Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 9*x^24 - 20*x^23 - 12*x^22 + 186*x^21 + 794*x^20 - 27*x^19 - 4404*x^18 - 5528*x^17 + 1038*x^16 + 13671*x^15 + 59660*x^14 + 64299*x^13 - 161565*x^12 - 318218*x^11 + 37881*x^10 + 189642*x^9 - 55703*x^8 + 873384*x^7 + 1978515*x^6 - 188640*x^5 - 4211379*x^4 - 4130865*x^3 + 1301535*x^2 + 4678236*x + 2259171)
gp: K = bnfinit(y^26 - 9*y^24 - 20*y^23 - 12*y^22 + 186*y^21 + 794*y^20 - 27*y^19 - 4404*y^18 - 5528*y^17 + 1038*y^16 + 13671*y^15 + 59660*y^14 + 64299*y^13 - 161565*y^12 - 318218*y^11 + 37881*y^10 + 189642*y^9 - 55703*y^8 + 873384*y^7 + 1978515*y^6 - 188640*y^5 - 4211379*y^4 - 4130865*y^3 + 1301535*y^2 + 4678236*y + 2259171, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - 9*x^24 - 20*x^23 - 12*x^22 + 186*x^21 + 794*x^20 - 27*x^19 - 4404*x^18 - 5528*x^17 + 1038*x^16 + 13671*x^15 + 59660*x^14 + 64299*x^13 - 161565*x^12 - 318218*x^11 + 37881*x^10 + 189642*x^9 - 55703*x^8 + 873384*x^7 + 1978515*x^6 - 188640*x^5 - 4211379*x^4 - 4130865*x^3 + 1301535*x^2 + 4678236*x + 2259171);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 9*x^24 - 20*x^23 - 12*x^22 + 186*x^21 + 794*x^20 - 27*x^19 - 4404*x^18 - 5528*x^17 + 1038*x^16 + 13671*x^15 + 59660*x^14 + 64299*x^13 - 161565*x^12 - 318218*x^11 + 37881*x^10 + 189642*x^9 - 55703*x^8 + 873384*x^7 + 1978515*x^6 - 188640*x^5 - 4211379*x^4 - 4130865*x^3 + 1301535*x^2 + 4678236*x + 2259171)
\( x^{26} - 9 x^{24} - 20 x^{23} - 12 x^{22} + 186 x^{21} + 794 x^{20} - 27 x^{19} - 4404 x^{18} + \cdots + 2259171 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $26$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-4634664002656599036916177705626061158525123\)
\(\medspace = -\,3^{13}\cdot 1093^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(43.75\) | 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: | $3^{1/2}1093^{1/2}\approx 57.262553208881634$ | ||
| Ramified primes: |
\(3\), \(1093\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{2}$, $\frac{1}{9}a^{9}-\frac{1}{3}a^{5}-\frac{4}{9}a^{3}-\frac{1}{3}a$, $\frac{1}{9}a^{10}-\frac{1}{9}a^{4}$, $\frac{1}{9}a^{11}-\frac{1}{9}a^{5}$, $\frac{1}{9}a^{12}-\frac{1}{9}a^{6}$, $\frac{1}{9}a^{13}-\frac{1}{9}a^{7}$, $\frac{1}{9}a^{14}-\frac{1}{9}a^{8}$, $\frac{1}{27}a^{15}+\frac{1}{27}a^{13}+\frac{1}{27}a^{11}-\frac{1}{27}a^{9}-\frac{1}{27}a^{7}+\frac{8}{27}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{27}a^{16}+\frac{1}{27}a^{14}+\frac{1}{27}a^{12}-\frac{1}{27}a^{10}-\frac{1}{27}a^{8}-\frac{1}{27}a^{6}$, $\frac{1}{81}a^{17}+\frac{1}{27}a^{13}+\frac{1}{81}a^{11}+\frac{1}{27}a^{9}-\frac{4}{27}a^{7}+\frac{25}{81}a^{5}+\frac{8}{27}a^{3}+\frac{4}{9}a$, $\frac{1}{891}a^{18}+\frac{2}{891}a^{17}-\frac{4}{297}a^{16}-\frac{5}{297}a^{15}-\frac{4}{99}a^{14}-\frac{1}{33}a^{13}+\frac{34}{891}a^{12}-\frac{13}{891}a^{11}-\frac{10}{297}a^{10}+\frac{7}{297}a^{9}-\frac{2}{33}a^{8}-\frac{8}{99}a^{7}+\frac{127}{891}a^{6}-\frac{151}{891}a^{5}-\frac{148}{297}a^{4}-\frac{10}{27}a^{3}+\frac{28}{99}a^{2}-\frac{34}{99}a+\frac{2}{11}$, $\frac{1}{891}a^{19}-\frac{5}{891}a^{17}+\frac{1}{99}a^{16}-\frac{2}{297}a^{15}+\frac{5}{99}a^{14}+\frac{2}{81}a^{13}+\frac{2}{99}a^{12}+\frac{7}{891}a^{11