Normalized defining polynomial

\( x^{30} + 6 x^{28} + 25 x^{26} + 45 x^{24} - 44 x^{22} - 197 x^{20} - 20 x^{18} + 411 x^{16} + 283 x^{14} + \cdots + 1 \) x^30 + 6*x^28 + 25*x^26 + 45*x^24 - 44*x^22 - 197*x^20 - 20*x^18 + 411*x^16 + 283*x^14 - 424*x^12 - 397*x^10 + 348*x^8 + 358*x^6 + 8*x^4 - 7*x^2 + 1

Invariants

Degree:		$30$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[0, 15]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(-1310417368511817590988945406032175169536\) -1310417368511817590988945406032175169536 \(\medspace = -\,2^{40}\cdot 11^{26}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(20.13\)	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:		$2^{4/3}11^{9/10}\approx 21.808547634345086$
Ramified primes:		\(2\), \(11\) 2, 11	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q(\sqrt{-1}) \)
$\card{ \Aut(K/\Q) }$:		$10$	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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{14}-\frac{1}{2}a^{8}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{26}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{27}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{80\!\cdots\!46}a^{28}-\frac{85\!\cdots\!05}{40\!\cdots\!23}a^{26}-\frac{16\!\cdots\!37}{80\!\cdots\!46}a^{24}-\frac{87\!\cdots\!57}{40\!\cdots\!23}a^{22}+\frac{14\!\cdots\!73}{80\!\cdots\!46}a^{20}-\frac{49\!\cdots\!93}{40\!\cdots\!23}a^{18}-\frac{87\!\cdots\!42}{40\!\cdots\!23}a^{16}-\frac{16\!\cdots\!11}{40\!\cdots\!23}a^{14}+\frac{11\!\cdots\!44}{40\!\cdots\!23}a^{12}-\frac{18\!\cdots\!93}{40\!\cdots\!23}a^{10}+\frac{18\!\cdots\!66}{40\!\cdots\!23}a^{8}+\frac{97\!\cdots\!40}{40\!\cdots\!23}a^{6}-\frac{15\!\cdots\!13}{80\!\cdots\!46}a^{4}+\frac{54\!\cdots\!55}{40\!\cdots\!23}a^{2}-\frac{20\!\cdots\!27}{40\!\cdots\!23}$, $\frac{1}{80\!\cdots\!46}a^{29}-\frac{85\!\cdots\!05}{40\!\cdots\!23}a^{27}-\frac{16\!\cdots\!37}{80\!\cdots\!46}a^{25}-\frac{87\!\cdots\!57}{40\!\cdots\!23}a^{23}+\frac{14\!\cdots\!73}{80\!\cdots\!46}a^{21}-\frac{49\!\cdots\!93}{40\!\cdots\!23}a^{19}-\frac{87\!\cdots\!42}{40\!\cdots\!23}a^{17}+\frac{81\!\cdots\!01}{80\!\cdots\!46}a^{15}-\frac{17\!\cdots\!35}{80\!\cdots\!46}a^{13}-\frac{1}{2}a^{12}-\frac{18\!\cdots\!93}{40\!\cdots\!23}a^{11}-\frac{1}{2}a^{10}+\frac{18\!\cdots\!66}{40\!\cdots\!23}a^{9}-\frac{1}{2}a^{8}-\frac{20\!\cdots\!43}{80\!\cdots\!46}a^{7}+\frac{12\!\cdots\!55}{40\!\cdots\!23}a^{5}+\frac{54\!\cdots\!55}{40\!\cdots\!23}a^{3}+\frac{58696604320469}{80\!\cdots\!46}a-\frac{1}{2}$ 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, 1/2*a^15 - 1/2*a^13 - 1/2*a^12 - 1/2*a^10 - 1/2*a^8 - 1/2*a^7 - 1/2*a^5 - 1/2*a - 1/2, 1/2*a^16 - 1/2*a^14 - 1/2*a^13 - 1/2*a^11 - 1/2*a^9 - 1/2*a^8 - 1/2*a^6 - 1/2*a^2 - 1/2*a, 1/2*a^17 - 1/2*a^14 - 1/2*a^13 - 1/2*a^9 - 1/2*a^8 - 1/2*a^5 - 1/2*a^3 - 1/2*a^2 - 1/2*a - 1/2, 1/2*a^18 - 1/2*a^14 - 1/2*a^13 - 1/2*a^12 - 1/2*a^9 - 1/2*a^8 - 1/2*a^7 - 1/2*a^6 - 1/2*a^5 - 1/2*a^4 - 1/2*a^3 - 1/2*a^2 - 