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Properties

Label 22.18.651...449.1
Degree $22$
Signature $[18, 2]$
Discriminant $6.511\times 10^{132}$
Root discriminant \(1\,088\,890.74\)
Ramified primes see page
Class number not computed
Class group not computed
Galois group $C_2^{10}.S_{11}$ (as 22T51)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 - 15789084607*x^20 + 157890846455*x^19 + 110746971971048806316*x^18 - 996722752239328383006*x^17 - 456686908335859663186856128341*x^16 + 3653495289279259709985789751062*x^15 + 1233340365680532560255064426238448307608*x^14 - 8633382623699895563245834354737997399860*x^13 - 2295306548915827818503229932405383312648591807203*x^12 + 13771839405728941185352057872535434682817913093144*x^11 + 3008235976397770175167222399027423306040909666725371574201*x^10 - 15041180008230712300780359161086680732594538490261818878188*x^9 - 2778656157484847280982897747877969488922506517585296579872733446640*x^8 + 11114624720186469324806098611947733941638935381034067180114602303069*x^7 + 1772974228559093358263195450183082391905157516634653024580978835244385317895*x^6 - 5318922724578466658615185173429817814200518248972628117560517528924164100228*x^5 - 744044115416507765021021196646690610169866572713152856848171728626532956098421434479*x^4 + 1488088239697886750639882361285264147203086069826044503097773504915513387520157567545*x^3 + 184729416370582242638251447645440388704667291009348175277482602807244002974054527942977543281*x^2 - 184729417114626363373681945580896803659929128262623837964712319402315440375343750732884448014*x - 20544135315942859966369326009302529501882658344474338458010659738621792061984845009538823872127218171)
 
gp: K = bnfinit(y^22 - 11*y^21 - 15789084607*y^20 + 157890846455*y^19 + 110746971971048806316*y^18 - 996722752239328383006*y^17 - 456686908335859663186856128341*y^16 + 3653495289279259709985789751062*y^15 + 1233340365680532560255064426238448307608*y^14 - 8633382623699895563245834354737997399860*y^13 - 2295306548915827818503229932405383312648591807203*y^12 + 13771839405728941185352057872535434682817913093144*y^11 + 3008235976397770175167222399027423306040909666725371574201*y^10 - 15041180008230712300780359161086680732594538490261818878188*y^9 - 2778656157484847280982897747877969488922506517585296579872733446640*y^8 + 11114624720186469324806098611947733941638935381034067180114602303069*y^7 + 1772974228559093358263195450183082391905157516634653024580978835244385317895*y^6 - 5318922724578466658615185173429817814200518248972628117560517528924164100228*y^5 - 744044115416507765021021196646690610169866572713152856848171728626532956098421434479*y^4 + 1488088239697886750639882361285264147203086069826044503097773504915513387520157567545*y^3 + 184729416370582242638251447645440388704667291009348175277482602807244002974054527942977543281*y^2 - 184729417114626363373681945580896803659929128262623837964712319402315440375343750732884448014*y - 20544135315942859966369326009302529501882658344474338458010659738621792061984845009538823872127218171, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 