}-\frac{2}{99}a^{10}+\frac{4}{99}a^{9}+\frac{4}{99}a^{8}-\frac{59}{891}a^{7}+\frac{10}{99}a^{6}+\frac{430}{891}a^{5}+\frac{40}{99}a^{4}-\frac{136}{297}a^{3}-\frac{8}{33}a^{2}-\frac{2}{99}a-\frac{4}{11}$, $\frac{1}{2673}a^{20}-\frac{1}{2673}a^{19}-\frac{1}{297}a^{17}+\frac{8}{891}a^{16}+\frac{1}{297}a^{15}+\frac{94}{2673}a^{14}-\frac{7}{2673}a^{13}-\frac{13}{891}a^{12}+\frac{4}{99}a^{11}-\frac{32}{891}a^{10}+\frac{8}{297}a^{9}-\frac{68}{2673}a^{8}-\frac{46}{2673}a^{7}+\frac{94}{891}a^{6}+\frac{8}{27}a^{5}+\frac{53}{297}a^{4}-\frac{7}{33}a^{3}-\frac{34}{99}a^{2}-\frac{13}{99}a+\frac{1}{11}$, $\frac{1}{8019}a^{21}+\frac{1}{8019}a^{20}-\frac{2}{8019}a^{19}+\frac{8}{2673}a^{17}+\frac{16}{2673}a^{16}+\frac{76}{8019}a^{15}-\frac{31}{729}a^{14}+\frac{100}{8019}a^{13}+\frac{145}{2673}a^{12}-\frac{65}{2673}a^{11}-\frac{64}{2673}a^{10}+\frac{166}{8019}a^{9}+\frac{421}{8019}a^{8}-\frac{854}{8019}a^{7}-\frac{388}{2673}a^{6}-\frac{98}{891}a^{5}+\frac{61}{891}a^{4}+\frac{5}{33}a^{3}+\frac{5}{11}a^{2}+\frac{112}{297}a+\frac{8}{33}$, $\frac{1}{8019}a^{22}-\frac{1}{8019}a^{19}-\frac{1}{2673}a^{18}+\frac{14}{2673}a^{17}+\frac{127}{8019}a^{16}+\frac{5}{2673}a^{15}-\frac{128}{2673}a^{14}-\frac{442}{8019}a^{13}-\frac{20}{891}a^{12}-\frac{38}{2673}a^{11}+\frac{26}{729}a^{10}+\frac{67}{2673}a^{9}-\frac{205}{2673}a^{8}-\frac{53}{729}a^{7}-\frac{371}{2673}a^{6}+\frac{62}{891}a^{5}+\frac{179}{891}a^{4}+\frac{49}{297}a^{3}+\frac{118}{297}a^{2}+\frac{62}{297}a+\frac{10}{33}$, $\frac{1}{24057}a^{23}-\frac{1}{24057}a^{22}-\frac{1}{24057}a^{20}+\frac{7}{24057}a^{19}+\frac{1}{2673}a^{18}-\frac{131}{24057}a^{17}+\frac{104}{24057}a^{16}+\frac{29}{8019}a^{15}-\frac{1327}{24057}a^{14}+\frac{1135}{24057}a^{13}-\frac{134}{8019}a^{12}-\frac{950}{24057}a^{11}+\frac{239}{24057}a^{10}+\frac{376}{8019}a^{9}-\frac{2749}{24057}a^{8}+\frac{46}{24057}a^{7}-\frac{4}{8019}a^{6}+\frac{259}{891}a^{5}-\frac{173}{2673}a^{4}+\frac{128}{297}a^{3}+\frac{229}{891}a^{2}+\frac{298}{891}a-\frac{13}{99}$, $\frac{1}{697653}a^{24}+\frac{1}{63423}a^{23}-\frac{13}{232551}a^{22}-\frac{2}{63423}a^{21}-\frac{107}{697653}a^{20}+\frac{1}{2871}a^{19}-\frac{212}{697653}a^{18}-\frac{1729}{697653}a^{17}-\frac{8}{232551}a^{16}+\frac{343}{24057}a^{15}+\frac{25339}{697653}a^{14}+\frac{8224}{232551}a^{13}+\frac{12442}{697653}a^{12}-\frac{15166}{697653}a^{11}-\frac{2855}{77517}a^{10}+\frac{11756}{697653}a^{9}+\frac{110740}{697653}a^{8}+\frac{1604}{232551}a^{7}+\frac{197}{2871}a^{6}-\frac{26393}{77517}a^{5}+\frac{1904}{8613}a^{4}+\frac{1399}{25839}a^{3}+\frac{2011}{25839}a^{2}-\frac{140}{957}a-\frac{84}{319}$, $\frac{1}{81\!\cdots\!11}a^{25}-\frac{56\!\cdots\!18}{81\!\cdots\!11}a^{24}-\frac{10\!\cdots\!97}{81\!\cdots\!11}a^{23}+\frac{14\!\cdots\!42}{81\!\cdots\!11}a^{22}-\frac{47\!\cdots\!10}{81\!\cdots\!11}a^{21}-\frac{12\!\cdots\!21}{81\!\cdots\!11}a^{20}-\frac{81\!\cdots\!01}{81\!\cdots\!11}a^{19}-\frac{31\!\cdots\!33}{81\!\cdots\!11}a^{18}-\frac{23\!\cdots\!68}{81\!\cdots\!11}a^{17}-\frac{33\!\cdots\!41}{81\!\cdots\!11}a^{16}+\frac{96\!\cdots\!28}{81\!\cdots\!11}a^{15}-\frac{32\!\cdots\!04}{81\!\cdots\!11}a^{14}-\frac{32\!\cdots\!30}{81\!\cdots\!11}a^{13}-\frac{19\!\cdots\!80}{74\!