Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^30 + 6*x^28 - x^27 + 14*x^26 - x^25 + 8*x^24 + 13*x^23 - 39*x^22 + 37*x^21 - 93*x^20 + 30*x^19 + 10*x^18 - 13*x^17 + 349*x^16 + 77*x^15 + 721*x^14 + 575*x^13 + 1026*x^12 + 1233*x^11 + 1137*x^10 + 993*x^9 + 501*x^8 - 41*x^7 - 253*x^6 - 186*x^5 - 58*x^4 + 6*x^3 + 14*x^2 + 6*x + 1)
gp: K = bnfinit(y^30 + 6*y^28 - y^27 + 14*y^26 - y^25 + 8*y^24 + 13*y^23 - 39*y^22 + 37*y^21 - 93*y^20 + 30*y^19 + 10*y^18 - 13*y^17 + 349*y^16 + 77*y^15 + 721*y^14 + 575*y^13 + 1026*y^12 + 1233*y^11 + 1137*y^10 + 993*y^9 + 501*y^8 - 41*y^7 - 253*y^6 - 186*y^5 - 58*y^4 + 6*y^3 + 14*y^2 + 6*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 + 6*x^28 - x^27 + 14*x^26 - x^25 + 8*x^24 + 13*x^23 - 39*x^22 + 37*x^21 - 93*x^20 + 30*x^19 + 10*x^18 - 13*x^17 + 349*x^16 + 77*x^15 + 721*x^14 + 575*x^13 + 1026*x^12 + 1233*x^11 + 1137*x^10 + 993*x^9 + 501*x^8 - 41*x^7 - 253*x^6 - 186*x^5 - 58*x^4 + 6*x^3 + 14*x^2 + 6*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 + 6*x^28 - x^27 + 14*x^26 - x^25 + 8*x^24 + 13*x^23 - 39*x^22 + 37*x^21 - 93*x^20 + 30*x^19 + 10*x^18 - 13*x^17 + 349*x^16 + 77*x^15 + 721*x^14 + 575*x^13 + 1026*x^12 + 1233*x^11 + 1137*x^10 + 993*x^9 + 501*x^8 - 41*x^7 - 253*x^6 - 186*x^5 - 58*x^4 + 6*x^3 + 14*x^2 + 6*x + 1)
\( x^{30} + 6 x^{28} - x^{27} + 14 x^{26} - x^{25} + 8 x^{24} + 13 x^{23} - 39 x^{22} + 37 x^{21} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
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: |
\(-45711723244122563996456104229882472771\)
\(\medspace = -\,3^{20}\cdot 11^{27}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.00\) | 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: | $3^{4/3}11^{9/10}\approx 37.44683262965639$ | ||
| Ramified primes: |
\(3\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $15$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{445}a^{25}+\frac{11}{445}a^{24}-\frac{219}{445}a^{23}+\frac{86}{445}a^{22}+\frac{39}{89}a^{21}+\frac{186}{445}a^{20}+\frac{82}{445}a^{19}-\frac{26}{89}a^{18}+\frac{203}{445}a^{17}+\frac{153}{445}a^{16}+\frac{167}{445}a^{15}+\frac{169}{445}a^{14}+\frac{23}{89}a^{13}-\frac{34}{89}a^{12}+\frac{51}{445}a^{11}-\frac{167}{445}a^{10}-\frac{164}{445}a^{9}-\frac{25}{89}a^{8}-\frac{10}{89}a^{7}+\frac{84}{445}a^{6}-\frac{4}{445}a^{5}+\frac{212}{445}a^{4}+\frac{18}{89}a^{3}-\frac{32}{89}a^{2}+\frac{18}{445}a+\frac{152}{445}$, $\frac{1}{445}a^{26}+\frac{21}{89}a^{24}-\frac{35}{89}a^{23}+\frac{139}{445}a^{22}-\frac{179}{445}a^{21}-\frac{184}{445}a^{20}-\frac{142}{445}a^{19}-\frac{147}{445}a^{18}+\frac{29}{89}a^{17}-\frac{181}{445}a^{16}+\frac{112}{445}a^{15}+\frac{36}{445}a^{14}-\frac{20}{89}a^{13}+\frac{141}{445}a^{12}+\frac{162}{445}a^{11}-\frac{107}{445}a^{10}-\frac{101}{445}a^{9}-\frac{2}{89}a^{8}+\frac{189}{445}a^{7}-\frac{38}{445}a^{6}-\frac{189}{445}a^{5}-\frac{17}{445}a^{4}+\frac{37}{89}a^{3}-\frac{2}{445}a^{2}-\frac{46}{445}a+\frac{108}{445}$, $\frac{1}{445}a^{27}+\frac{1}{89}a^{24}-\frac{6}{445}a^{23}+\frac{136}{445}a^{22}-\frac{189}{445}a^{21}-\frac{92}{445}a^{20}+\frac{143}{445}a^{19}-\frac{136}{445}a^{17}+\frac{67}{445}a^{16}-\frac{144}{445}a^{15}-\frac{9}{89}a^{14}+\frac{81}{445}a^{13}+\frac{212}{445}a^{12}-\frac{122}{445}a^{11}+\frac{79}{445}a^{10}-\frac{29}{89}a^{9}-\frac{36}{445}a^{8}-\frac{128}{445}a^{7}-\frac{109}{445}a^{6}-\frac{42}{445}a^{5}+\frac{35}{89}a^{4}-\frac{107}{445}a^{3}-\frac{156}{445}a^{2}-\frac{2}{445}a+\frac{12}{89}$, $\frac{1}{13387012429555}a^{28}+\frac{11292184506}{13387012429555}a^{27}+\frac{239979340}{2677402485911}a^{26}-\frac{11621511658}{13387012429555}a^{25}-\frac{4902792104044}{13387012429555}a^{24}-\frac{6166493836393}{13387012429555}a^{23}-\frac{5758271901301}{13387012429555}a^{22}-\frac{1341906105356}{13387012429555}a^{21}-\frac{5466840494667}{13387012429555}a^{20}+\frac{6444606444167}{13387012429555}a^{19}+\frac{6175612326114}{13387012429555}a^{18}-\frac{1231175297543}{13387012429555}a^{17}+\frac{1579876048774}{13387012429555}a^{16}-\frac{896338943675}{2677402485911}a^{15}+\frac{4836948846369}{13387012429555}a^{14}+\frac{4601689343788}{13387012429555}a^{13}-\frac{511854111159}{2677402485