1/2, 1/2*a^19 - 1/2*a^14 - 1/2*a^12 - 1/2*a^9 - 1/2*a^6 - 1/2*a^4 - 1/2*a^3 - 1/2, 1/2*a^20 - 1/2*a^12 - 1/2*a^8 - 1/2*a^4 - 1/2, 1/2*a^21 - 1/2*a^13 - 1/2*a^9 - 1/2*a^5 - 1/2*a, 1/2*a^22 - 1/2*a^14 - 1/2*a^10 - 1/2*a^6 - 1/2*a^2, 1/2*a^23 - 1/2*a^13 - 1/2*a^12 - 1/2*a^11 - 1/2*a^10 - 1/2*a^8 - 1/2*a^5 - 1/2*a^3 - 1/2*a - 1/2, 1/2*a^24 - 1/2*a^14 - 1/2*a^13 - 1/2*a^12 - 1/2*a^11 - 1/2*a^9 - 1/2*a^6 - 1/2*a^4 - 1/2*a^2 - 1/2*a, 1/2*a^25 - 1/2*a^14 - 1/2*a^8 - 1/2*a^3 - 1/2*a^2 - 1/2*a - 1/2, 1/2*a^26 - 1/2*a^13 - 1/2*a^12 - 1/2*a^10 - 1/2*a^9 - 1/2*a^8 - 1/2*a^7 - 1/2*a^5 - 1/2*a^4 - 1/2*a^3 - 1/2*a^2 - 1/2, 1/2*a^27 - 1/2*a^14 - 1/2*a^13 - 1/2*a^11 - 1/2*a^10 - 1/2*a^9 - 1/2*a^8 - 1/2*a^6 - 1/2*a^5 - 1/2*a^4 - 1/2*a^3 - 1/2*a, 1/806438113501246246*a^28 - 8547019296480505/403219056750623123*a^26 - 161059070594040737/806438113501246246*a^24 - 87862912953318957/403219056750623123*a^22 + 147397953066571973/806438113501246246*a^20 - 4901431061596493/403219056750623123*a^18 - 87429740741882642/403219056750623123*a^16 - 160823540610087811/403219056750623123*a^14 + 113679147513814544/403219056750623123*a^12 - 180267560084387693/403219056750623123*a^10 + 182171665704510766/403219056750623123*a^8 + 97550603202531440/403219056750623123*a^6 - 151433037274049413/806438113501246246*a^4 + 54798993027099655/403219056750623123*a^2 - 201580180073151327/403219056750623123, 1/806438113501246246*a^29 - 8547019296480505/403219056750623123*a^27 - 161059070594040737/806438113501246246*a^25 - 87862912953318957/403219056750623123*a^23 + 147397953066571973/806438113501246246*a^21 - 4901431061596493/403219056750623123*a^19 - 87429740741882642/403219056750623123*a^17 + 81571975530447501/806438113501246246*a^15 - 175860761722994035/806438113501246246*a^13 - 1/2*a^12 - 180267560084387693/403219056750623123*a^11 - 1/2*a^10 + 182171665704510766/403219056750623123*a^9 - 1/2*a^8 - 208117850345560243/806438113501246246*a^7 + 125893009738286855/403219056750623123*a^5 + 54798993027099655/403219056750623123*a^3 + 58696604320469/806438113501246246*a - 1/2

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$14$	sage: UK.rank()   gp: K.fu   magma: UnitRank(K);   oscar: rank(UK)
Torsion generator:		\( -\frac{90395440773}{291211635746} a^{29} - \frac{277418055635}{145605817873} a^{27} - \frac{1164367872014}{145605817873} a^{25} - \frac{2176299246142}{145605817873} a^{23} + \frac{1760559259543}{145605817873} a^{21} + \frac{9223935112516}{145605817873} a^{19} + \frac{3783920726191}{291211635746} a^{17} - \frac{18737732908828}{145605817873} a^{15} - \frac{14874663307821}{145605817873} a^{13} + \frac{18141256073509}{145605817873} a^{11} + \frac{20163995858949}{145605817873} a^{9} - \frac{14393368446386}{145605817873} a^{7} - \frac{35782616547667}{291211635746} a^{5} - \frac{1743953091134}{145605817873} a^{3} - \frac{7625965931}{291211635746} a \) -(90395440773)/(291211635746)*a^(29) - (277418055635)/(145605817873)*a^(27) - (1164367872014)/(145605817873)*a^(25) - (2176299246142)/(145605817873)*a^(23) + (1760559259543)/(145605817873)*a^(21) + (9223935112516)/(145605817873)*a^(19) + (3783920726191)/(291211635746)*a^(17) - (18737732908828)/(145605817873)*a^(15) - (14874663307821)/(145605817873)*a^(13) + (18141256073509)/(145605817873)*a^(11) + (20163995858949)/(145605817873)*a^(9) - (14393368446386)/(145605817873)*a^(7) - (35782616547667)/(291211635746)*a^(5) - (1743953091134)/(145605817873)*a^(3) - (7625965931)/(291211635746)*a  (order $4$)	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:		$\frac{13\!\cdots\!33}{18\!\cdots\!22}a^{28}+\frac{83\!\cdots\!57}{18\!\cdots\!22}a^{26}+\frac{35\!\cdots\!15}{18\!\cdots\!22}a^{24}+\frac{34\!\cdots\!11}{93\!\cdots\!61}a^{22}-\frac{41\!\cdots\!53}{18\!\cdots\!22}a^{20}-\frac{27\!\cdots\!69}{18\!\cdots\!22}a^{18}-\frac{48\!\cdots\!33}{93\!\cdots\!61}a^{16}+\frac{52\!\cdots\!25}{18\!\cdots\!22}a^{14}+\frac{25\!\cdots\!51}{93\!\cdots\!61}a^{12}-\frac{44\!\cdots\!43}{18\!\cdots\!22}a^{10}-\frac{33\!\cdots\!76}{93\!\cdots\!61}a^{8}+\frac{30\!\cdots\!47}{18\!\cdots\!22}a^{6}+\frac{58\!\cdots\!03}{18\!\cdots\!22}a^{4}+\frac{79\!\cdots\!71}{93\!\cdots\!61}a^{2}+\frac{331179604370172}{93\!\cdots\!61}$, $\frac{62\!\cdots\!02}{40\!\cdots\!23}a^{28}+\frac{75\!\cdots\!47}{80\!\cdots\!46}a^{26}+\frac{15\!\cdots\!05}{40\!\cdots\!23}a^{24}+\frac{28\!\cdots\!13}{40\!\cdots\!23}a^{22}-\frac{26\!\cdots\!64}{40\!\cdots\!23}a^{20}-\frac{24\!\cdots\!13}{80\!\cdots\!46}a^{18}-\frac{15\!\cdots\!96}{40\!\cdots\!23}a^{16}+\frac{51\!\cdots\!81}{80\!\cdots\!46}a^{14}+\frac{18\!\cdots\!12}{40\!\cdots\!23}a^{12}-\frac{51\!\cdots\!39}{80\!\cdots\!46}a^{10}-\frac{25\!\cdots\!71}{40\!\cdots\!23}a^{8}+\frac{42\!\cdots\!53}{80\!\cdots\!46}a^{6}+\frac{22\!\cdots\!88}{40\!\cdots\!23}a^{4}+\frac{16\!\cdots\!07}{40\!\cdots\!23}a^{2}-\frac{67\!\cdots\!50}{40\!\cdots\!23}$, $\frac{30\!\cdots\!25}{80\!\cdots\!46}a^{28}+\frac{10\!\cdots\!14}{40\!\cdots\!23}a^{26}+\frac{89\!\cdots\!95}{80\!\cdots\!46}a^{24}+\frac{94\!\cdots\!66}{40\!\cdots\!23}a^{22}-\frac{41\!\cdots\!53}{40\!\cdots\!23}a^{20}-\frac{41\!\cdots\!63}{40\!\cdots\!23}a^{18}-\frac{18\!\cdots\!69}{40\!\cdots\!23}a^{16}+\frac{82\!\cdots\!08}{40\!\cdots\!23}a^{14}+\frac{16\!\cdots\!75}{80\!\cdots\!46}a^{12}-\frac{76\!\cdots\!59}{40\!\cdots\!23}a^{10}-\frac{23\!\cdots\!29}{80\!\cdots\!46}a^{8}+\frac{63\!\cdots\!57}{40\!\cdots\!23}a^{6}+\frac{94\!\cdots\!11}{40\!\cdots\!23}a^{4}-\frac{21\!\cdots\!68}{40\!\cdots\!23}a^{2}-\frac{29\!\cdots\!13}{80\!\cdots\!46}$, $\frac{59\!\cdots\!93}{80\!\cdots\!46}a^{28}+\frac{18\!\cdots\!13}{40\!\cdots\!23}a^{26}+\frac{75\!\cdots\!79}{40\!\cdots\!23}a^{24}+\frac{13\!\cdots\!18}{40\!\cdots\!23}a^{22}-\frac{12\!\cdots\!77}{40\!\cdots\!23}a^{20}-\frac{60\!\cdots\!26}{40\!\cdots\!23}a^{18}-\frac{17\!\cdots\!31}{80\!\cdots\!46}a^{16}+\frac{12\!\cdots\!87}{40\!\cdots\!23}a^{14}+\frac{90\!\cdots\!09}{40\!\cdots\!23}a^{12}-\frac{12\!\cdots\!78}{40\!