11*x^21 - 15789084607*x^20 + 157890846455*x^19 + 110746971971048806316*x^18 - 996722752239328383006*x^17 - 456686908335859663186856128341*x^16 + 3653495289279259709985789751062*x^15 + 1233340365680532560255064426238448307608*x^14 - 8633382623699895563245834354737997399860*x^13 - 2295306548915827818503229932405383312648591807203*x^12 + 13771839405728941185352057872535434682817913093144*x^11 + 3008235976397770175167222399027423306040909666725371574201*x^10 - 15041180008230712300780359161086680732594538490261818878188*x^9 - 2778656157484847280982897747877969488922506517585296579872733446640*x^8 + 11114624720186469324806098611947733941638935381034067180114602303069*x^7 + 1772974228559093358263195450183082391905157516634653024580978835244385317895*x^6 - 5318922724578466658615185173429817814200518248972628117560517528924164100228*x^5 - 744044115416507765021021196646690610169866572713152856848171728626532956098421434479*x^4 + 1488088239697886750639882361285264147203086069826044503097773504915513387520157567545*x^3 + 184729416370582242638251447645440388704667291009348175277482602807244002974054527942977543281*x^2 - 184729417114626363373681945580896803659929128262623837964712319402315440375343750732884448014*x - 20544135315942859966369326009302529501882658344474338458010659738621792061984845009538823872127218171);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 11*x^21 - 15789084607*x^20 + 157890846455*x^19 + 110746971971048806316*x^18 - 996722752239328383006*x^17 - 456686908335859663186856128341*x^16 + 3653495289279259709985789751062*x^15 + 1233340365680532560255064426238448307608*x^14 - 8633382623699895563245834354737997399860*x^13 - 2295306548915827818503229932405383312648591807203*x^12 + 13771839405728941185352057872535434682817913093144*x^11 + 3008235976397770175167222399027423306040909666725371574201*x^10 - 15041180008230712300780359161086680732594538490261818878188*x^9 - 2778656157484847280982897747877969488922506517585296579872733446640*x^8 + 11114624720186469324806098611947733941638935381034067180114602303069*x^7 + 1772974228559093358263195450183082391905157516634653024580978835244385317895*x^6 - 5318922724578466658615185173429817814200518248972628117560517528924164100228*x^5 - 744044115416507765021021196646690610169866572713152856848171728626532956098421434479*x^4 + 1488088239697886750639882361285264147203086069826044503097773504915513387520157567545*x^3 + 184729416370582242638251447645440388704667291009348175277482602807244002974054527942977543281*x^2 - 184729417114626363373681945580896803659929128262623837964712319402315440375343750732884448014*x - 20544135315942859966369326009302529501882658344474338458010659738621792061984845009538823872127218171)
 

\( x^{22} - 11 x^{21} - 15789084607 x^{20} + 157890846455 x^{19} + \cdots - 20\!\cdots\!71 \) 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:  $[18, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(651\!\cdots\!449\) \(\medspace = 29^{10}\cdot 151^{2}\cdot 2311^{2}\cdot 24910163^{2}\cdot 1702694681^{10}\) 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:  \(1\,088\,890.74\)