911}a^{12}-\frac{5347110096196}{13387012429555}a^{11}-\frac{991991156498}{2677402485911}a^{10}-\frac{2241961602974}{13387012429555}a^{9}+\frac{2053307832501}{13387012429555}a^{8}-\frac{759610988112}{13387012429555}a^{7}+\frac{4602001636602}{13387012429555}a^{6}-\frac{848191643384}{2677402485911}a^{5}-\frac{606966174518}{13387012429555}a^{4}-\frac{1820764963473}{13387012429555}a^{3}+\frac{589929716967}{13387012429555}a^{2}-\frac{3254376689291}{13387012429555}a-\frac{2112461484861}{13387012429555}$, $\frac{1}{13387012429555}a^{29}+\frac{1002026522}{2677402485911}a^{27}+\frac{14689278362}{13387012429555}a^{26}+\frac{13605534833}{13387012429555}a^{25}+\frac{324983930549}{2677402485911}a^{24}+\frac{682285836191}{2677402485911}a^{23}-\frac{1164366560091}{2677402485911}a^{22}-\frac{1045981908}{2677402485911}a^{21}+\frac{4249606413601}{13387012429555}a^{20}+\frac{3656525880493}{13387012429555}a^{19}+\frac{1039894128888}{13387012429555}a^{18}-\frac{5903150631877}{13387012429555}a^{17}-\frac{326707186607}{2677402485911}a^{16}-\frac{3339284003187}{13387012429555}a^{15}+\frac{1170255526547}{2677402485911}a^{14}-\frac{6669905464812}{13387012429555}a^{13}+\frac{6663867702186}{13387012429555}a^{12}+\frac{3666619169923}{13387012429555}a^{11}-\frac{2379384253328}{13387012429555}a^{10}-\frac{4240524289601}{13387012429555}a^{9}-\frac{1682088816724}{13387012429555}a^{8}+\frac{293229730896}{13387012429555}a^{7}+\frac{451143399178}{2677402485911}a^{6}+\frac{2876129479224}{13387012429555}a^{5}+\frac{1849453151018}{13387012429555}a^{4}+\frac{604978445978}{13387012429555}a^{3}-\frac{2196819749374}{13387012429555}a^{2}+\frac{883452233921}{2677402485911}a-\frac{2116054177081}{13387012429555}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $14$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{12653957152607}{2677402485911} a^{29} - \frac{6159480532392}{2677402485911} a^{28} + \frac{78477429740129}{2677402485911} a^{27} - \frac{50349640529454}{2677402485911} a^{26} + \frac{198772937530862}{2677402485911} a^{25} - \frac{105849092098559}{2677402485911} a^{24} + \frac{144663718380552}{2677402485911} a^{23} + \frac{102353127789123}{2677402485911} a^{22} - \frac{550735971791860}{2677402485911} a^{21} + \frac{736058773339872}{2677402485911} a^{20} - \frac{1513402505071755}{2677402485911} a^{19} + \frac{1078104915700476}{2677402485911} a^{18} - \frac{328577462792968}{2677402485911} a^{17} - \frac{76704618826301}{2677402485911} a^{16} + \frac{4489202749714251}{2677402485911} a^{15} - \frac{1215995815292423}{2677402485911} a^{14} + \frac{9556496914311311}{2677402485911} a^{13} + \frac{2768316816085662}{2677402485911} a^{12} + \frac{11274958798156982}{2677402485911} a^{11} + \frac{10235722776091104}{2677402485911} a^{10} + \frac{9077290907100770}{2677402485911} a^{9} + \frac{8052604766471538}{2677402485911} a^{8} + \frac{2342098007255521}{2677402485911} a^{7} - \frac{1717269073290881}{2677402485911} a^{6} - \frac{2251867087989700}{2677402485911} a^{5} - \frac{1126642969502881}{2677402485911} a^{4} - \frac{133995717584920}{2677402485911} a^{3} + \frac{136551788434983}{2677402485911} a^{2} + \frac{93114123118404}{2677402485911} a + \frac{22939094334598}{2677402485911} \)
(order $22$)
| 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: |
$\frac{10789396149643}{2677402485911}a^{29}-\frac{6209009900155}{2677402485911}a^{28}+\frac{68069252536155}{2677402485911}a^{27}-\frac{49752505208909}{2677402485911}a^{26}+\frac{178218159148945}{2677402485911}a^{25}-\frac{111778223935115}{2677402485911}a^{24}+\frac{2191710381284}{39961231133}a^{23}+\frac{59462656799041}{2677402485911}a^{22}-\frac{458068295311766}{2677402485911}a^{21}+\frac{662896898215296}{2677402485911}a^{20}-\frac{13\!\cdots\!88}{2677402485911}a^{19}+\frac{10\!\cdots\!26}{2677402485911}a^{18}-\frac{493537286062922}{2677402485911}a^{17}+\frac{572177848391}{13590875563}a^{16}+\frac{37\!\cdots\!43}{2677402485911}a^{15}-\frac{13\!\cdots\!80}{2677402485911}a^{14}+\frac{84\!\cdots\!27}{2677402485911}a^{13}+\frac{13\!\cdots\!82}{2677402485911}a^{12}+\frac{150928106925311}{39961231133}a^{11}+\frac{75\!\cdots\!36}{2677402485911}a^{10}+\frac{77\!