\cdots\!01}a^{12}+\frac{29\!\cdots\!52}{81\!\cdots\!11}a^{11}-\frac{20\!\cdots\!32}{81\!\cdots\!11}a^{10}+\frac{22\!\cdots\!28}{18\!\cdots\!77}a^{9}+\frac{10\!\cdots\!04}{81\!\cdots\!11}a^{8}+\frac{22\!\cdots\!71}{27\!\cdots\!37}a^{7}-\frac{64\!\cdots\!20}{90\!\cdots\!79}a^{6}+\frac{10\!\cdots\!24}{90\!\cdots\!79}a^{5}-\frac{32\!\cdots\!07}{30\!\cdots\!93}a^{4}+\frac{22\!\cdots\!74}{10\!\cdots\!31}a^{3}-\frac{56\!\cdots\!46}{30\!\cdots\!93}a^{2}+\frac{16\!\cdots\!18}{33\!\cdots\!77}a-\frac{85\!\cdots\!41}{37\!\cdots\!53}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: | $12$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{36208078418198207877821}{4458879115785042788098908891} a^{25} - \frac{359615535410293894115}{45039182987727704930292009} a^{24} - \frac{31262516208027746085383}{495431012865004754233212099} a^{23} - \frac{425990883981092931175735}{4458879115785042788098908891} a^{22} - \frac{22199755028563812957317}{1486293038595014262699636297} a^{21} + \frac{2169545031902070785718007}{1486293038595014262699636297} a^{20} + \frac{1966809765563692650586139}{405352646889549344372628081} a^{19} - \frac{2436372673414889562011809}{495431012865004754233212099} a^{18} - \frac{43113806312603747638121330}{1486293038595014262699636297} a^{17} - \frac{55496681384568909869166544}{4458879115785042788098908891} a^{16} + \frac{27681912439440856669971023}{1486293038595014262699636297} a^{15} + \frac{12712401746592951227598830}{165143670955001584744404033} a^{14} + \frac{1692552223897118071205295928}{4458879115785042788098908891} a^{13} + \frac{168498381437229074001685235}{1486293038595014262699636297} a^{12} - \frac{2028905847961002529244927770}{1486293038595014262699636297} a^{11} - \frac{4313244339113148598844038264}{4458879115785042788098908891} a^{10} + \frac{720907275691409161955384764}{495431012865004754233212099} a^{9} - \frac{19088612661357788993348372}{135117548963183114790876027} a^{8} - \frac{260934342760137406650214223}{405352646889549344372628081} a^{7} + \frac{10759599472953414157159779574}{1486293038595014262699636297} a^{6} + \frac{336473067120924159363559183}{45039182987727704930292009} a^{5} - \frac{4829827678413257911798921813}{495431012865004754233212099} a^{4} - \frac{3635339375171236108272860357}{165143670955001584744404033} a^{3} - \frac{1174528933481695845663433988}{165143670955001584744404033} a^{2} + \frac{3149028082692286688645040485}{165143670955001584744404033} a + \frac{296381255429909873085936859}{18349296772777953860489337} \)
(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: |
$\frac{39\!\cdots\!42}{81\!\cdots\!11}a^{25}-\frac{59\!\cdots\!79}{81\!\cdots\!11}a^{24}+\frac{44\!\cdots\!66}{81\!\cdots\!11}a^{23}-\frac{25\!\cdots\!23}{81\!\cdots\!11}a^{22}-\frac{28\!\cdots\!95}{81\!\cdots\!11}a^{21}-\frac{64\!\cdots\!05}{81\!\cdots\!11}a^{20}-\frac{82\!\cdots\!26}{81\!\cdots\!11}a^{19}+\frac{52\!\cdots\!91}{74\!\cdots\!01}a^{18}+\frac{27\!\cdots\!31}{81\!\cdots\!11}a^{17}-\frac{16\!\cdots\!44}{81\!\cdots\!11}a^{16}-\frac{10\!\cdots\!19}{81\!\cdots\!11}a^{15}-\frac{96\!\cdots\!11}{81\!\cdots\!11}a^{14}-\frac{13\!\cdots\!86}{81\!\cdots\!11}a^{13}-\frac{15\!\cdots\!35}{81\!\cdots\!11}a^{12}+\frac{16\!