\cdots\!23}a^{10}-\frac{12\!\cdots\!49}{40\!\cdots\!23}a^{8}+\frac{10\!\cdots\!30}{40\!\cdots\!23}a^{6}+\frac{23\!\cdots\!99}{80\!\cdots\!46}a^{4}-\frac{26\!\cdots\!77}{40\!\cdots\!23}a^{2}-\frac{42\!\cdots\!97}{80\!\cdots\!46}$, $\frac{36\!\cdots\!26}{40\!\cdots\!23}a^{28}+\frac{32\!\cdots\!73}{40\!\cdots\!23}a^{26}+\frac{14\!\cdots\!66}{40\!\cdots\!23}a^{24}+\frac{77\!\cdots\!47}{80\!\cdots\!46}a^{22}+\frac{30\!\cdots\!13}{80\!\cdots\!46}a^{20}-\frac{14\!\cdots\!08}{40\!\cdots\!23}a^{18}-\frac{17\!\cdots\!07}{40\!\cdots\!23}a^{16}+\frac{47\!\cdots\!81}{80\!\cdots\!46}a^{14}+\frac{96\!\cdots\!91}{80\!\cdots\!46}a^{12}-\frac{22\!\cdots\!03}{80\!\cdots\!46}a^{10}-\frac{12\!\cdots\!71}{80\!\cdots\!46}a^{8}+\frac{29\!\cdots\!05}{80\!\cdots\!46}a^{6}+\frac{11\!\cdots\!21}{80\!\cdots\!46}a^{4}+\frac{16\!\cdots\!15}{80\!\cdots\!46}a^{2}-\frac{77\!\cdots\!79}{80\!\cdots\!46}$, $\frac{13\!\cdots\!35}{80\!\cdots\!46}a^{28}+\frac{39\!\cdots\!81}{40\!\cdots\!23}a^{26}+\frac{16\!\cdots\!75}{40\!\cdots\!23}a^{24}+\frac{29\!\cdots\!50}{40\!\cdots\!23}a^{22}-\frac{57\!\cdots\!45}{80\!\cdots\!46}a^{20}-\frac{12\!\cdots\!50}{40\!\cdots\!23}a^{18}-\frac{25\!\cdots\!65}{80\!\cdots\!46}a^{16}+\frac{26\!\cdots\!37}{40\!\cdots\!23}a^{14}+\frac{36\!\cdots\!31}{80\!\cdots\!46}a^{12}-\frac{27\!\cdots\!59}{40\!\cdots\!23}a^{10}-\frac{50\!\cdots\!83}{80\!\cdots\!46}a^{8}+\frac{22\!\cdots\!13}{40\!\cdots\!23}a^{6}+\frac{22\!\cdots\!36}{40\!\cdots\!23}a^{4}+\frac{14\!\cdots\!69}{40\!\cdots\!23}a^{2}-\frac{30\!\cdots\!29}{40\!\cdots\!23}$, $\frac{95\!\cdots\!31}{80\!\cdots\!46}a^{28}+\frac{25\!\cdots\!28}{40\!\cdots\!23}a^{26}+\frac{10\!\cdots\!26}{40\!\cdots\!23}a^{24}+\frac{15\!\cdots\!64}{40\!\cdots\!23}a^{22}-\frac{59\!\cdots\!51}{80\!\cdots\!46}a^{20}-\frac{78\!\cdots\!82}{40\!\cdots\!23}a^{18}+\frac{60\!\cdots\!19}{80\!\cdots\!46}a^{16}+\frac{18\!\cdots\!04}{40\!\cdots\!23}a^{14}+\frac{91\!\cdots\!97}{80\!\cdots\!46}a^{12}-\frac{22\!\cdots\!42}{40\!\cdots\!23}a^{10}-\frac{16\!\cdots\!85}{80\!\cdots\!46}a^{8}+\frac{20\!\cdots\!31}{40\!\cdots\!23}a^{6}+\frac{81\!\cdots\!42}{40\!\cdots\!23}a^{4}-\frac{37\!\cdots\!87}{40\!\cdots\!23}a^{2}-\frac{28\!\cdots\!30}{40\!\cdots\!23}$, $\frac{56\!\cdots\!55}{40\!\cdots\!23}a^{28}+\frac{36\!\cdots\!19}{40\!\cdots\!23}a^{26}+\frac{15\!\cdots\!29}{40\!\cdots\!23}a^{24}+\frac{62\!\cdots\!83}{80\!\cdots\!46}a^{22}-\frac{17\!\cdots\!62}{40\!\cdots\!23}a^{20}-\frac{12\!\cdots\!31}{40\!\cdots\!23}a^{18}-\frac{48\!\cdots\!74}{40\!\cdots\!23}a^{16}+\frac{50\!\cdots\!55}{80\!\cdots\!46}a^{14}+\frac{24\!\cdots\!35}{40\!\cdots\!23}a^{12}-\frac{45\!\cdots\!61}{80\!\cdots\!46}a^{10}-\frac{33\!\cdots\!09}{40\!\cdots\!23}a^{8}+\frac{34\!\cdots\!53}{80\!\cdots\!46}a^{6}+\frac{28\!\cdots\!84}{40\!\cdots\!23}a^{4}+\frac{70\!\cdots\!93}{80\!\cdots\!46}a^{2}-\frac{52\!\cdots\!46}{40\!\cdots\!23}$, $\frac{35\!\cdots\!93}{40\!\cdots\!23}a^{28}+\frac{23\!\cdots\!37}{40\!\cdots\!23}a^{26}+\frac{20\!\cdots\!63}{80\!