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:  $29^{1/2}151^{1/2}2311^{1/2}24910163^{1/2}1702694681^{1/2}\approx 655155090331.9882$
Ramified primes:   \(29\), \(151\), \(2311\), \(24910163\), \(1702694681\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{49378145749}a^{4}-\frac{2}{49378145749}a^{3}-\frac{3444548226}{49378145749}a^{2}+\frac{3444548227}{49378145749}a-\frac{22464075822}{49378145749}$, $\frac{1}{49378145749}a^{5}-\frac{3444548230}{49378145749}a^{3}-\frac{118777525}{1702694681}a^{2}-\frac{15574979368}{49378145749}a+\frac{4449994105}{49378145749}$, $\frac{1}{24\!\cdots\!01}a^{6}-\frac{3}{24\!\cdots\!01}a^{5}-\frac{15789084661}{24\!\cdots\!01}a^{4}+\frac{31578169327}{24\!\cdots\!01}a^{3}-\frac{49\!\cdots\!12}{24\!\cdots\!01}a^{2}+\frac{49\!\cdots\!48}{24\!\cdots\!01}a+\frac{75\!\cdots\!08}{24\!\cdots\!01}$, $\frac{1}{24\!\cdots\!01}a^{7}-\frac{15789084670}{24\!\cdots\!01}a^{5}-\frac{15789084656}{24\!\cdots\!01}a^{4}-\frac{49\!\cdots\!31}{24\!\cdots\!01}a^{3}-\frac{99\!\cdots\!88}{24\!\cdots\!01}a^{2}-\frac{19\!\cdots\!49}{24\!\cdots\!01}a-\frac{18\!\cdots\!77}{24\!\cdots\!01}$, $\frac{1}{12\!\cdots\!49}a^{8}-\frac{4}{12\!\cdots\!49}a^{7}+\frac{21244524654}{12\!\cdots\!49}a^{6}-\frac{63733573948}{12\!\cdots\!49}a^{5}-\frac{10\!\cdots\!71}{12\!\cdots\!49}a^{4}+\frac{21\!\cdots\!84}{12\!\cdots\!49}a^{3}+\frac{20\!\cdots\!74}{12\!\cdots\!49}a^{2}-\frac{20\!\cdots\!90}{12\!\cdots\!49}a-\frac{22\!\cdots\!89}{12\!\cdots\!49}$, $\frac{1}{12\!\cdots\!49}a^{9}+\frac{21244524638}{12\!\cdots\!49}a^{7}+\frac{21244524668}{12\!\cdots\!49}a^{6}-\frac{10\!\cdots\!63}{12\!\cdots\!49}a^{5}+\frac{27\!\cdots\!01}{12\!\cdots\!49}a^{4}+\frac{20\!\cdots\!08}{12\!\cdots\!49}a^{3}+\frac{53\!\cdots\!80}{12\!\cdots\!49}a^{2}+\frac{23\!\cdots\!27}{12\!\cdots\!49}a-\frac{24\!\cdots\!29}{12\!\cdots\!49}$, $\frac{1}{59\!\cdots\!01}a^{10}-\frac{5}{59\!\cdots\!01}a^{9}+\frac{8899988221}{59\!\cdots\!01}a^{8}-\frac{35599952854}{59\!\cdots\!01}a^{7}+\frac{10\!\cdots\!80}{59\!\cdots\!01}a^{6}-\frac{32\!\cdots\!72}{59\!\cdots\!01}a^{5}-\frac{43\!\cdots\!44}{59\!\cdots\!01}a^{4}+\frac{86\!\cdots\!55}{59\!\cdots\!01}a^{3}+\frac{97\!\cdots\!26}{59\!\cdots\!01}a^{2}-\frac{97\!\cdots\!08}{59\!\cdots\!01}a-\frac{17\!\cdots\!56}{59\!\cdots\!01}$, $\frac{1}{59\!\cdots\!01}a^{11}+\frac{8899988196}{59\!\cdots\!01}a^{9}+\frac{8899988251}{59\!\cdots\!01}a^{8}+\frac{10\!\cdots\!10}{59\!\cdots\!01}a^{7}-\frac{25\!\cdots\!73}{59\!\cdots\!01}a^{6}-\frac{43\!\cdots\!01}{59\!\cdots\!01}a^{5}+\frac{25\!\cdots\!96}{59\!\cdots\!01}a^{4}+\frac{97\!\cdots\!74}{59\!\cdots\!01}a^{3}-\frac{83\!\cdots\!67}{59\!\cdots\!01}a^{2}-\frac{18\!\cdots\!43}{59\!\cdots\!01}a+\frac{13\!\cdots\!14}{59\!\cdots\!01}$, $\frac{1}{29\!\cdots\!49}a^{12}-\frac{6}{29\!\cdots\!49}a^{11}-\frac{3444548211}{29\!\cdots\!49}a^{10}+\frac{17222741110}{29\!\cdots\!49}a^{9}+\frac{98\!\cdots\!57}{29\!\cdots\!49}a^{8}-\frac{39\!\cdots\!54}{29\!