\cdots\!10}{81\!\cdots\!11}a^{11}+\frac{14\!\cdots\!77}{74\!\cdots\!01}a^{10}-\frac{78\!\cdots\!95}{18\!\cdots\!77}a^{9}-\frac{21\!\cdots\!68}{81\!\cdots\!11}a^{8}+\frac{38\!\cdots\!89}{90\!\cdots\!79}a^{7}-\frac{10\!\cdots\!42}{90\!\cdots\!79}a^{6}-\frac{13\!\cdots\!38}{90\!\cdots\!79}a^{5}+\frac{10\!\cdots\!76}{33\!\cdots\!77}a^{4}+\frac{37\!\cdots\!86}{10\!\cdots\!31}a^{3}-\frac{49\!\cdots\!84}{30\!\cdots\!93}a^{2}-\frac{28\!\cdots\!95}{10\!\cdots\!31}a-\frac{53\!\cdots\!29}{11\!\cdots\!59}$, $\frac{13\!\cdots\!70}{81\!\cdots\!11}a^{25}+\frac{37\!\cdots\!77}{81\!\cdots\!11}a^{24}-\frac{24\!\cdots\!68}{81\!\cdots\!11}a^{23}-\frac{38\!\cdots\!34}{81\!\cdots\!11}a^{22}-\frac{17\!\cdots\!08}{81\!\cdots\!11}a^{21}+\frac{28\!\cdots\!62}{81\!\cdots\!11}a^{20}+\frac{17\!\cdots\!01}{81\!\cdots\!11}a^{19}+\frac{74\!\cdots\!39}{81\!\cdots\!11}a^{18}-\frac{11\!\cdots\!83}{81\!\cdots\!11}a^{17}-\frac{12\!\cdots\!27}{81\!\cdots\!11}a^{16}+\frac{13\!\cdots\!49}{81\!\cdots\!11}a^{15}+\frac{14\!\cdots\!57}{81\!\cdots\!11}a^{14}+\frac{11\!\cdots\!89}{81\!\cdots\!11}a^{13}+\frac{19\!\cdots\!88}{81\!\cdots\!11}a^{12}-\frac{47\!\cdots\!18}{81\!\cdots\!11}a^{11}-\frac{78\!\cdots\!12}{81\!\cdots\!11}a^{10}+\frac{14\!\cdots\!33}{18\!\cdots\!77}a^{9}+\frac{46\!\cdots\!34}{81\!\cdots\!11}a^{8}-\frac{32\!\cdots\!40}{27\!\cdots\!37}a^{7}+\frac{27\!\cdots\!25}{90\!\cdots\!79}a^{6}+\frac{52\!\cdots\!03}{90\!\cdots\!79}a^{5}-\frac{86\!\cdots\!45}{30\!\cdots\!93}a^{4}-\frac{11\!\cdots\!65}{91\!\cdots\!21}a^{3}-\frac{25\!\cdots\!70}{30\!\cdots\!93}a^{2}+\frac{32\!\cdots\!07}{33\!\cdots\!77}a+\frac{41\!\cdots\!32}{37\!\cdots\!53}$, $\frac{14\!\cdots\!17}{27\!\cdots\!37}a^{25}-\frac{12\!\cdots\!30}{27\!\cdots\!37}a^{24}-\frac{27\!\cdots\!07}{37\!\cdots\!53}a^{23}-\frac{26\!\cdots\!60}{27\!\cdots\!37}a^{22}-\frac{11\!\cdots\!73}{27\!\cdots\!37}a^{21}+\frac{28\!\cdots\!32}{10\!\cdots\!31}a^{20}+\frac{45\!\cdots\!65}{27\!\cdots\!37}a^{19}+\frac{49\!\cdots\!71}{27\!\cdots\!37}a^{18}+\frac{12\!\cdots\!13}{90\!\cdots\!79}a^{17}+\frac{36\!\cdots\!78}{24\!\cdots\!67}a^{16}-\frac{32\!\cdots\!89}{27\!\cdots\!37}a^{15}-\frac{11\!\cdots\!53}{90\!\cdots\!79}a^{14}+\frac{15\!\cdots\!91}{27\!\cdots\!37}a^{13}-\frac{25\!\cdots\!22}{27\!\cdots\!37}a^{12}+\frac{10\!\cdots\!25}{90\!\cdots\!79}a^{11}+\frac{25\!\cdots\!47}{93\!\cdots\!53}a^{10}-\frac{13\!\cdots\!87}{63\!\cdots\!59}a^{9}-\frac{49\!\cdots\!20}{90\!\cdots\!79}a^{8}-\frac{63\!\cdots\!71}{90\!\cdots\!79}a^{7}-\frac{60\!\cdots\!51}{10\!\cdots\!31}a^{6}-\frac{93\!\cdots\!05}{10\!\cdots\!31}a^{5}+\frac{23\!\cdots\!15}{10\!\cdots\!31}a^{4}+\frac{12\!\cdots\!46}{34\!\cdots\!39}a^{3}+\frac{34\!\cdots\!00}{33\!\cdots\!77}a^{2}-\frac{13\!\cdots\!29}{33\!\cdots\!77}a-\frac{15\!\cdots\!91}{37\!\cdots\!53}$, $\frac{19\!\cdots\!12}{27\!\cdots\!37}a^{25}+\frac{11\!\cdots\!83}{27\!\cdots\!37}a^{24}-\frac{58\!\cdots\!15}{27\!\cdots\!37}a^{23}-\frac{50\!\cdots\!39}{27\!\cdots\!37}a^{22}-\frac{52\!\cdots\!02}{27\!\cdots\!37}a^{21}+\frac{50\!\cdots\!51}{27\!\cdots\!37}a^{20}+\frac{35\!\cdots\!68}{27\!\cdots\!37}a^{19}+\frac{16\!\cdots\!88}{27\!\cdots\!37}a^{18}-\frac{24\!\cdots\!72}{27\!\cdots\!37}a^{17}-\frac{19\!