\cdots\!78}{2677402485911}a^{9}+\frac{62\!\cdots\!12}{2677402485911}a^{8}+\frac{17\!\cdots\!01}{2677402485911}a^{7}-\frac{14\!\cdots\!09}{2677402485911}a^{6}-\frac{18\!\cdots\!11}{2677402485911}a^{5}-\frac{831819137188284}{2677402485911}a^{4}-\frac{71618559635646}{2677402485911}a^{3}+\frac{97771517789218}{2677402485911}a^{2}+\frac{71078602571932}{2677402485911}a+\frac{13261190527233}{2677402485911}$, $\frac{42007054911324}{13387012429555}a^{29}-\frac{22675878196668}{13387012429555}a^{28}+\frac{262773843399943}{13387012429555}a^{27}-\frac{182763378244703}{13387012429555}a^{26}+\frac{135359689497363}{2677402485911}a^{25}-\frac{398783921543694}{13387012429555}a^{24}+\frac{523240508501217}{13387012429555}a^{23}+\frac{284537300105769}{13387012429555}a^{22}-\frac{18\!\cdots\!42}{13387012429555}a^{21}+\frac{25\!\cdots\!31}{13387012429555}a^{20}-\frac{52\!\cdots\!86}{13387012429555}a^{19}+\frac{39\!\cdots\!17}{13387012429555}a^{18}-\frac{15\!\cdots\!23}{13387012429555}a^{17}+\frac{61737431159688}{13387012429555}a^{16}+\frac{14\!\cdots\!89}{13387012429555}a^{15}-\frac{48\!\cdots\!67}{13387012429555}a^{14}+\frac{32\!\cdots\!34}{13387012429555}a^{13}+\frac{13\!\cdots\!38}{2677402485911}a^{12}+\frac{76\!\cdots\!99}{2677402485911}a^{11}+\frac{31\!\cdots\!16}{13387012429555}a^{10}+\frac{29\!\cdots\!48}{13387012429555}a^{9}+\frac{50\!\cdots\!92}{2677402485911}a^{8}+\frac{62\!\cdots\!68}{13387012429555}a^{7}-\frac{60\!\cdots\!11}{13387012429555}a^{6}-\frac{77\!\cdots\!43}{13387012429555}a^{5}-\frac{36\!\cdots\!94}{13387012429555}a^{4}-\frac{365815902811211}{13387012429555}a^{3}+\frac{498389740604109}{13387012429555}a^{2}+\frac{336963092467679}{13387012429555}a+\frac{81064847986459}{13387012429555}$, $\frac{67219901717457}{13387012429555}a^{29}-\frac{7126288002627}{13387012429555}a^{28}+\frac{391563042616521}{13387012429555}a^{27}-\frac{19259994968910}{2677402485911}a^{26}+\frac{867080398279693}{13387012429555}a^{25}-\frac{66465290753786}{13387012429555}a^{24}+\frac{299380846041724}{13387012429555}a^{23}+\frac{214535402800339}{2677402485911}a^{22}-\frac{30\!\cdots\!99}{13387012429555}a^{21}+\frac{28\!\cdots\!32}{13387012429555}a^{20}-\frac{60\!\cdots\!49}{13387012429555}a^{19}+\frac{16\!\cdots\!96}{13387012429555}a^{18}+\frac{25\!\cdots\!87}{13387012429555}a^{17}-\frac{649992573395986}{2677402485911}a^{16}+\frac{25\!\cdots\!38}{13387012429555}a^{15}+\frac{18\!\cdots\!37}{13387012429555}a^{14}+\frac{44\!\cdots\!73}{13387012429555}a^{13}+\frac{37\!\cdots\!54}{13387012429555}a^{12}+\frac{10\!\cdots\!23}{2677402485911}a^{11}+\frac{80\!\cdots\!02}{13387012429555}a^{10}+\frac{55\!\cdots\!81}{13387012429555}a^{9}+\frac{57\!\cdots\!37}{13387012429555}a^{8}+\frac{20\!\cdots\!96}{13387012429555}a^{7}-\frac{28004243922717}{39961231133}a^{6}-\frac{16\!\cdots\!61}{13387012429555}a^{5}-\frac{92\!\cdots\!13}{13387012429555}a^{4}-\frac{15\!\cdots\!89}{13387012429555}a^{3}+\frac{981454440044648}{13387012429555}a^{2}+\frac{778788840691258}{13387012429555}a+\frac{46634290305740}{2677402485911}$, $\frac{11826100080436}{13387012429555}a^{29}+\frac{52085378911}{67954377815}a^{28}+\frac{57535011508638}{13387012429555}a^{27}+\frac{60027700972756}{13387012429555}a^{26}+\frac{69186180630498}{13387012429555}a^{25}+\frac{209392128100773}{13387012429555}a^{24}-\frac{148195811715041}{13387012429555}a^{23}+\frac{413622750710206}{13387012429555}a^{22}-\frac{5967710218223}{150415869995}a^{21}-\frac{2620153184796}{2677402485911}a^{20}-\frac{122006526574176}{13387012429555}a^{19}-\frac{15\!\cdots\!22}{13387012429555}a^{18}+\frac{22\!\cdots\!49}{13387012429555}a^{17}-\frac{17\!\cdots\!11}{13387012429555}a^{16}+\frac{48\!\cdots\!09}{13387012429555}a^{15}+\frac{43\!\cdots\!86}{13387012429555}a^{14}+\frac{920627911524680}{2677402485911}a^{13}+\frac{16\!\cdots\!47}{13387012429555}a^{12}+\frac{71\!\cdots\!68}{13387012429555}a^{11}+\frac{25\!\cdots\!46}{13387012429555}a^{10}+\frac{210329843377249}{199806155665}a^{9}+\frac{15\!\cdots\!27}{13387012429555}a^{8}+\frac{17\!\cdots\!03}{2677402485911}a^{7}-\frac{17\!\cdots\!13}{13387012429555}a^{6}-\frac{39\!\cdots\!98}{13387012429555}a^{5}-\frac{27\!