\cdots\!46}a^{24}+\frac{21\!\cdots\!15}{40\!\cdots\!23}a^{22}-\frac{78\!\cdots\!43}{40\!\cdots\!23}a^{20}-\frac{83\!\cdots\!44}{40\!\cdots\!23}a^{18}-\frac{88\!\cdots\!71}{80\!\cdots\!46}a^{16}+\frac{15\!\cdots\!58}{40\!\cdots\!23}a^{14}+\frac{36\!\cdots\!69}{80\!\cdots\!46}a^{12}-\frac{12\!\cdots\!53}{40\!\cdots\!23}a^{10}-\frac{47\!\cdots\!51}{80\!\cdots\!46}a^{8}+\frac{77\!\cdots\!69}{40\!\cdots\!23}a^{6}+\frac{41\!\cdots\!73}{80\!\cdots\!46}a^{4}+\frac{50\!\cdots\!85}{40\!\cdots\!23}a^{2}-\frac{23\!\cdots\!38}{40\!\cdots\!23}$, $\frac{39\!\cdots\!73}{80\!\cdots\!46}a^{29}+\frac{994860012356039}{93\!\cdots\!61}a^{28}+\frac{13\!\cdots\!55}{40\!\cdots\!23}a^{27}+\frac{12\!\cdots\!79}{18\!\cdots\!22}a^{26}+\frac{11\!\cdots\!19}{80\!\cdots\!46}a^{25}+\frac{25\!\cdots\!08}{93\!\cdots\!61}a^{24}+\frac{12\!\cdots\!36}{40\!\cdots\!23}a^{23}+\frac{46\!\cdots\!64}{93\!\cdots\!61}a^{22}-\frac{27\!\cdots\!92}{40\!\cdots\!23}a^{21}-\frac{80\!\cdots\!93}{18\!\cdots\!22}a^{20}-\frac{44\!\cdots\!25}{40\!\cdots\!23}a^{19}-\frac{19\!\cdots\!01}{93\!\cdots\!61}a^{18}-\frac{57\!\cdots\!67}{80\!\cdots\!46}a^{17}-\frac{34\!\cdots\!12}{93\!\cdots\!61}a^{16}+\frac{15\!\cdots\!59}{80\!\cdots\!46}a^{15}+\frac{81\!\cdots\!05}{18\!\cdots\!22}a^{14}+\frac{10\!\cdots\!27}{40\!\cdots\!23}a^{13}+\frac{61\!\cdots\!01}{18\!\cdots\!22}a^{12}-\frac{46\!\cdots\!71}{40\!\cdots\!23}a^{11}-\frac{39\!\cdots\!05}{93\!\cdots\!61}a^{10}-\frac{25\!\cdots\!21}{80\!\cdots\!46}a^{9}-\frac{42\!\cdots\!43}{93\!\cdots\!61}a^{8}+\frac{20\!\cdots\!58}{40\!\cdots\!23}a^{7}+\frac{31\!\cdots\!01}{93\!\cdots\!61}a^{6}+\frac{21\!\cdots\!91}{80\!\cdots\!46}a^{5}+\frac{37\!\cdots\!61}{93\!\cdots\!61}a^{4}+\frac{48\!\cdots\!55}{40\!\cdots\!23}a^{3}+\frac{32\!\cdots\!27}{93\!\cdots\!61}a^{2}+\frac{17\!\cdots\!91}{80\!\cdots\!46}a-\frac{12\!\cdots\!39}{93\!\cdots\!61}$, $\frac{22\!\cdots\!36}{40\!\cdots\!23}a^{29}+\frac{18\!\cdots\!08}{40\!\cdots\!23}a^{28}+\frac{13\!\cdots\!81}{40\!\cdots\!23}a^{27}+\frac{11\!\cdots\!07}{40\!\cdots\!23}a^{26}+\frac{57\!\cdots\!82}{40\!\cdots\!23}a^{25}+\frac{50\!\cdots\!95}{40\!\cdots\!23}a^{24}+\frac{21\!\cdots\!25}{80\!\cdots\!46}a^{23}+\frac{20\!\cdots\!93}{80\!\cdots\!46}a^{22}-\frac{17\!\cdots\!59}{80\!\cdots\!46}a^{21}-\frac{98\!\cdots\!53}{80\!\cdots\!46}a^{20}-\frac{45\!\cdots\!15}{40\!\cdots\!23}a^{19}-\frac{40\!\cdots\!00}{40\!\cdots\!23}a^{18}-\frac{85\!\cdots\!38}{40\!\cdots\!23}a^{17}-\frac{18\!\cdots\!78}{40\!\cdots\!23}a^{16}+\frac{93\!\cdots\!19}{40\!\cdots\!23}a^{15}+\frac{15\!\cdots\!69}{80\!\cdots\!46}a^{14}+\frac{71\!\cdots\!04}{40\!\cdots\!23}a^{13}+\frac{83\!\cdots\!32}{40\!\cdots\!23}a^{12}-\frac{18\!\cdots\!17}{80\!\cdots\!46}a^{11}-\frac{63\!\cdots\!16}{40\!\cdots\!23}a^{10}-\frac{19\!\cdots\!57}{80\!\cdots\!46}a^{9}-\frac{10\!\cdots\!22}{40\!\cdots\!23}a^{8}+\frac{75\!\cdots\!77}{40\!\cdots\!23}a^{7}+\frac{83\!\cdots\!93}{80\!\cdots\!46}a^{6}+\frac{79\!\cdots\!70}{40\!\cdots\!23}a^{5}+\frac{18\!