\cdots\!49}a^{7}-\frac{17\!\cdots\!32}{29\!\cdots\!49}a^{6}+\frac{53\!\cdots\!63}{29\!\cdots\!49}a^{5}+\frac{10\!\cdots\!99}{29\!\cdots\!49}a^{4}-\frac{20\!\cdots\!69}{29\!\cdots\!49}a^{3}-\frac{31\!\cdots\!55}{29\!\cdots\!49}a^{2}+\frac{31\!\cdots\!97}{29\!\cdots\!49}a+\frac{90\!\cdots\!98}{29\!\cdots\!49}$, $\frac{1}{29\!\cdots\!49}a^{13}-\frac{3444548247}{29\!\cdots\!49}a^{11}-\frac{3444548156}{29\!\cdots\!49}a^{10}+\frac{98\!\cdots\!17}{29\!\cdots\!49}a^{9}-\frac{47\!\cdots\!13}{29\!\cdots\!49}a^{8}-\frac{17\!\cdots\!52}{29\!\cdots\!49}a^{7}+\frac{15\!\cdots\!66}{29\!\cdots\!49}a^{6}+\frac{10\!\cdots\!78}{29\!\cdots\!49}a^{5}-\frac{11\!\cdots\!94}{29\!\cdots\!49}a^{4}-\frac{31\!\cdots\!28}{29\!\cdots\!49}a^{3}+\frac{47\!\cdots\!80}{29\!\cdots\!49}a^{2}+\frac{74\!\cdots\!46}{29\!\cdots\!49}a-\frac{61\!\cdots\!56}{29\!\cdots\!49}$, $\frac{1}{14\!\cdots\!01}a^{14}-\frac{7}{14\!\cdots\!01}a^{13}-\frac{15789084642}{14\!\cdots\!01}a^{12}+\frac{94734507943}{14\!\cdots\!01}a^{11}+\frac{10\!\cdots\!54}{14\!\cdots\!01}a^{10}-\frac{51\!\cdots\!81}{14\!\cdots\!01}a^{9}-\frac{29\!\cdots\!87}{14\!\cdots\!01}a^{8}+\frac{11\!\cdots\!93}{14\!\cdots\!01}a^{7}+\frac{12\!\cdots\!20}{14\!\cdots\!01}a^{6}-\frac{37\!\cdots\!99}{14\!\cdots\!01}a^{5}-\frac{43\!\cdots\!49}{14\!\cdots\!01}a^{4}+\frac{87\!\cdots\!05}{14\!\cdots\!01}a^{3}+\frac{16\!\cdots\!89}{14\!\cdots\!01}a^{2}-\frac{16\!\cdots\!40}{14\!\cdots\!01}a-\frac{44\!\cdots\!38}{14\!\cdots\!01}$, $\frac{1}{14\!\cdots\!01}a^{15}-\frac{15789084691}{14\!\cdots\!01}a^{13}-\frac{15789084551}{14\!\cdots\!01}a^{12}+\frac{10\!\cdots\!55}{14\!\cdots\!01}a^{11}-\frac{38\!\cdots\!04}{14\!\cdots\!01}a^{10}-\frac{29\!\cdots\!49}{14\!\cdots\!01}a^{9}+\frac{88\!\cdots\!12}{14\!\cdots\!01}a^{8}+\frac{12\!\cdots\!29}{14\!\cdots\!01}a^{7}-\frac{10\!\cdots\!94}{14\!\cdots\!01}a^{6}-\frac{43\!\cdots\!19}{14\!\cdots\!01}a^{5}+\frac{47\!\cdots\!37}{14\!\cdots\!01}a^{4}+\frac{16\!\cdots\!60}{14\!\cdots\!01}a^{3}-\frac{24\!\cdots\!80}{14\!\cdots\!01}a^{2}-\frac{36\!\cdots\!44}{14\!\cdots\!01}a+\frac{30\!\cdots\!46}{14\!\cdots\!01}$, $\frac{1}{71\!\cdots\!49}a^{16}-\frac{8}{71\!\cdots\!49}a^{15}+\frac{21244524677}{71\!\cdots\!49}a^{14}-\frac{148711672599}{71\!\cdots\!49}a^{13}+\frac{44\!\cdots\!07}{71\!\cdots\!49}a^{12}-\frac{26\!\cdots\!19}{71\!\cdots\!49}a^{11}+\frac{80\!\cdots\!42}{71\!\cdots\!49}a^{10}-\frac{40\!\cdots\!92}{71\!\cdots\!49}a^{9}+\frac{13\!\cdots\!83}{71\!\cdots\!49}a^{8}-\frac{54\!\cdots\!03}{71\!\cdots\!49}a^{7}+\frac{22\!\cdots\!90}{71\!\cdots\!49}a^{6}-\frac{66\!\cdots\!34}{71\!\cdots\!49}a^{5}+\frac{35\!\cdots\!96}{71\!\cdots\!49}a^{4}-\frac{70\!\cdots\!69}{71\!\cdots\!49}a^{3}+\frac{53\!\cdots\!73}{71\!\cdots\!49}a^{2}-\frac{53\!\cdots\!45}{71\!\cdots\!49}a+\frac{14\!\cdots\!57}{71\!\cdots\!49}$, $\frac{1}{71\!\cdots\!49}a^{17}+\frac{21244524613}{71\!\cdots\!49}a^{15}+\frac{21244524817}{71\!\cdots\!49}a^{14}+\frac{44\!\cdots\!15}{71\!\cdots\!49}a^{13}+\frac{88\!\cdots\!37}{71\!\cdots\!49}a^{12}+\frac{80\!\cdots\!90}{71\!\cdots\!49}a^{11}+\frac{24\!