdots\!51}{14\!\cdots\!29}a^{10}-\frac{60\!\cdots\!14}{14\!\cdots\!29}a^{9}+\frac{14\!\cdots\!19}{14\!\cdots\!29}a^{8}-\frac{58\!\cdots\!41}{14\!\cdots\!29}a^{7}+\frac{17\!\cdots\!98}{14\!\cdots\!29}a^{6}-\frac{51\!\cdots\!25}{14\!\cdots\!29}a^{5}+\frac{19\!\cdots\!60}{14\!\cdots\!29}a^{4}-\frac{39\!\cdots\!08}{14\!\cdots\!29}a^{3}+\frac{22\!\cdots\!89}{14\!\cdots\!29}a^{2}-\frac{22\!\cdots\!83}{14\!\cdots\!29}a+\frac{78\!\cdots\!82}{14\!\cdots\!29}$, $\frac{1}{14\!\cdots\!29}a^{17}+\frac{578206765}{14\!\cdots\!29}a^{15}+\frac{578206969}{14\!\cdots\!29}a^{14}+\frac{10\!\cdots\!28}{14\!\cdots\!29}a^{13}+\frac{20\!\cdots\!19}{14\!\cdots\!29}a^{12}+\frac{12\!\cdots\!55}{14\!\cdots\!29}a^{11}+\frac{36\!\cdots\!94}{14\!\cdots\!29}a^{10}+\frac{14\!\cdots\!07}{14\!\cdots\!29}a^{9}+\frac{58\!\cdots\!11}{14\!\cdots\!29}a^{8}+\frac{17\!\cdots\!70}{14\!\cdots\!29}a^{7}+\frac{86\!\cdots\!59}{14\!\cdots\!29}a^{6}+\frac{19\!\cdots\!60}{14\!\cdots\!29}a^{5}+\frac{11\!\cdots\!72}{14\!\cdots\!29}a^{4}+\frac{22\!\cdots\!25}{14\!\cdots\!29}a^{3}+\frac{15\!\cdots\!29}{14\!\cdots\!29}a^{2}-\frac{17\!\cdots\!82}{14\!\cdots\!29}a+\frac{62\!\cdots\!56}{14\!\cdots\!29}$, $\frac{1}{55\!\cdots\!81}a^{18}-\frac{9}{55\!\cdots\!81}a^{17}-\frac{400193060}{55\!\cdots\!81}a^{16}+\frac{3201544684}{55\!\cdots\!81}a^{15}+\frac{46\!\cdots\!82}{55\!\cdots\!81}a^{14}-\frac{32\!\cdots\!58}{55\!\cdots\!81}a^{13}+\frac{20\!\cdots\!84}{55\!\cdots\!81}a^{12}-\frac{12\!\cdots\!76}{55\!\cdots\!81}a^{11}+\frac{27\!\cdots\!86}{55\!\cdots\!81}a^{10}-\frac{13\!\cdots\!02}{55\!\cdots\!81}a^{9}+\frac{28\!\cdots\!96}{55\!\cdots\!81}a^{8}-\frac{11\!\cdots\!46}{55\!\cdots\!81}a^{7}+\frac{30\!\cdots\!79}{55\!\cdots\!81}a^{6}-\frac{91\!\cdots\!43}{55\!\cdots\!81}a^{5}+\frac{31\!\cdots\!77}{55\!\cdots\!81}a^{4}-\frac{63\!\cdots\!96}{55\!\cdots\!81}a^{3}+\frac{33\!\cdots\!36}{55\!\cdots\!81}a^{2}-\frac{33\!\cdots\!35}{55\!\cdots\!81}a-\frac{12\!\cdots\!03}{55\!\cdots\!81}$, $\frac{1}{55\!\cdots\!81}a^{19}-\frac{400193141}{55\!\cdots\!81}a^{17}-\frac{400192856}{55\!\cdots\!81}a^{16}+\frac{46\!\cdots\!38}{55\!\cdots\!81}a^{15}+\frac{92\!\cdots\!80}{55\!\cdots\!81}a^{14}+\frac{20\!\cdots\!62}{55\!\cdots\!81}a^{13}+\frac{62\!\cdots\!80}{55\!\cdots\!81}a^{12}+\frac{27\!\cdots\!02}{55\!\cdots\!81}a^{11}+\frac{11\!\cdots\!72}{55\!\cdots\!81}a^{10}+\frac{28\!\cdots\!78}{55\!\cdots\!81}a^{9}+\frac{14\!\cdots\!18}{55\!\cdots\!81}a^{8}+\frac{30\!\cdots\!65}{55\!\cdots\!81}a^{7}+\frac{18\!\cdots\!68}{55\!\cdots\!81}a^{6}+\frac{31\!\cdots\!90}{55\!\cdots\!81}a^{5}+\frac{22\!\cdots\!97}{55\!\cdots\!81}a^{4}+\frac{33\!\cdots\!72}{55\!\cdots\!81}a^{3}+\frac{26\!\cdots\!89}{55\!\cdots\!81}a^{2}-\frac{31\!\cdots\!18}{55\!\cdots\!81}a-\frac{11\!\cdots\!27}{55\!\cdots\!81}$, $\frac{1}{21\!\cdots\!09}a^{20}-\frac{10}{21\!\cdots\!09}a^{19}-\frac{1378592948}{21\!\cdots\!09}a^{18}+\frac{12407336817}{21\!\cdots\!09}a^{17}+\frac{85\!\cdots\!18}{21\!\cdots\!09}a^{16}-\frac{68\!\cdots\!88}{21\!\cdots\!09}a^{15}-\frac{24\!\cdots\!12}{21\!\cdots\!09}a^{14}+\frac{17\!\cdots\!66}{21\!\cdots\!09}a^{13}+\frac{75\!\cdots\!14}{21\!\cdots\!09}a^{12}-\frac{45\!\cdots\!16}{21\!\cdots\!09}a^{11}+\frac{15\!\cdots\!56}{21\!\cdots\!09}a^{10}-\frac{76\!\cdots\!48}{21\!\cdots\!09}a^{9}+\frac{22\!\cdots\!13}{21\!\cdots\!09}a^{8}-\frac{88\!\cdots\!60}{21\!\cdots\!09}a^{7}+\frac{21\!\cdots\!57}{21\!\cdots\!09}a^{6}-\frac{63\!\cdots\!02}{21\!\cdots\!09}a^{5}+\frac{20\!\cdots\!63}{21\!\cdots\!09}a^{4}-\frac{41\!\cdots\!59}{21\!\cdots\!09}a^{3}+\frac{20\!\cdots\!92}{21\!\cdots\!09}a^{2}-\frac{20\!\cdots\!54}{21\!\cdots\!09}a+\frac{21\!\cdots\!04}{21\!\cdots\!09}$, $\frac{1}{21\!\cdots\!09}a^{21}-\frac{1378593048}{21\!\cdots\!09}a^{19}-\frac{1378592663}{21\!\cdots\!09}a^{18}+\frac{85\!\cdots\!88}{21\!\cdots\!09}a^{17}+\frac{17\!\cdots\!92}{21\!\cdots\!09}a^{16}-\frac{24\!\cdots\!92}{21\!\cdots\!09}a^{15}-\frac{73\!\cdots\!54}{21\!\cdots\!09}a^{14}+\frac{75\!\cdots\!74}{21\!\cdots\!09}a^{13}+\frac{30\!\cdots\!24}{21\!\cdots\!09}a^{12}+\frac{15\!\cdots\!96}{21\!\cdots\!09}a^{11}+\frac{76\!\cdots\!12}{21\!\cdots\!09}a^{10}+\frac{22\!\cdots\!33}{21\!\cdots\!09}a^{9}+\frac{13\!\cdots\!70}{21\!\cdots\!09}a^{8}+\frac{21\!\cdots\!57}{21\!\cdots\!09}a^{7}+\frac{14\!\cdots\!68}{21\!\cdots\!09}a^{6}+\frac{20\!\cdots\!43}{21\!\cdots\!09}a^{5}+\frac{16\!\cdots\!71}{21\!\cdots\!09}a^{4}+\frac{20\!\cdots\!02}{21\!\cdots\!09}a^{3}+\frac{18\!\cdots\!66}{21\!\cdots\!09}a^{2}+\frac{79\!\cdots\!64}{21\!\cdots\!09}a+\frac{21\!\cdots\!40}{21\!\cdots\!09}$ Copy content Toggle raw display