\cdots\!10}{27\!\cdots\!37}a^{16}+\frac{32\!\cdots\!81}{27\!\cdots\!37}a^{15}+\frac{12\!\cdots\!73}{24\!\cdots\!67}a^{14}+\frac{23\!\cdots\!96}{24\!\cdots\!67}a^{13}+\frac{37\!\cdots\!90}{27\!\cdots\!37}a^{12}-\frac{10\!\cdots\!28}{27\!\cdots\!37}a^{11}-\frac{14\!\cdots\!01}{27\!\cdots\!37}a^{10}+\frac{34\!\cdots\!98}{63\!\cdots\!59}a^{9}+\frac{21\!\cdots\!47}{93\!\cdots\!53}a^{8}-\frac{19\!\cdots\!04}{27\!\cdots\!37}a^{7}+\frac{18\!\cdots\!33}{90\!\cdots\!79}a^{6}+\frac{90\!\cdots\!29}{30\!\cdots\!93}a^{5}-\frac{60\!\cdots\!62}{30\!\cdots\!93}a^{4}-\frac{25\!\cdots\!27}{33\!\cdots\!77}a^{3}-\frac{12\!\cdots\!83}{30\!\cdots\!07}a^{2}+\frac{63\!\cdots\!91}{10\!\cdots\!31}a+\frac{66\!\cdots\!36}{11\!\cdots\!59}$, $\frac{41\!\cdots\!11}{18\!\cdots\!77}a^{25}-\frac{17\!\cdots\!01}{18\!\cdots\!77}a^{24}-\frac{39\!\cdots\!24}{18\!\cdots\!77}a^{23}-\frac{65\!\cdots\!18}{18\!\cdots\!77}a^{22}+\frac{10\!\cdots\!27}{65\!\cdots\!13}a^{21}+\frac{80\!\cdots\!60}{18\!\cdots\!77}a^{20}+\frac{29\!\cdots\!55}{18\!\cdots\!77}a^{19}-\frac{16\!\cdots\!08}{17\!\cdots\!07}a^{18}-\frac{19\!\cdots\!43}{18\!\cdots\!77}a^{17}-\frac{13\!\cdots\!25}{18\!\cdots\!77}a^{16}+\frac{19\!\cdots\!72}{17\!\cdots\!07}a^{15}+\frac{52\!\cdots\!57}{18\!\cdots\!77}a^{14}+\frac{21\!\cdots\!29}{18\!\cdots\!77}a^{13}+\frac{15\!\cdots\!89}{18\!\cdots\!77}a^{12}-\frac{85\!\cdots\!92}{18\!\cdots\!77}a^{11}-\frac{10\!\cdots\!59}{18\!\cdots\!77}a^{10}+\frac{11\!\cdots\!04}{18\!\cdots\!77}a^{9}+\frac{68\!\cdots\!33}{17\!\cdots\!07}a^{8}-\frac{17\!\cdots\!36}{21\!\cdots\!53}a^{7}+\frac{47\!\cdots\!73}{21\!\cdots\!53}a^{6}+\frac{88\!\cdots\!84}{21\!\cdots\!53}a^{5}-\frac{32\!\cdots\!79}{80\!\cdots\!73}a^{4}-\frac{76\!\cdots\!41}{78\!\cdots\!39}a^{3}-\frac{82\!\cdots\!01}{70\!\cdots\!51}a^{2}+\frac{17\!\cdots\!71}{23\!\cdots\!17}a+\frac{10\!\cdots\!58}{23\!\cdots\!83}$, $\frac{25\!\cdots\!18}{81\!\cdots\!11}a^{25}-\frac{22\!\cdots\!48}{81\!\cdots\!11}a^{24}-\frac{19\!\cdots\!84}{81\!\cdots\!11}a^{23}-\frac{28\!\cdots\!09}{81\!\cdots\!11}a^{22}-\frac{23\!\cdots\!69}{81\!\cdots\!11}a^{21}+\frac{44\!\cdots\!48}{81\!\cdots\!11}a^{20}+\frac{14\!\cdots\!54}{74\!\cdots\!01}a^{19}-\frac{11\!\cdots\!80}{74\!\cdots\!01}a^{18}-\frac{82\!\cdots\!68}{81\!\cdots\!11}a^{17}-\frac{13\!\cdots\!56}{28\!\cdots\!59}a^{16}-\frac{14\!\cdots\!30}{81\!\cdots\!11}a^{15}+\frac{17\!\cdots\!38}{81\!\cdots\!11}a^{14}+\frac{12\!\cdots\!09}{81\!\cdots\!11}a^{13}+\frac{45\!\cdots\!27}{81\!\cdots\!11}a^{12}-\frac{34\!\cdots\!81}{81\!\cdots\!11}a^{11}-\frac{18\!\cdots\!04}{81\!\cdots\!11}a^{10}+\frac{39\!\cdots\!06}{18\!\cdots\!77}a^{9}-\frac{37\!\cdots\!21}{81\!\cdots\!11}a^{8}-\frac{24\!\cdots\!93}{30\!\cdots\!93}a^{7}+\frac{21\!\cdots\!02}{90\!\cdots\!79}a^{6}+\frac{24\!\cdots\!39}{90\!\cdots\!79}a^{5}-\frac{14\!\cdots\!30}{91\!\cdots\!21}a^{4}-\frac{56\!\cdots\!29}{10\!\cdots\!31}a^{3}-\frac{72\!\cdots\!39}{30\!\cdots\!93}a^{2}+\frac{28\!\cdots\!63}{10\!\cdots\!31}a+\frac{83\!\cdots\!89}{38\!\cdots\!71}$, $\frac{62\!\cdots\!60}{74\!\cdots\!01}a^{25}-\frac{13\!\cdots\!48}{81\!\cdots\!11}a^{24}-\frac{19\!\cdots\!52}{81\!\cdots\!11}a^{23}+\frac{98\!\cdots\!57}{81\!\cdots\!11}a^{22}+\frac{71\!