\cdots\!44}{71\!\cdots\!49}a^{10}+\frac{13\!\cdots\!47}{71\!\cdots\!49}a^{9}+\frac{54\!\cdots\!61}{71\!\cdots\!49}a^{8}+\frac{22\!\cdots\!66}{71\!\cdots\!49}a^{7}+\frac{11\!\cdots\!86}{71\!\cdots\!49}a^{6}+\frac{35\!\cdots\!24}{71\!\cdots\!49}a^{5}+\frac{21\!\cdots\!99}{71\!\cdots\!49}a^{4}+\frac{53\!\cdots\!21}{71\!\cdots\!49}a^{3}+\frac{37\!\cdots\!39}{71\!\cdots\!49}a^{2}-\frac{41\!\cdots\!03}{71\!\cdots\!49}a+\frac{11\!\cdots\!56}{71\!\cdots\!49}$, $\frac{1}{35\!\cdots\!01}a^{18}-\frac{9}{35\!\cdots\!01}a^{17}+\frac{8899988248}{35\!\cdots\!01}a^{16}-\frac{71199905780}{35\!\cdots\!01}a^{15}+\frac{17\!\cdots\!57}{35\!\cdots\!01}a^{14}-\frac{12\!\cdots\!63}{35\!\cdots\!01}a^{13}+\frac{25\!\cdots\!02}{35\!\cdots\!01}a^{12}-\frac{15\!\cdots\!31}{35\!\cdots\!01}a^{11}+\frac{37\!\cdots\!21}{35\!\cdots\!01}a^{10}-\frac{18\!\cdots\!82}{35\!\cdots\!01}a^{9}+\frac{53\!\cdots\!22}{35\!\cdots\!01}a^{8}-\frac{21\!\cdots\!13}{35\!\cdots\!01}a^{7}+\frac{75\!\cdots\!00}{35\!\cdots\!01}a^{6}-\frac{22\!\cdots\!07}{35\!\cdots\!01}a^{5}+\frac{10\!\cdots\!90}{35\!\cdots\!01}a^{4}-\frac{20\!\cdots\!65}{35\!\cdots\!01}a^{3}+\frac{14\!\cdots\!53}{35\!\cdots\!01}a^{2}-\frac{14\!\cdots\!44}{35\!\cdots\!01}a+\frac{22\!\cdots\!54}{35\!\cdots\!01}$, $\frac{1}{35\!\cdots\!01}a^{19}+\frac{8899988167}{35\!\cdots\!01}a^{17}+\frac{8899988452}{35\!\cdots\!01}a^{16}+\frac{17\!\cdots\!37}{35\!\cdots\!01}a^{15}+\frac{35\!\cdots\!50}{35\!\cdots\!01}a^{14}+\frac{25\!\cdots\!35}{35\!\cdots\!01}a^{13}+\frac{77\!\cdots\!87}{35\!\cdots\!01}a^{12}+\frac{37\!\cdots\!42}{35\!\cdots\!01}a^{11}+\frac{15\!\cdots\!07}{35\!\cdots\!01}a^{10}+\frac{53\!\cdots\!84}{35\!\cdots\!01}a^{9}+\frac{26\!\cdots\!85}{35\!\cdots\!01}a^{8}+\frac{75\!\cdots\!83}{35\!\cdots\!01}a^{7}+\frac{45\!\cdots\!93}{35\!\cdots\!01}a^{6}+\frac{10\!\cdots\!27}{35\!\cdots\!01}a^{5}+\frac{72\!\cdots\!45}{35\!\cdots\!01}a^{4}+\frac{14\!\cdots\!68}{35\!\cdots\!01}a^{3}+\frac{11\!\cdots\!33}{35\!\cdots\!01}a^{2}-\frac{12\!\cdots\!42}{35\!\cdots\!01}a+\frac{19\!\cdots\!86}{35\!\cdots\!01}$, $\frac{1}{17\!\cdots\!49}a^{20}-\frac{10}{17\!\cdots\!49}a^{19}-\frac{3444548180}{17\!\cdots\!49}a^{18}+\frac{31000933905}{17\!\cdots\!49}a^{17}+\frac{68\!\cdots\!61}{17\!\cdots\!49}a^{16}-\frac{54\!\cdots\!60}{17\!\cdots\!49}a^{15}+\frac{38\!\cdots\!56}{17\!\cdots\!49}a^{14}-\frac{26\!\cdots\!02}{17\!\cdots\!49}a^{13}+\frac{59\!\cdots\!78}{17\!\cdots\!49}a^{12}-\frac{35\!\cdots\!56}{17\!\cdots\!49}a^{11}+\frac{71\!\cdots\!27}{17\!\cdots\!49}a^{10}-\frac{35\!\cdots\!01}{17\!\cdots\!49}a^{9}+\frac{88\!\cdots\!99}{17\!\cdots\!49}a^{8}-\frac{35\!\cdots\!26}{17\!\cdots\!49}a^{7}+\frac{10\!\cdots\!97}{17\!\cdots\!49}a^{6}-\frac{32\!\cdots\!96}{17\!\cdots\!49}a^{5}+\frac{13\!\cdots\!88}{17\!\cdots\!49}a^{4}-\frac{27\!\cdots\!92}{17\!\cdots\!49}a^{3}+\frac{16\!\cdots\!85}{17\!\cdots\!49}a^{2}-\frac{16\!\cdots\!74}{17\!\cdots\!49}a+\frac{20\!\cdots\!64}{17\!\cdots\!49}$, $\frac{1}{17\!\cdots\!49}a^{21}-\frac{3444548280}{17\!\cdots\!49}a^{19}-\frac{3444547895}{17\!\cdots\!49}a^{18}+\frac{68\!