\cdots\!71}{80\!\cdots\!46}a^{4}+\frac{76\!\cdots\!25}{80\!\cdots\!46}a^{3}+\frac{39\!\cdots\!05}{80\!\cdots\!46}a^{2}+\frac{79\!\cdots\!04}{40\!\cdots\!23}a-\frac{41\!\cdots\!61}{40\!\cdots\!23}$, $\frac{36\!\cdots\!93}{80\!\cdots\!46}a^{29}+\frac{58\!\cdots\!44}{40\!\cdots\!23}a^{28}+\frac{21\!\cdots\!83}{80\!\cdots\!46}a^{27}+\frac{36\!\cdots\!92}{40\!\cdots\!23}a^{26}+\frac{45\!\cdots\!02}{40\!\cdots\!23}a^{25}+\frac{30\!\cdots\!83}{80\!\cdots\!46}a^{24}+\frac{82\!\cdots\!80}{40\!\cdots\!23}a^{23}+\frac{28\!\cdots\!01}{40\!\cdots\!23}a^{22}-\frac{77\!\cdots\!38}{40\!\cdots\!23}a^{21}-\frac{46\!\cdots\!81}{80\!\cdots\!46}a^{20}-\frac{71\!\cdots\!83}{80\!\cdots\!46}a^{19}-\frac{12\!\cdots\!22}{40\!\cdots\!23}a^{18}-\frac{47\!\cdots\!95}{40\!\cdots\!23}a^{17}-\frac{23\!\cdots\!29}{40\!\cdots\!23}a^{16}+\frac{73\!\cdots\!11}{40\!\cdots\!23}a^{15}+\frac{25\!\cdots\!74}{40\!\cdots\!23}a^{14}+\frac{10\!\cdots\!15}{80\!\cdots\!46}a^{13}+\frac{39\!\cdots\!13}{80\!\cdots\!46}a^{12}-\frac{73\!\cdots\!52}{40\!\cdots\!23}a^{11}-\frac{52\!\cdots\!57}{80\!\cdots\!46}a^{10}-\frac{73\!\cdots\!03}{40\!\cdots\!23}a^{9}-\frac{55\!\cdots\!53}{80\!\cdots\!46}a^{8}+\frac{57\!\cdots\!11}{40\!\cdots\!23}a^{7}+\frac{44\!\cdots\!75}{80\!\cdots\!46}a^{6}+\frac{13\!\cdots\!97}{80\!\cdots\!46}a^{5}+\frac{24\!\cdots\!24}{40\!\cdots\!23}a^{4}+\frac{11\!\cdots\!19}{80\!\cdots\!46}a^{3}-\frac{16\!\cdots\!74}{40\!\cdots\!23}a^{2}-\frac{19\!\cdots\!67}{40\!\cdots\!23}a-\frac{14\!\cdots\!43}{80\!\cdots\!46}$, $\frac{39\!\cdots\!19}{80\!\cdots\!46}a^{29}+\frac{994860012356039}{93\!\cdots\!61}a^{28}+\frac{11\!\cdots\!56}{40\!\cdots\!23}a^{27}+\frac{12\!\cdots\!79}{18\!\cdots\!22}a^{26}+\frac{99\!\cdots\!81}{80\!\cdots\!46}a^{25}+\frac{25\!\cdots\!08}{93\!\cdots\!61}a^{24}+\frac{90\!\cdots\!70}{40\!\cdots\!23}a^{23}+\frac{46\!\cdots\!64}{93\!\cdots\!61}a^{22}-\frac{16\!\cdots\!29}{80\!\cdots\!46}a^{21}-\frac{80\!\cdots\!93}{18\!\cdots\!22}a^{20}-\frac{39\!\cdots\!37}{40\!\cdots\!23}a^{19}-\frac{19\!\cdots\!01}{93\!\cdots\!61}a^{18}-\frac{10\!\cdots\!45}{80\!\cdots\!46}a^{17}-\frac{34\!\cdots\!12}{93\!\cdots\!61}a^{16}+\frac{16\!\cdots\!75}{80\!\cdots\!46}a^{15}+\frac{81\!\cdots\!05}{18\!\cdots\!22}a^{14}+\frac{11\!\cdots\!57}{80\!\cdots\!46}a^{13}+\frac{61\!\cdots\!01}{18\!\cdots\!22}a^{12}-\frac{81\!\cdots\!61}{40\!\cdots\!23}a^{11}-\frac{39\!\cdots\!05}{93\!\cdots\!61}a^{10}-\frac{80\!\cdots\!46}{40\!\cdots\!23}a^{9}-\frac{42\!\cdots\!43}{93\!\cdots\!61}a^{8}+\frac{66\!\cdots\!89}{40\!\cdots\!23}a^{7}+\frac{31\!\cdots\!01}{93\!\cdots\!61}a^{6}+\frac{71\!\cdots\!00}{40\!\cdots\!23}a^{5}+\frac{37\!\cdots\!61}{93\!\cdots\!61}a^{4}+\frac{39\!\cdots\!59}{40\!\cdots\!23}a^{3}+\frac{32\!\cdots\!27}{93\!\cdots\!61}a^{2}-\frac{37\!\cdots\!70}{40\!\cdots\!23}a-\frac{12\!\cdots\!39}{93\!\cdots\!61}$, $\frac{88\!\cdots\!27}{80\!\cdots\!46}a^{29}+\frac{11\!\cdots\!61}{80\!\cdots\!46}a^{28}+\frac{20\!\cdots\!92}{40\!\cdots\!23}a^{27}+\frac{33\!