\cdots\!06}{81\!\cdots\!11}a^{21}+\frac{35\!\cdots\!90}{81\!\cdots\!11}a^{20}+\frac{63\!\cdots\!91}{81\!\cdots\!11}a^{19}-\frac{27\!\cdots\!55}{81\!\cdots\!11}a^{18}-\frac{10\!\cdots\!21}{81\!\cdots\!11}a^{17}+\frac{19\!\cdots\!35}{74\!\cdots\!01}a^{16}+\frac{25\!\cdots\!41}{81\!\cdots\!11}a^{15}+\frac{34\!\cdots\!98}{81\!\cdots\!11}a^{14}+\frac{95\!\cdots\!83}{81\!\cdots\!11}a^{13}-\frac{35\!\cdots\!15}{81\!\cdots\!11}a^{12}-\frac{55\!\cdots\!25}{81\!\cdots\!11}a^{11}-\frac{11\!\cdots\!27}{28\!\cdots\!59}a^{10}+\frac{17\!\cdots\!20}{18\!\cdots\!77}a^{9}+\frac{25\!\cdots\!76}{81\!\cdots\!11}a^{8}-\frac{44\!\cdots\!06}{10\!\cdots\!31}a^{7}+\frac{28\!\cdots\!00}{90\!\cdots\!79}a^{6}+\frac{35\!\cdots\!85}{90\!\cdots\!79}a^{5}-\frac{73\!\cdots\!11}{10\!\cdots\!31}a^{4}-\frac{38\!\cdots\!54}{34\!\cdots\!39}a^{3}+\frac{12\!\cdots\!97}{30\!\cdots\!93}a^{2}+\frac{10\!\cdots\!26}{10\!\cdots\!31}a+\frac{31\!\cdots\!94}{10\!\cdots\!69}$, $\frac{27\!\cdots\!54}{81\!\cdots\!11}a^{25}+\frac{19\!\cdots\!57}{81\!\cdots\!11}a^{24}-\frac{16\!\cdots\!86}{81\!\cdots\!11}a^{23}-\frac{61\!\cdots\!51}{81\!\cdots\!11}a^{22}-\frac{12\!\cdots\!35}{81\!\cdots\!11}a^{21}+\frac{25\!\cdots\!32}{81\!\cdots\!11}a^{20}+\frac{20\!\cdots\!67}{81\!\cdots\!11}a^{19}+\frac{22\!\cdots\!52}{81\!\cdots\!11}a^{18}-\frac{49\!\cdots\!70}{81\!\cdots\!11}a^{17}-\frac{13\!\cdots\!13}{81\!\cdots\!11}a^{16}-\frac{23\!\cdots\!54}{81\!\cdots\!11}a^{15}-\frac{21\!\cdots\!01}{81\!\cdots\!11}a^{14}+\frac{92\!\cdots\!94}{81\!\cdots\!11}a^{13}+\frac{22\!\cdots\!71}{81\!\cdots\!11}a^{12}+\frac{29\!\cdots\!34}{81\!\cdots\!11}a^{11}-\frac{18\!\cdots\!03}{81\!\cdots\!11}a^{10}-\frac{17\!\cdots\!74}{18\!\cdots\!77}a^{9}-\frac{62\!\cdots\!75}{81\!\cdots\!11}a^{8}-\frac{48\!\cdots\!95}{27\!\cdots\!63}a^{7}-\frac{20\!\cdots\!26}{90\!\cdots\!79}a^{6}+\frac{25\!\cdots\!91}{90\!\cdots\!79}a^{5}+\frac{10\!\cdots\!57}{33\!\cdots\!77}a^{4}+\frac{63\!\cdots\!74}{10\!\cdots\!31}a^{3}-\frac{16\!\cdots\!27}{30\!\cdots\!93}a^{2}+\frac{19\!\cdots\!29}{10\!\cdots\!31}a+\frac{21\!\cdots\!88}{10\!\cdots\!69}$, $\frac{12\!\cdots\!09}{35\!\cdots\!93}a^{25}-\frac{19\!\cdots\!89}{35\!\cdots\!93}a^{24}-\frac{13\!\cdots\!32}{35\!\cdots\!93}a^{23}-\frac{11\!\cdots\!09}{32\!\cdots\!63}a^{22}+\frac{93\!\cdots\!14}{12\!\cdots\!17}a^{21}+\frac{32\!\cdots\!69}{35\!\cdots\!93}a^{20}+\frac{82\!\cdots\!66}{35\!\cdots\!93}a^{19}-\frac{15\!\cdots\!28}{35\!\cdots\!93}a^{18}-\frac{64\!\cdots\!83}{32\!\cdots\!63}a^{17}-\frac{30\!\cdots\!02}{35\!\cdots\!93}a^{16}+\frac{96\!\cdots\!14}{35\!\cdots\!93}a^{15}+\frac{30\!\cdots\!37}{35\!\cdots\!93}a^{14}+\frac{83\!\cdots\!60}{32\!\cdots\!63}a^{13}+\frac{19\!\cdots\!60}{35\!\cdots\!93}a^{12}-\frac{34\!\cdots\!01}{35\!\cdots\!93}a^{11}-\frac{37\!\cdots\!20}{35\!\cdots\!93}a^{10}+\frac{39\!\cdots\!28}{83\!\cdots\!51}a^{9}+\frac{10\!\cdots\!24}{32\!\cdots\!63}a^{8}+\frac{81\!\cdots\!34}{11\!\cdots\!31}a^{7}+\frac{81\!\cdots\!87}{13\!\cdots\!59}a^{6}+\frac{29\!\cdots\!25}{39\!\cdots\!77}a^{5}-\frac{21\!\cdots\!59}{45\!\cdots\!71}a^{4}-\frac{24\!\cdots\!49}{14\!\cdots\!51}a^{3}-\frac{16\!\cdots\!72}{13\!\cdots\!59}a^{2}+\frac{14\!\cdots\!35}{44\!\cdots\!53}a+\frac{29\!