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

not computed

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 $2$) Copy content Toggle raw display
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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 - 2356992835*x^20 + 23569928735*x^19 + 2204501155911423557*x^18 - 19840511074945783155*x^17 - 1082872885988662125300812370*x^16 + 8662983537627551079745723914*x^15 + 316068217715312920637575328756082112*x^14 - 2212477675609403926959065035382244332*x^13 - 58767377684409992170556055760995021176639986*x^12 + 352604294868670130111344193424712514895639548*x^11 + 7226966902317690915617765163171122089181106876101229*x^10 - 36134837743794543604950652866191954074296615184490417*x^9 - 595718839192069496369987165264666540405203201509932504884044*x^8 + 2382875573577308326894544241652333406345733476067116734044261*x^7 + 32584603183355485909634946704681447022832056003466832943811521734636*x^6 - 97753817890131117015806386829370798310090825227154792176518881828628*x^5 - 1135082737337117340979212736589910913095311476546586257204562141505863404960*x^4 + 2270165637597267278865195276060941498200493335290757444902509490795622424977*x^3 + 22798723985900078548327022529561358312453616137357242512589877765454307023391451962*x^2 - 22798725120982913639264040796618268980326961172862719464781569239076972125427474194*x - 200957389535301956135132770569928284254466122629162705805201402317826353092603069971791081)
 