\cdots\!04}{13387012429555}a^{4}-\frac{129044171625233}{2677402485911}a^{3}+\frac{178393670765071}{13387012429555}a^{2}+\frac{41194870441408}{2677402485911}a+\frac{80767532630006}{13387012429555}$, $\frac{17094875756948}{13387012429555}a^{29}-\frac{2739024913921}{13387012429555}a^{28}+\frac{98592172321184}{13387012429555}a^{27}-\frac{5771217172283}{2677402485911}a^{26}+\frac{43023798939345}{2677402485911}a^{25}-\frac{21681521184888}{13387012429555}a^{24}+\frac{59737850088856}{13387012429555}a^{23}+\frac{56739012195392}{2677402485911}a^{22}-\frac{157865090286762}{2677402485911}a^{21}+\frac{756030101485417}{13387012429555}a^{20}-\frac{303169999120818}{2677402485911}a^{19}+\frac{419374969792167}{13387012429555}a^{18}+\frac{764430592013586}{13387012429555}a^{17}-\frac{993227911331254}{13387012429555}a^{16}+\frac{64\!\cdots\!88}{13387012429555}a^{15}+\frac{151951262184567}{13387012429555}a^{14}+\frac{10\!\cdots\!41}{13387012429555}a^{13}+\frac{91\!\cdots\!97}{13387012429555}a^{12}+\frac{139079969292229}{150415869995}a^{11}+\frac{19\!\cdots\!27}{13387012429555}a^{10}+\frac{24\!\cdots\!14}{2677402485911}a^{9}+\frac{13\!\cdots\!47}{13387012429555}a^{8}+\frac{42\!\cdots\!61}{13387012429555}a^{7}-\frac{29\!\cdots\!22}{13387012429555}a^{6}-\frac{37\!\cdots\!06}{13387012429555}a^{5}-\frac{360570498321059}{2677402485911}a^{4}-\frac{133436459907038}{13387012429555}a^{3}+\frac{242526276439653}{13387012429555}a^{2}+\frac{132265553158686}{13387012429555}a+\frac{33417111869476}{13387012429555}$, $\frac{97529897696703}{13387012429555}a^{29}-\frac{7713177825590}{2677402485911}a^{28}+\frac{593063089563429}{13387012429555}a^{27}-\frac{326521477051892}{13387012429555}a^{26}+\frac{14\!\cdots\!06}{13387012429555}a^{25}-\frac{627895524810778}{13387012429555}a^{24}+\frac{901406368708288}{13387012429555}a^{23}+\frac{10\!\cdots\!34}{13387012429555}a^{22}-\frac{21901724960437}{67954377815}a^{21}+\frac{52\!\cdots\!94}{13387012429555}a^{20}-\frac{10\!\cdots\!49}{13387012429555}a^{19}+\frac{13\!\cdots\!45}{2677402485911}a^{18}-\frac{662887492532534}{13387012429555}a^{17}-\frac{19\!\cdots\!73}{13387012429555}a^{16}+\frac{35\!\cdots\!96}{13387012429555}a^{15}-\frac{13\!\cdots\!42}{2677402485911}a^{14}+\frac{70\!\cdots\!43}{13387012429555}a^{13}+\frac{29\!\cdots\!78}{13387012429555}a^{12}+\frac{82\!\cdots\!42}{13387012429555}a^{11}+\frac{87\!\cdots\!39}{13387012429555}a^{10}+\frac{69\!\cdots\!17}{13387012429555}a^{9}+\frac{65\!\cdots\!84}{13387012429555}a^{8}+\frac{18\!\cdots\!19}{13387012429555}a^{7}-\frac{14\!\cdots\!64}{13387012429555}a^{6}-\frac{19\!\cdots\!32}{13387012429555}a^{5}-\frac{93\!\cdots\!31}{13387012429555}a^{4}-\frac{937845151994079}{13387012429555}a^{3}+\frac{253103972378841}{2677402485911}a^{2}+\frac{797713953147521}{13387012429555}a+\frac{194519865294752}{13387012429555}$, $\frac{40258899005446}{13387012429555}a^{29}-\frac{20974745794573}{13387012429555}a^{28}+\frac{253048149132137}{13387012429555}a^{27}-\frac{172267683297982}{13387012429555}a^{26}+\frac{656613260855748}{13387012429555}a^{25}-\frac{383896269083752}{13387012429555}a^{24}+\frac{105836065042937}{2677402485911}a^{23}+\frac{245173198019878}{13387012429555}a^{22}-\frac{16\!\cdots\!76}{13387012429555}a^{21}+\frac{23\!\cdots\!79}{13387012429555}a^{20}-\frac{50\!\cdots\!13}{13387012429555}a^{19}+\frac{38\!\cdots\!89}{13387012429555}a^{18}-\frac{330267097910695}{2677402485911}a^{17}+\frac{72243652126591}{2677402485911}a^{16}+\frac{13\!\cdots\!22}{13387012429555}a^{15}-\frac{41\!\cdots\!97}{13387012429555}a^{14}+\frac{31\!\cdots\!94}{13387012429555}a^{13}+\frac{67\!\cdots\!42}{13387012429555}a^{12}+\frac{38\!\cdots\!34}{13387012429555}a^{11}+\frac{29\!\cdots\!18}{13387012429555}a^{10}+\frac{30\!\cdots\!06}{13387012429555}a^{9}+\frac{24\!\cdots\!73}{13387012429555}a^{8}+\frac{76\!\cdots\!36}{13387012429555}a^{7}-\frac{10\!\cdots\!30}{2677402485911}a^{6}-\frac{73\!\cdots\!59}{13387012429555}a^{5}-\frac{38\!