\cdots\!11}{17\!\cdots\!49}a^{17}+\frac{13\!\cdots\!50}{17\!\cdots\!49}a^{16}+\frac{38\!\cdots\!56}{17\!\cdots\!49}a^{15}+\frac{11\!\cdots\!58}{17\!\cdots\!49}a^{14}+\frac{20\!\cdots\!02}{60\!\cdots\!81}a^{13}+\frac{23\!\cdots\!24}{17\!\cdots\!49}a^{12}+\frac{71\!\cdots\!67}{17\!\cdots\!49}a^{11}+\frac{35\!\cdots\!69}{17\!\cdots\!49}a^{10}+\frac{88\!\cdots\!89}{17\!\cdots\!49}a^{9}+\frac{53\!\cdots\!64}{17\!\cdots\!49}a^{8}+\frac{10\!\cdots\!37}{17\!\cdots\!49}a^{7}+\frac{76\!\cdots\!74}{17\!\cdots\!49}a^{6}+\frac{13\!\cdots\!28}{17\!\cdots\!49}a^{5}+\frac{37\!\cdots\!72}{60\!\cdots\!81}a^{4}+\frac{16\!\cdots\!65}{17\!\cdots\!49}a^{3}+\frac{15\!\cdots\!76}{17\!\cdots\!49}a^{2}-\frac{14\!\cdots\!76}{17\!\cdots\!49}a+\frac{20\!\cdots\!40}{17\!\cdots\!49}$ 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:  $19$
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 - 15789084607*x^20 + 157890846455*x^19 + 110746971971048806316*x^18 - 996722752239328383006*x^17 - 456686908335859663186856128341*x^16 + 3653495289279259709985789751062*x^15 + 1233340365680532560255064426238448307608*x^14 - 8633382623699895563245834354737997399860*x^13 - 2295306548915827818503229932405383312648591807203*x^12 + 13771839405728941185352057872535434682817913093144*x^11 + 3008235976397770175167222399027423306040909666725371574201*x^10 - 15041180008230712300780359161086680732594538490261818878188*x^9 - 2778656157484847280982897747877969488922506517585296579872733446640*x^8 + 11114624720186469324806098611947733941638935381034067180114602303069*x^7 + 1772974228559093358263195450183082391905157516634653024580978835244385317895*x^6 - 5318922724578466658615185173429817814200518248972628117560517528924164100228*x^5 - 744044115416507765021021196646690610169866572713152856848171728626532956098421434479*x^4 + 1488088239697886750639882361285264147203086069826044503097773504915513387520157567545*x^3 + 184729416370582242638251447645440388704667291009348175277482602807244002974054527942977543281*x^2 - 184729417114626363373681945580896803659929128262623837964712319402315440375343750732884448014*x - 20544135315942859966369326009302529501882658344474338458010659738621792061984845009538823872127218171)
 
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 - 15789084607*x^20 + 157890846455*x^19 + 110746971971048806316*x^18 - 996722752239328383006*x^17 - 456686908335859663186856128341*x^16 + 3653495289279259709985789751062*x^15 + 1233340365680532560255064426238448307608*x^14 - 8633382623699895563245834354737997399860*x^13 - 2295306548915827818503229932405383312648591807203*x^12 + 13771839405728941185352057872535434682817913093144*x^11 + 3008235976397770175167222399027423306040909666725371574201*x^10 - 15041180008230712300780359161086680732594538490261818878188*x^9 - 2778656157484847280982897747877969488922506517585296579872733446640*x^8 + 11114624720186469324806098611947733941638935381034067180114602303069*x^7 + 1772974228559093358263195450183082391905157516634653024580978835244385317895*x^6 - 