\cdots\!99}{49\!\cdots\!17}$, $\frac{84\!\cdots\!05}{81\!\cdots\!11}a^{25}-\frac{92\!\cdots\!84}{81\!\cdots\!11}a^{24}-\frac{65\!\cdots\!82}{81\!\cdots\!11}a^{23}-\frac{97\!\cdots\!04}{81\!\cdots\!11}a^{22}+\frac{39\!\cdots\!53}{81\!\cdots\!11}a^{21}+\frac{15\!\cdots\!51}{81\!\cdots\!11}a^{20}+\frac{49\!\cdots\!20}{81\!\cdots\!11}a^{19}-\frac{56\!\cdots\!69}{81\!\cdots\!11}a^{18}-\frac{30\!\cdots\!57}{81\!\cdots\!11}a^{17}-\frac{12\!\cdots\!85}{81\!\cdots\!11}a^{16}+\frac{23\!\cdots\!69}{81\!\cdots\!11}a^{15}+\frac{31\!\cdots\!01}{28\!\cdots\!59}a^{14}+\frac{39\!\cdots\!14}{81\!\cdots\!11}a^{13}+\frac{32\!\cdots\!95}{28\!\cdots\!59}a^{12}-\frac{14\!\cdots\!71}{81\!\cdots\!11}a^{11}-\frac{10\!\cdots\!44}{81\!\cdots\!11}a^{10}+\frac{35\!\cdots\!74}{18\!\cdots\!77}a^{9}+\frac{78\!\cdots\!32}{81\!\cdots\!11}a^{8}-\frac{14\!\cdots\!87}{27\!\cdots\!37}a^{7}+\frac{86\!\cdots\!53}{90\!\cdots\!79}a^{6}+\frac{27\!\cdots\!90}{28\!\cdots\!41}a^{5}-\frac{40\!\cdots\!21}{30\!\cdots\!93}a^{4}-\frac{92\!\cdots\!32}{30\!\cdots\!07}a^{3}-\frac{30\!\cdots\!23}{30\!\cdots\!93}a^{2}+\frac{86\!\cdots\!53}{33\!\cdots\!77}a+\frac{28\!\cdots\!53}{12\!\cdots\!51}$, $\frac{39\!\cdots\!80}{81\!\cdots\!11}a^{25}-\frac{19\!\cdots\!81}{28\!\cdots\!59}a^{24}-\frac{29\!\cdots\!03}{74\!\cdots\!01}a^{23}-\frac{36\!\cdots\!88}{81\!\cdots\!11}a^{22}+\frac{43\!\cdots\!80}{81\!\cdots\!11}a^{21}+\frac{79\!\cdots\!39}{81\!\cdots\!11}a^{20}+\frac{21\!\cdots\!53}{81\!\cdots\!11}a^{19}-\frac{38\!\cdots\!00}{81\!\cdots\!11}a^{18}-\frac{55\!\cdots\!15}{28\!\cdots\!59}a^{17}-\frac{26\!\cdots\!38}{81\!\cdots\!11}a^{16}+\frac{21\!\cdots\!61}{74\!\cdots\!01}a^{15}+\frac{56\!\cdots\!89}{81\!\cdots\!11}a^{14}+\frac{18\!\cdots\!54}{81\!\cdots\!11}a^{13}-\frac{24\!\cdots\!60}{81\!\cdots\!11}a^{12}-\frac{86\!\cdots\!38}{81\!\cdots\!11}a^{11}-\frac{55\!\cdots\!05}{81\!\cdots\!11}a^{10}+\frac{27\!\cdots\!43}{18\!\cdots\!77}a^{9}+\frac{54\!\cdots\!65}{81\!\cdots\!11}a^{8}-\frac{25\!\cdots\!94}{90\!\cdots\!79}a^{7}+\frac{43\!\cdots\!92}{90\!\cdots\!79}a^{6}+\frac{38\!\cdots\!00}{82\!\cdots\!89}a^{5}-\frac{10\!\cdots\!63}{10\!\cdots\!31}a^{4}-\frac{19\!\cdots\!91}{10\!\cdots\!31}a^{3}-\frac{55\!\cdots\!49}{27\!\cdots\!63}a^{2}+\frac{21\!\cdots\!90}{10\!\cdots\!31}a+\frac{17\!\cdots\!36}{11\!\cdots\!59}$, $\frac{12\!\cdots\!75}{81\!\cdots\!11}a^{25}+\frac{11\!\cdots\!81}{81\!\cdots\!11}a^{24}+\frac{79\!\cdots\!37}{81\!\cdots\!11}a^{23}-\frac{29\!\cdots\!55}{81\!\cdots\!11}a^{22}-\frac{98\!\cdots\!81}{81\!\cdots\!11}a^{21}-\frac{54\!\cdots\!05}{81\!\cdots\!11}a^{20}+\frac{63\!\cdots\!96}{81\!\cdots\!11}a^{19}+\frac{19\!\cdots\!63}{81\!\cdots\!11}a^{18}+\frac{31\!\cdots\!84}{81\!\cdots\!11}a^{17}-\frac{16\!\cdots\!10}{81\!\cdots\!11}a^{16}-\frac{18\!\cdots\!56}{74\!\cdots\!01}a^{15}-\frac{41\!\cdots\!32}{81\!\cdots\!11}a^{14}-\frac{34\!\cdots\!16}{81\!\cdots\!11}a^{13}+\frac{42\!\cdots\!74}{81\!\cdots\!11}a^{12}+\frac{21\!\cdots\!12}{81\!\cdots\!11}a^{11}+\frac{12\!\cdots\!08}{28\!\cdots\!59}a^{10}+\frac{46\!\cdots\!16}{18\!\cdots\!77}a^{9}-\frac{24\!\cdots\!68}{81\!\cdots\!11}a^{8}-\frac{23\!\cdots\!76}{27\!\cdots\!37}a^{7}-\frac{17\!\cdots\!69}{90\!\cdots\!79}a^{6}-\frac{22\!\cdots\!58}{90\!\cdots\!79}a^{5}+\frac{42\!\cdots\!77}{30\!\cdots\!93}a^{4}+\frac{12\!\cdots\!34}{31\!\cdots\!49}a^{3}+\frac{14\!\cdots\!27}{30\!\cdots\!93}a^{2}+\frac{27\!\cdots\!03}{11\!\cdots\!59}a-\frac{21\!\cdots\!25}{37\!\cdots\!53}$