\cdots\!63}{13387012429555}a^{4}-\frac{7682110786774}{199806155665}a^{3}+\frac{480326197183558}{13387012429555}a^{2}+\frac{67252228826568}{2677402485911}a+\frac{81169188417917}{13387012429555}$, $\frac{27563838967856}{13387012429555}a^{29}-\frac{18861847106534}{13387012429555}a^{28}+\frac{177839134539321}{13387012429555}a^{27}-\frac{29277780461155}{2677402485911}a^{26}+\frac{480987399724706}{13387012429555}a^{25}-\frac{338076317379993}{13387012429555}a^{24}+\frac{428295290219153}{13387012429555}a^{23}+\frac{113972280820947}{13387012429555}a^{22}-\frac{11\!\cdots\!63}{13387012429555}a^{21}+\frac{18\!\cdots\!63}{13387012429555}a^{20}-\frac{38\!\cdots\!29}{13387012429555}a^{19}+\frac{32\!\cdots\!91}{13387012429555}a^{18}-\frac{17\!\cdots\!87}{13387012429555}a^{17}+\frac{411045151357959}{13387012429555}a^{16}+\frac{96\!\cdots\!31}{13387012429555}a^{15}-\frac{46\!\cdots\!89}{13387012429555}a^{14}+\frac{22\!\cdots\!81}{13387012429555}a^{13}+\frac{12\!\cdots\!59}{13387012429555}a^{12}+\frac{26\!\cdots\!39}{13387012429555}a^{11}+\frac{17\!\cdots\!09}{13387012429555}a^{10}+\frac{18\!\cdots\!03}{13387012429555}a^{9}+\frac{16\!\cdots\!17}{13387012429555}a^{8}+\frac{36\!\cdots\!36}{13387012429555}a^{7}-\frac{28\!\cdots\!48}{13387012429555}a^{6}-\frac{40\!\cdots\!67}{13387012429555}a^{5}-\frac{23\!\cdots\!64}{13387012429555}a^{4}-\frac{85227254697025}{2677402485911}a^{3}+\frac{238376568121197}{13387012429555}a^{2}+\frac{200172586426084}{13387012429555}a+\frac{56597877082394}{13387012429555}$, $\frac{94130203183329}{13387012429555}a^{29}-\frac{56224789005017}{13387012429555}a^{28}+\frac{119442524672796}{2677402485911}a^{27}-\frac{449876979177703}{13387012429555}a^{26}+\frac{15\!\cdots\!54}{13387012429555}a^{25}-\frac{10\!\cdots\!56}{13387012429555}a^{24}+\frac{268538425221723}{2677402485911}a^{23}+\frac{443260029326641}{13387012429555}a^{22}-\frac{39\!\cdots\!82}{13387012429555}a^{21}+\frac{58\!\cdots\!24}{13387012429555}a^{20}-\frac{12\!\cdots\!57}{13387012429555}a^{19}+\frac{10\!\cdots\!41}{13387012429555}a^{18}-\frac{48\!\cdots\!86}{13387012429555}a^{17}+\frac{294759510579908}{2677402485911}a^{16}+\frac{32\!\cdots\!39}{13387012429555}a^{15}-\frac{24\!\cdots\!09}{2677402485911}a^{14}+\frac{74\!\cdots\!48}{13387012429555}a^{13}+\frac{96\!\cdots\!77}{13387012429555}a^{12}+\frac{89\!\cdots\!89}{13387012429555}a^{11}+\frac{62\!\cdots\!86}{13387012429555}a^{10}+\frac{67\!\cdots\!46}{13387012429555}a^{9}+\frac{10\!\cdots\!69}{2677402485911}a^{8}+\frac{14\!\cdots\!43}{13387012429555}a^{7}-\frac{26\!\cdots\!54}{2677402485911}a^{6}-\frac{16\!\cdots\!38}{13387012429555}a^{5}-\frac{78\!\cdots\!83}{13387012429555}a^{4}-\frac{739884115261486}{13387012429555}a^{3}+\frac{1063276950858}{13590875563}a^{2}+\frac{724996230900102}{13387012429555}a+\frac{154541359039872}{13387012429555}$, $\frac{26405723044562}{13387012429555}a^{29}-\frac{4969019976201}{13387012429555}a^{28}+\frac{30994702533268}{2677402485911}a^{27}-\frac{10336187832034}{2677402485911}a^{26}+\frac{350849671469396}{13387012429555}a^{25}-\frac{63608065196429}{13387012429555}a^{24}+\frac{144199118522349}{13387012429555}a^{23}+\frac{384112599667378}{13387012429555}a^{22}-\frac{11\!\cdots\!67}{13387012429555}a^{21}+\frac{238575219159747}{2677402485911}a^{20}-\frac{12621908333203}{67954377815}a^{19}+\frac{186744035022434}{2677402485911}a^{18}+\frac{730503554814264}{13387012429555}a^{17}-\frac{10\!\cdots\!72}{13387012429555}a^{16}+\frac{97\!\cdots\!88}{13387012429555}a^{15}+\frac{17499424076961}{2677402485911}a^{14}+\frac{17\!\cdots\!34}{13387012429555}a^{13}+\frac{25\!\cdots\!08}{2677402485911}a^{12}+\frac{21\!\cdots\!59}{13387012429555}a^{11}+\frac{29\!\cdots\!32}{13387012429555}a^{10}+\frac{20\!\cdots\!73}{13387012429555}a^{9}+\frac{41\!\cdots\!35}{2677402485911}a^{8}+\frac{14\!\cdots\!01}{2677402485911}a^{7}-\frac{39\!\cdots\!46}{13387012429555}a^{6}-\frac{88908195713943}{199806155665}a^{5}-\frac{28\!\cdots\!24}{13387012429555}a^{4}-\frac{419135941140363}{13387012429555}a^{3}+\frac{395537519555392}{13387012429555}a^{2}+\frac{249467793872279}{13387012429555}a+\frac{52302911946176}{13387012429555}$, $\frac{95759169505053}{13387012429555}a^{29}-\frac{17593080557021}{2677402485911}a^{28}+\frac{636340653827757}{13387012429555}a^{27}-\frac{7419409133422}{150415869995}a^{26}+\frac{18\!\cdots\!01}{13387012429555}a^{25}-\frac{16\!