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 - 11*x^21 - 2356992835*x^20 + 23569928735*x^19 + 2204501155911423557*x^18 - 19840511074945783155*x^17 - 1082872885988662125300812370*x^16 + 8662983537627551079745723914*x^15 + 316068217715312920637575328756082112*x^14 - 2212477675609403926959065035382244332*x^13 - 58767377684409992170556055760995021176639986*x^12 + 352604294868670130111344193424712514895639548*x^11 + 7226966902317690915617765163171122089181106876101229*x^10 - 36134837743794543604950652866191954074296615184490417*x^9 - 595718839192069496369987165264666540405203201509932504884044*x^8 + 2382875573577308326894544241652333406345733476067116734044261*x^7 + 32584603183355485909634946704681447022832056003466832943811521734636*x^6 - 97753817890131117015806386829370798310090825227154792176518881828628*x^5 - 1135082737337117340979212736589910913095311476546586257204562141505863404960*x^4 + 2270165637597267278865195276060941498200493335290757444902509490795622424977*x^3 + 22798723985900078548327022529561358312453616137357242512589877765454307023391451962*x^2 - 22798725120982913639264040796618268980326961172862719464781569239076972125427474194*x - 200957389535301956135132770569928284254466122629162705805201402317826353092603069971791081, 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 - 11*x^21 - 2356992835*x^20 + 23569928735*x^19 + 2204501155911423557*x^18 - 19840511074945783155*x^17 - 1082872885988662125300812370*x^16 + 8662983537627551079745723914*x^15 + 316068217715312920637575328756082112*x^14 - 2212477675609403926959065035382244332*x^13 - 58767377684409992170556055760995021176639986*x^12 + 352604294868670130111344193424712514895639548*x^11 + 7226966902317690915617765163171122089181106876101229*x^10 - 36134837743794543604950652866191954074296615184490417*x^9 - 595718839192069496369987165264666540405203201509932504884044*x^8 + 2382875573577308326894544241652333406345733476067116734044261*x^7 + 32584603183355485909634946704681447022832056003466832943811521734636*x^6 - 97753817890131117015806386829370798310090825227154792176518881828628*x^5 - 1135082737337117340979212736589910913095311476546586257204562141505863404960*x^4 + 2270165637597267278865195276060941498200493335290757444902509490795622424977*x^3 + 22798723985900078548327022529561358312453616137357242512589877765454307023391451962*x^2 - 22798725120982913639264040796618268980326961172862719464781569239076972125427474194*x - 200957389535301956135132770569928284254466122629162705805201402317826353092603069971791081);
 
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 - 11*x^21 - 2356992835*x^20 + 23569928735*x^19 + 2204501155911423557*x^18 - 19840511074945783155*x^17 - 1082872885988662125300812370*x^16 + 8662983537627551079745723914*x^15 + 316068217715312920637575328756082112*x^14 - 2212477675609403926959065035382244332*x^13 - 58767377684409992170556055760995021176639986*x^12 + 352604294868670130111344193424712514895639548*x^11 + 7226966902317690915617765163171122089181106876101229*x^10 - 36134837743794543604950652866191954074296615184490417*x^9 - 595718839192069496369987165264666540405203201509932504884044*x^8 + 2382875573577308326894544241652333406345733476067116734044261*x^7 + 32584603183355485909634946704681447022832056003466832943811521734636*x^6 - 97753817890131117015806386829370798310090825227154792176518881828628*x^5 - 1135082737337117340979212736589910913095311476546586257204562141505863404960*x^4 + 2270165637597267278865195276060941498200493335290757444902509490795622424977*x^3 + 22798723985900078548327022529561358312453616137357242512589877765454307023391451962*x^2 - 22798725120982913639264040796618268980326961172862719464781569239076972125427474194*x - 200957389535301956135132770569928284254466122629162705805201402317826353092603069971791081);
 
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

$C_2^{10}.S_{11}$ (as 22T51):

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 40874803200
The 376 conjugacy class representatives for $C_2^{10}.S_{11}$
Character table for $C_2^{10}.S_{11}$

Intermediate fields

11.11.48706494267293.1

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: 22.22.3084746536040392057834423520781425728271788353261903536454522694362224648434376820066850017183801218370499765821966379524657522809463693823777476581948227706828497025274248056960013839516828667105809406926499481575723283744096906820849165128748157.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}{,}\,{\href{/padicField/2.3.0.1}{3} }^{2}$ ${\href{/padicField/3.7.0.1}{7} }^{2}{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ ${\href{/padicField/5.7.0.1}{7} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}$ ${\href{/padicField/7.11.0.1}{11} }^{2}$ ${\href{/padicField/11.11.0.1}{11} }^{2}$ ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.8.0.1}{8} }$ ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.4.0.1}{4} }$ ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.10.0.1}{10} }$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ ${\href{/padicField/53.11.0.1}{11} }^{2}$ ${\href{/padicField/59.9.0.1}{9} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(421\) Copy content Toggle raw display $\Q_{421}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{421}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $4$$2$$2$$2$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
\(3913599589\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$2$$2$$2$
Deg $14$$2$$7$$7$
\(115692385433\) Copy content Toggle raw display $\Q_{115692385433}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{115692385433}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{115692385433}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{115692385433}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $4$$2$$2$$2$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$