5318922724578466658615185173429817814200518248972628117560517528924164100228*x^5 - 744044115416507765021021196646690610169866572713152856848171728626532956098421434479*x^4 + 1488088239697886750639882361285264147203086069826044503097773504915513387520157567545*x^3 + 184729416370582242638251447645440388704667291009348175277482602807244002974054527942977543281*x^2 - 184729417114626363373681945580896803659929128262623837964712319402315440375343750732884448014*x - 20544135315942859966369326009302529501882658344474338458010659738621792061984845009538823872127218171, 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 - 15789084607*x^20 + 157890846455*x^19 + 110746971971048806316*x^18 - 996722752239328383006*x^17 - 456686908335859663186856128341*x^16 + 3653495289279259709985789751062*x^15 + 1233340365680532560255064426238448307608*x^14 - 8633382623699895563245834354737997399860*x^13 - 2295306548915827818503229932405383312648591807203*x^12 + 13771839405728941185352057872535434682817913093144*x^11 + 3008235976397770175167222399027423306040909666725371574201*x^10 - 15041180008230712300780359161086680732594538490261818878188*x^9 - 2778656157484847280982897747877969488922506517585296579872733446640*x^8 + 11114624720186469324806098611947733941638935381034067180114602303069*x^7 + 1772974228559093358263195450183082391905157516634653024580978835244385317895*x^6 - 5318922724578466658615185173429817814200518248972628117560517528924164100228*x^5 - 744044115416507765021021196646690610169866572713152856848171728626532956098421434479*x^4 + 1488088239697886750639882361285264147203086069826044503097773504915513387520157567545*x^3 + 184729416370582242638251447645440388704667291009348175277482602807244002974054527942977543281*x^2 - 184729417114626363373681945580896803659929128262623837964712319402315440375343750732884448014*x - 20544135315942859966369326009302529501882658344474338458010659738621792061984845009538823872127218171);
 
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 - 15789084607*x^20 + 157890846455*x^19 + 110746971971048806316*x^18 - 996722752239328383006*x^17 - 456686908335859663186856128341*x^16 + 3653495289279259709985789751062*x^15 + 1233340365680532560255064426238448307608*x^14 - 8633382623699895563245834354737997399860*x^13 - 2295306548915827818503229932405383312648591807203*x^12 + 13771839405728941185352057872535434682817913093144*x^11 + 3008235976397770175167222399027423306040909666725371574201*x^10 - 15041180008230712300780359161086680732594538490261818878188*x^9 - 2778656157484847280982897747877969488922506517585296579872733446640*x^8 + 11114624720186469324806098611947733941638935381034067180114602303069*x^7 + 1772974228559093358263195450183082391905157516634653024580978835244385317895*x^6 - 5318922724578466658615185173429817814200518248972628117560517528924164100228*x^5 - 744044115416507765021021196646690610169866572713152856848171728626532956098421434479*x^4 + 1488088239697886750639882361285264147203086069826044503097773504915513387520157567545*x^3 + 184729416370582242638251447645440388704667291009348175277482602807244002974054527942977543281*x^2 - 184729417114626363373681945580896803659929128262623837964712319402315440375343750732884448014*x - 20544135315942859966369326009302529501882658344474338458010659738621792061984845009538823872127218171);
 