\cdots\!93}{13387012429555}a^{24}+\frac{375454538585191}{2677402485911}a^{23}-\frac{112412788647457}{13387012429555}a^{22}-\frac{40\!\cdots\!92}{13387012429555}a^{21}+\frac{73\!\cdots\!72}{13387012429555}a^{20}-\frac{14\!\cdots\!08}{13387012429555}a^{19}+\frac{14\!\cdots\!12}{13387012429555}a^{18}-\frac{96\!\cdots\!67}{13387012429555}a^{17}+\frac{42\!\cdots\!29}{13387012429555}a^{16}+\frac{31\!\cdots\!92}{13387012429555}a^{15}-\frac{22\!\cdots\!06}{13387012429555}a^{14}+\frac{83\!\cdots\!21}{13387012429555}a^{13}-\frac{16\!\cdots\!72}{13387012429555}a^{12}+\frac{96\!\cdots\!79}{13387012429555}a^{11}+\frac{35\!\cdots\!23}{13387012429555}a^{10}+\frac{57\!\cdots\!48}{13387012429555}a^{9}+\frac{37\!\cdots\!86}{13387012429555}a^{8}+\frac{36146482168589}{150415869995}a^{7}-\frac{12\!\cdots\!33}{13387012429555}a^{6}-\frac{11\!\cdots\!99}{13387012429555}a^{5}-\frac{39\!\cdots\!89}{13387012429555}a^{4}+\frac{65143108357098}{2677402485911}a^{3}+\frac{841698333178224}{13387012429555}a^{2}+\frac{400557995544531}{13387012429555}a+\frac{20164710183359}{13387012429555}$, $\frac{37024752893528}{13387012429555}a^{29}-\frac{14278148072644}{13387012429555}a^{28}+\frac{224579673235431}{13387012429555}a^{27}-\frac{122842578711847}{13387012429555}a^{26}+\frac{547762141182457}{13387012429555}a^{25}-\frac{241005889364278}{13387012429555}a^{24}+\frac{348414508435953}{13387012429555}a^{23}+\frac{357319150820083}{13387012429555}a^{22}-\frac{15\!\cdots\!79}{13387012429555}a^{21}+\frac{19\!\cdots\!84}{13387012429555}a^{20}-\frac{40\!\cdots\!01}{13387012429555}a^{19}+\frac{25\!\cdots\!94}{13387012429555}a^{18}-\frac{317799628009506}{13387012429555}a^{17}-\frac{96289217828993}{2677402485911}a^{16}+\frac{26\!\cdots\!15}{2677402485911}a^{15}-\frac{411024674237814}{2677402485911}a^{14}+\frac{26\!\cdots\!67}{13387012429555}a^{13}+\frac{22\!\cdots\!03}{2677402485911}a^{12}+\frac{31\!\cdots\!33}{13387012429555}a^{11}+\frac{64\!\cdots\!29}{2677402485911}a^{10}+\frac{27\!\cdots\!02}{13387012429555}a^{9}+\frac{23\!\cdots\!04}{13387012429555}a^{8}+\frac{73\!\cdots\!44}{13387012429555}a^{7}-\frac{12\!\cdots\!59}{2677402485911}a^{6}-\frac{15\!\cdots\!16}{2677402485911}a^{5}-\frac{33\!\cdots\!76}{13387012429555}a^{4}-\frac{247194113126379}{13387012429555}a^{3}+\frac{564409621277436}{13387012429555}a^{2}+\frac{63821167308660}{2677402485911}a+\frac{57750682630384}{13387012429555}$, $\frac{12980134134192}{2677402485911}a^{29}-\frac{4862789174310}{2677402485911}a^{28}+\frac{395732483611212}{13387012429555}a^{27}-\frac{210514480464804}{13387012429555}a^{26}+\frac{969408995591541}{13387012429555}a^{25}-\frac{408651373022329}{13387012429555}a^{24}+\frac{622154027148774}{13387012429555}a^{23}+\frac{656574557989727}{13387012429555}a^{22}-\frac{28\!\cdots\!27}{13387012429555}a^{21}+\frac{34\!\cdots\!53}{13387012429555}a^{20}-\frac{72\!\cdots\!49}{13387012429555}a^{19}+\frac{44\!\cdots\!53}{13387012429555}a^{18}-\frac{592733022942399}{13387012429555}a^{17}-\frac{10\!\cdots\!74}{13387012429555}a^{16}+\frac{23\!\cdots\!91}{13387012429555}a^{15}-\frac{755890068179878}{2677402485911}a^{14}+\frac{47\!\cdots\!32}{13387012429555}a^{13}+\frac{40\!\cdots\!60}{2677402485911}a^{12}+\frac{56\!\cdots\!44}{13387012429555}a^{11}+\frac{59\!\cdots\!69}{13387012429555}a^{10}+\frac{98\!\cdots\!82}{2677402485911}a^{9}+\frac{45\!\cdots\!28}{13387012429555}a^{8}+\frac{14\!\cdots\!43}{13387012429555}a^{7}-\frac{88\!\cdots\!07}{13387012429555}a^{6}-\frac{12\!\cdots\!02}{13387012429555}a^{5}-\frac{13\!\cdots\!73}{2677402485911}a^{4}-\frac{859337515699214}{13387012429555}a^{3}+\frac{826355987558006}{13387012429555}a^{2}+\frac{541040162736538}{13387012429555}a+\frac{25352819131906}{2677402485911}$, $\frac{29492351076456}{13387012429555}a^{29}-\frac{13342198239206}{13387012429555}a^{28}+\frac{182784547773298}{13387012429555}a^{27}-\frac{112146806455206}{13387012429555}a^{26}+\frac{92381558452263}{2677402485911}a^{25}-\frac{237872005673042}{13387012429555}a^{24}+\frac{337661341141038}{13387012429555}a^{23}+\frac{232816761417372}{13387012429555}a^{22}-\frac{12\!\cdots\!33}{13387012429555}a^{21}+\frac{16\!\cdots\!86}{13387012429555}a^{20}-\frac{34\!