| 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: | \( 16355807538960.088 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 16355807538960.088 \cdot 1}{6\cdot\sqrt{4634664002656599036916177705626061158525123}}\cr\approx \mathstrut & 30.1196487161707 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^26 - 9*x^24 - 20*x^23 - 12*x^22 + 186*x^21 + 794*x^20 - 27*x^19 - 4404*x^18 - 5528*x^17 + 1038*x^16 + 13671*x^15 + 59660*x^14 + 64299*x^13 - 161565*x^12 - 318218*x^11 + 37881*x^10 + 189642*x^9 - 55703*x^8 + 873384*x^7 + 1978515*x^6 - 188640*x^5 - 4211379*x^4 - 4130865*x^3 + 1301535*x^2 + 4678236*x + 2259171)
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^26 - 9*x^24 - 20*x^23 - 12*x^22 + 186*x^21 + 794*x^20 - 27*x^19 - 4404*x^18 - 5528*x^17 + 1038*x^16 + 13671*x^15 + 59660*x^14 + 64299*x^13 - 161565*x^12 - 318218*x^11 + 37881*x^10 + 189642*x^9 - 55703*x^8 + 873384*x^7 + 1978515*x^6 - 188640*x^5 - 4211379*x^4 - 4130865*x^3 + 1301535*x^2 + 4678236*x + 2259171, 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^26 - 9*x^24 - 20*x^23 - 12*x^22 + 186*x^21 + 794*x^20 - 27*x^19 - 4404*x^18 - 5528*x^17 + 1038*x^16 + 13671*x^15 + 59660*x^14 + 64299*x^13 - 161565*x^12 - 318218*x^11 + 37881*x^10 + 189642*x^9 - 55703*x^8 + 873384*x^7 + 1978515*x^6 - 188640*x^5 - 4211379*x^4 - 4130865*x^3 + 1301535*x^2 + 4678236*x + 2259171);
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^26 - 9*x^24 - 20*x^23 - 12*x^22 + 186*x^21 + 794*x^20 - 27*x^19 - 4404*x^18 - 5528*x^17 + 1038*x^16 + 13671*x^15 + 59660*x^14 + 64299*x^13 - 161565*x^12 - 318218*x^11 + 37881*x^10 + 189642*x^9 - 55703*x^8 + 873384*x^7 + 1978515*x^6 - 188640*x^5 - 4211379*x^4 - 4130865*x^3 + 1301535*x^2 + 4678236*x + 2259171);
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
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 solvable group of order 52 |
| The 16 conjugacy class representatives for $D_{26}$ |
| Character table for $D_{26}$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 13.1.1242935235998051916321.1 |
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 26 sibling: | deg 26 |
| 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 | $26$ | R | $26$ | ${\href{/padicField/7.13.0.1}{13} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{13}$ | ${\href{/padicField/13.13.0.1}{13} }^{2}$ | $26$ | ${\href{/padicField/19.2.0.1}{2} }^{12}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{13}$ | ${\href{/padicField/29.2.0.1}{2} }^{13}$ | ${\href{/padicField/31.13.0.1}{13} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{12}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{13}$ | ${\href{/padicField/43.2.0.1}{2} }^{12}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{13}$ | $26$ | $26$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(1093\)
| $\Q_{1093}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1093}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.