\cdots\!48}{40\!\cdots\!23}a^{26}+\frac{15\!\cdots\!19}{80\!\cdots\!46}a^{25}+\frac{13\!\cdots\!55}{40\!\cdots\!23}a^{24}+\frac{59\!\cdots\!90}{40\!\cdots\!23}a^{23}+\frac{24\!\cdots\!49}{40\!\cdots\!23}a^{22}-\frac{43\!\cdots\!41}{40\!\cdots\!23}a^{21}-\frac{52\!\cdots\!23}{80\!\cdots\!46}a^{20}-\frac{10\!\cdots\!59}{80\!\cdots\!46}a^{19}-\frac{21\!\cdots\!23}{80\!\cdots\!46}a^{18}+\frac{12\!\cdots\!90}{40\!\cdots\!23}a^{17}-\frac{55\!\cdots\!55}{40\!\cdots\!23}a^{16}+\frac{31\!\cdots\!13}{80\!\cdots\!46}a^{15}+\frac{45\!\cdots\!79}{80\!\cdots\!46}a^{14}-\frac{18\!\cdots\!15}{40\!\cdots\!23}a^{13}+\frac{14\!\cdots\!79}{40\!\cdots\!23}a^{12}-\frac{60\!\cdots\!19}{80\!\cdots\!46}a^{11}-\frac{47\!\cdots\!23}{80\!\cdots\!46}a^{10}+\frac{21\!\cdots\!25}{40\!\cdots\!23}a^{9}-\frac{20\!\cdots\!66}{40\!\cdots\!23}a^{8}+\frac{37\!\cdots\!76}{40\!\cdots\!23}a^{7}+\frac{38\!\cdots\!41}{80\!\cdots\!46}a^{6}-\frac{21\!\cdots\!22}{40\!\cdots\!23}a^{5}+\frac{18\!\cdots\!83}{40\!\cdots\!23}a^{4}-\frac{15\!\cdots\!94}{40\!\cdots\!23}a^{3}+\frac{10\!\cdots\!17}{40\!\cdots\!23}a^{2}+\frac{22\!\cdots\!67}{80\!\cdots\!46}a-\frac{63\!\cdots\!75}{80\!\cdots\!46}$ 1326576254979433/18754374732587122*a^28 + 8306286150833357/18754374732587122*a^26 + 35307424091996615/18754374732587122*a^24 + 34377203743704211/9377187366293561*a^22 - 41145277102823553/18754374732587122*a^20 - 273521500932551969/18754374732587122*a^18 - 48642970614083733/9377187366293561*a^16 + 525107525986560025/18754374732587122*a^14 + 257602554726402251/9377187366293561*a^12 - 442257634298146243/18754374732587122*a^10 - 331833835712344276/9377187366293561*a^8 + 302477397796410247/18754374732587122*a^6 + 588094923802096703/18754374732587122*a^4 + 79411307076985771/9377187366293561*a^2 + 331179604370172/9377187366293561, 62172617982818102/403219056750623123*a^28 + 751046170523241747/806438113501246246*a^26 + 1567240858760021405/403219056750623123*a^24 + 2854252083138208613/403219056750623123*a^22 - 2645300528959563064/403219056750623123*a^20 - 24658836747239886113/806438113501246246*a^18 - 1518553969119112296/403219056750623123*a^16 + 51042074214668450981/806438113501246246*a^14 + 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132783325116105535797/806438113501246246*a^5 + 24586891493759843024/403219056750623123*a^4 + 11641441153664804719/806438113501246246*a^3 - 167542852696273974/403219056750623123*a^2 - 1921645751204874867/403219056750623123*a - 1432312203375380643/806438113501246246, 392535263631290319/806438113501246246*a^29 + 994860012356039/9377187366293561*a^28 + 1185967419808793356/403219056750623123*a^27 + 12092099470130079/18754374732587122*a^26 + 9904687381671756981/806438113501246246*a^25 + 25338983356380108/9377187366293561*a^24 + 9023580874337134570/403219056750623123*a^23 + 46747009369505964/9377187366293561*a^22 - 16656070575188831129/806438113501246246*a^21 - 80096093858975293/18754374732587122*a^20 - 39058046417664512237/403219056750623123*a^19 - 198635665420746001/9377187366293561*a^18 - 10380956487758076845/806438113501246246*a^17 - 34540844884616512/9377187366293561*a^16 + 161407504444524945175/806438113501246246*a^15 + 810103656146961405/18754374732587122*a^14 + 116296406244698564657/806438113501246246*a^13 + 618019097522412901/18754374732587122*a^12 - 81401651302772757061/403219056750623123*a^11 - 398489723690218305/9377187366293561*a^10 - 80265420992711759046/403219056750623123*a^9 - 420747605475803043/9377187366293561*a^8 + 66074298446245003389/403219056750623123*a^7 + 316940204321380701/9377187366293561*a^6 + 71526400918338541700/403219056750623123*a^5 + 378412545500415161/9377187366293561*a^4 + 3909086181047114959/403219056750623123*a^3 + 32559292125796427/9377187366293561*a^2 - 375064360266454470/403219056750623123*a - 1208553524409339/9377187366293561, 8878395817795227/806438113501246246*a^29 + 113692416776715561/806438113501246246*a^28 + 20850976133175792/403219056750623123*a^27 + 336332205458634448/403219056750623123*a^26 + 151574622831108419/806438113501246246*a^25 + 1397450130546819555/403219056750623123*a^24 + 59520288567699890/403219056750623123*a^23 + 2464960368749679549/403219056750623123*a^22 - 433853984369884641/403219056750623123*a^21 - 5230324474576881723/806438113501246246*a^20 - 1006374465752291959/806438113501246246*a^19 - 21708067080704512023/806438113501246246*a^18 + 1226591142211202690/403219056750623123*a^17 - 557556522783552455/403219056750623123*a^16 + 3124126268642381713/806438113501246246*a^15 + 45373699656617990179/806438113501246246*a^14 - 1816994803897491815/403219056750623123*a^13 + 14746026139006089679/403219056750623123*a^12 - 6040806010698033519/806438113501246246*a^11 - 47074881597365284823/806438113501246246*a^10 + 2119563328375465725/403219056750623123*a^9 - 20602963196492654866/403219056750623123*a^8 + 3713291531739116176/403219056750623123*a^7 + 38180567499051153341/806438113501246246*a^6 - 2193355155527019822/403219056750623123*a^5 + 18353178994944075183/403219056750623123*a^4 - 1576758817003050294/403219056750623123*a^3 + 1094278982372336117/403219056750623123*a^2 + 2289513918668694167/806438113501246246*a - 638829885238258675/806438113501246246 (assuming GRH)	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:		\( 18931497.119988322 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 18931497.119988322 \cdot 1}{4\cdot\sqrt{1310417368511817590988945406032175169536}}\cr\approx \mathstrut & 0.122777326819508 \end{aligned}\] (assuming GRH)

Galois group

$S_3\times C_{10}$ (as 30T12):

Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Frobenius cycle types

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
\(2\) 2		Deg $30$	$6$	$5$	$40$
\(11\) 11	11.10.8.5	$x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$	$5$	$2$	$8$	$C_{10}$	$[\ ]_{5}^{2}$
	11.20.18.1	$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$	$10$	$2$	$18$	20T3	$[\ ]_{10}^{2}$

Artin representations