\cdots\!53}{13387012429555}a^{19}+\frac{490267922225700}{2677402485911}a^{18}-\frac{774209428621736}{13387012429555}a^{17}-\frac{67037680467141}{13387012429555}a^{16}+\frac{10\!\cdots\!19}{13387012429555}a^{15}-\frac{24\!\cdots\!41}{13387012429555}a^{14}+\frac{22\!\cdots\!24}{13387012429555}a^{13}+\frac{68\!\cdots\!76}{13387012429555}a^{12}+\frac{26\!\cdots\!94}{13387012429555}a^{11}+\frac{24\!\cdots\!86}{13387012429555}a^{10}+\frac{65844946893910}{39961231133}a^{9}+\frac{18\!\cdots\!56}{13387012429555}a^{8}+\frac{55\!\cdots\!39}{13387012429555}a^{7}-\frac{43\!\cdots\!23}{13387012429555}a^{6}-\frac{60\!\cdots\!38}{13387012429555}a^{5}-\frac{623083553916764}{2677402485911}a^{4}-\frac{446930265670927}{13387012429555}a^{3}+\frac{417739633289371}{13387012429555}a^{2}+\frac{287138392178128}{13387012429555}a+\frac{76334083611404}{13387012429555}$
| 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: | \( 27376181.549941503 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 27376181.549941503 \cdot 1}{22\cdot\sqrt{45711723244122563996456104229882472771}}\cr\approx \mathstrut & 0.172836084881715 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^30 + 6*x^28 - x^27 + 14*x^26 - x^25 + 8*x^24 + 13*x^23 - 39*x^22 + 37*x^21 - 93*x^20 + 30*x^19 + 10*x^18 - 13*x^17 + 349*x^16 + 77*x^15 + 721*x^14 + 575*x^13 + 1026*x^12 + 1233*x^11 + 1137*x^10 + 993*x^9 + 501*x^8 - 41*x^7 - 253*x^6 - 186*x^5 - 58*x^4 + 6*x^3 + 14*x^2 + 6*x + 1)
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^30 + 6*x^28 - x^27 + 14*x^26 - x^25 + 8*x^24 + 13*x^23 - 39*x^22 + 37*x^21 - 93*x^20 + 30*x^19 + 10*x^18 - 13*x^17 + 349*x^16 + 77*x^15 + 721*x^14 + 575*x^13 + 1026*x^12 + 1233*x^11 + 1137*x^10 + 993*x^9 + 501*x^8 - 41*x^7 - 253*x^6 - 186*x^5 - 58*x^4 + 6*x^3 + 14*x^2 + 6*x + 1, 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^30 + 6*x^28 - x^27 + 14*x^26 - x^25 + 8*x^24 + 13*x^23 - 39*x^22 + 37*x^21 - 93*x^20 + 30*x^19 + 10*x^18 - 13*x^17 + 349*x^16 + 77*x^15 + 721*x^14 + 575*x^13 + 1026*x^12 + 1233*x^11 + 1137*x^10 + 993*x^9 + 501*x^8 - 41*x^7 - 253*x^6 - 186*x^5 - 58*x^4 + 6*x^3 + 14*x^2 + 6*x + 1);
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^30 + 6*x^28 - x^27 + 14*x^26 - x^25 + 8*x^24 + 13*x^23 - 39*x^22 + 37*x^21 - 93*x^20 + 30*x^19 + 10*x^18 - 13*x^17 + 349*x^16 + 77*x^15 + 721*x^14 + 575*x^13 + 1026*x^12 + 1233*x^11 + 1137*x^10 + 993*x^9 + 501*x^8 - 41*x^7 - 253*x^6 - 186*x^5 - 58*x^4 + 6*x^3 + 14*x^2 + 6*x + 1);
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
$S_3\times C_{15}$ (as 30T15):
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 90 |
| The 45 conjugacy class representatives for $S_3\times C_{15}$ |
| Character table for $S_3\times C_{15}$ |
Intermediate fields
| \(\Q(\sqrt{-11}) \), \(\Q(\zeta_{11})^+\), 6.0.107811.1, \(\Q(\zeta_{11})\) |
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 45 sibling: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $30$ | R | $15{,}\,{\href{/padicField/5.5.0.1}{5} }^{3}$ | $30$ | R | $30$ | ${\href{/padicField/17.10.0.1}{10} }^{3}$ | ${\href{/padicField/19.10.0.1}{10} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }^{10}$ | $30$ | $15{,}\,{\href{/padicField/31.5.0.1}{5} }^{3}$ | $15^{2}$ | $30$ | ${\href{/padicField/43.6.0.1}{6} }^{5}$ | $15{,}\,{\href{/padicField/47.5.0.1}{5} }^{3}$ | $15^{2}$ | $15{,}\,{\href{/padicField/59.5.0.1}{5} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.15.20.99 | $x^{15} + 30 x^{14} + 360 x^{13} + 2220 x^{12} + 7920 x^{11} + 20736 x^{10} + 54576 x^{9} + 127008 x^{8} + 295488 x^{7} + 682614 x^{6} + 1099656 x^{5} + 3298968 x^{4} + 2128842 x^{3} + 12773052 x^{2} + 9410175$ | $3$ | $5$ | $20$ | $C_{15}$ | $[2]^{5}$ |
| 3.15.0.1 | $x^{15} + 2 x^{8} + x^{5} + 2 x^{2} + x + 1$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | |
|
\(11\)
| Deg $30$ | $10$ | $3$ | $27$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.11.2t1.a.a | $1$ | $ 11 $ | \(\Q(\sqrt{-11}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.99.6t1.a.a | $1$ | $ 3^{2} \cdot 11 $ | 6.0.8732691.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.99.6t1.a.b | $1$ | $ 3^{2} \cdot 11 $ | 6.0.8732691.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| * | 1.11.10t1.a.a | $1$ | $ 11 $ | \(\Q(\zeta_{11})\) | $C_{10}$ (as 10T1) | $0$ | $-1$ |