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 $C_2^{10}.S_{11}$ are not computed
Character table for $C_2^{10}.S_{11}$ is not computed

Intermediate fields

11.9.8692675390643.1

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: 22.0.1845167752419714627516945872278220189098084549702727475982340844115734708868405875281492700920983462045242300114593560894328072719676249021010558289302491771941906149664031851477206209224036755179404839354987086066262904628358632851815247801124361107.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }^{2}{,}\,{\href{/padicField/2.5.0.1}{5} }^{2}$ ${\href{/padicField/3.11.0.1}{11} }^{2}$ ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.10.0.1}{10} }$ ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.11.0.1}{11} }^{2}$ ${\href{/padicField/13.11.0.1}{11} }^{2}$ $18{,}\,{\href{/padicField/17.4.0.1}{4} }$ ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ R ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.10.0.1}{10} }$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.11.0.1}{11} }^{2}$ ${\href{/padicField/43.11.0.1}{11} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ $20{,}\,{\href{/padicField/59.2.0.1}{2} }$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(29\) Copy content Toggle raw display 29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.2$x^{4} - 696 x^{2} + 1682$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.16.8.1$x^{16} + 5568 x^{15} + 13563880 x^{14} + 18881728320 x^{13} + 16428480372226 x^{12} + 9148801553592528 x^{11} + 3184735792552646192 x^{10} + 633724384469595120286 x^{9} + 55249468166464640094225 x^{8} + 18378034980956595686936 x^{7} + 2688132351641987586456 x^{6} + 2199337349954841447326 x^{5} + 180719728922044973705920 x^{4} + 1366890777105246403746256 x^{3} + 1613855339675279772970993 x^{2} + 1393970274901245210856928 x + 165405867866431160098509$$2$$8$$8$$C_8\times C_2$$[\ ]_{2}^{8}$
\(151\) Copy content Toggle raw display $\Q_{151}$$x + 145$$1$$1$$0$Trivial$[\ ]$
$\Q_{151}$$x + 145$$1$$1$$0$Trivial$[\ ]$
151.2.1.1$x^{2} + 453$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.1$x^{2} + 453$$2$$1$$1$$C_2$$[\ ]_{2}$
151.8.0.1$x^{8} + 9 x^{4} + 140 x^{3} + 122 x^{2} + 43 x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
151.8.0.1$x^{8} + 9 x^{4} + 140 x^{3} + 122 x^{2} + 43 x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
\(2311\) Copy content Toggle raw display 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 $7$$1$$7$$0$$C_7$$[\ ]^{7}$
Deg $7$$1$$7$$0$$C_7$$[\ ]^{7}$
\(24910163\) Copy content Toggle raw display 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 $7$$1$$7$$0$$C_7$$[\ ]^{7}$
Deg $7$$1$$7$$0$$C_7$$[\ ]^{7}$
\(1702694681\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $4$$2$$2$$2$
Deg $16$$2$$8$$8$