| * | 1.11.5t1.a.a | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.11.10t1.a.b | $1$ | $ 11 $ | \(\Q(\zeta_{11})\) | $C_{10}$ (as 10T1) | $0$ | $-1$ |
| * | 1.11.10t1.a.c | $1$ | $ 11 $ | \(\Q(\zeta_{11})\) | $C_{10}$ (as 10T1) | $0$ | $-1$ |
| * | 1.11.5t1.a.b | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.11.5t1.a.c | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.11.5t1.a.d | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.11.10t1.a.d | $1$ | $ 11 $ | \(\Q(\zeta_{11})\) | $C_{10}$ (as 10T1) | $0$ | $-1$ |
| 1.99.30t1.a.a | $1$ | $ 3^{2} \cdot 11 $ | 30.0.159386923550435671074967363509984324121230045171.1 | $C_{30}$ (as 30T1) | $0$ | $-1$ | |
| 1.99.30t1.a.b | $1$ | $ 3^{2} \cdot 11 $ | 30.0.159386923550435671074967363509984324121230045171.1 | $C_{30}$ (as 30T1) | $0$ | $-1$ | |
| 1.99.15t1.a.a | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ | |
| 1.99.30t1.a.c | $1$ | $ 3^{2} \cdot 11 $ | 30.0.159386923550435671074967363509984324121230045171.1 | $C_{30}$ (as 30T1) | $0$ | $-1$ | |
| 1.99.30t1.a.d | $1$ | $ 3^{2} \cdot 11 $ | 30.0.159386923550435671074967363509984324121230045171.1 | $C_{30}$ (as 30T1) | $0$ | $-1$ | |
| 1.99.15t1.a.b | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ | |
| 1.99.30t1.a.e | $1$ | $ 3^{2} \cdot 11 $ | 30.0.159386923550435671074967363509984324121230045171.1 | $C_{30}$ (as 30T1) | $0$ | $-1$ | |
| 1.99.15t1.a.c | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ | |
| 1.99.30t1.a.f | $1$ | $ 3^{2} \cdot 11 $ | 30.0.159386923550435671074967363509984324121230045171.1 | $C_{30}$ (as 30T1) | $0$ | $-1$ | |
| 1.99.15t1.a.d | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ | |
| 1.99.30t1.a.g | $1$ | $ 3^{2} \cdot 11 $ | 30.0.159386923550435671074967363509984324121230045171.1 | $C_{30}$ (as 30T1) | $0$ | $-1$ | |
| 1.99.15t1.a.e | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ | |
| 1.99.30t1.a.h | $1$ | $ 3^{2} \cdot 11 $ | 30.0.159386923550435671074967363509984324121230045171.1 | $C_{30}$ (as 30T1) | $0$ | $-1$ | |
| 1.99.15t1.a.f | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ | |
| 1.99.15t1.a.g | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ | |
| 1.99.15t1.a.h | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ | |
| 2.891.3t2.b.a | $2$ | $ 3^{4} \cdot 11 $ | 3.1.891.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| * | 2.99.6t5.a.a | $2$ | $ 3^{2} \cdot 11 $ | 6.0.107811.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.99.6t5.a.b | $2$ | $ 3^{2} \cdot 11 $ | 6.0.107811.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| 2.9801.15t4.a.a | $2$ | $ 3^{4} \cdot 11^{2}$ | 15.5.120373254183671877620331.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ | |
| 2.9801.15t4.a.b | $2$ | $ 3^{4} \cdot 11^{2}$ | 15.5.120373254183671877620331.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ | |
| 2.9801.15t4.a.c | $2$ | $ 3^{4} \cdot 11^{2}$ | 15.5.120373254183671877620331.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ | |
| 2.9801.15t4.a.d | $2$ | $ 3^{4} \cdot 11^{2}$ | 15.5.120373254183671877620331.1 | $S_3 \times C_5$ (as 15T4) | $0$ | $0$ | |
| * | 2.1089.30t15.a.a | $2$ | $ 3^{2} \cdot 11^{2}$ | 30.0.45711723244122563996456104229882472771.1 | $S_3\times C_{15}$ (as 30T15) | $0$ | $0$ |
| * | 2.1089.30t15.a.b | $2$ | $ 3^{2} \cdot 11^{2}$ | 30.0.45711723244122563996456104229882472771.1 | $S_3\times C_{15}$ (as 30T15) | $0$ | $0$ |
| * | 2.1089.30t15.a.c | $2$ | $ 3^{2} \cdot 11^{2}$ | 30.0.45711723244122563996456104229882472771.1 | $S_3\times C_{15}$ (as 30T15) | $0$ | $0$ |
| * | 2.1089.30t15.a.d | $2$ | $ 3^{2} \cdot 11^{2}$ | 30.0.45711723244122563996456104229882472771.1 | $S_3\times C_{15}$ (as 30T15) | $0$ | $0$ |
| * | 2.1089.30t15.a.e | $2$ | $ 3^{2} \cdot 11^{2}$ | 30.0.45711723244122563996456104229882472771.1 | $S_3\times C_{15}$ (as 30T15) | $0$ | $0$ |
| * | 2.1089.30t15.a.f | $2$ | $ 3^{2} \cdot 11^{2}$ | 30.0.45711723244122563996456104229882472771.1 | $S_3\times C_{15}$ (as 30T15) | $0$ | $0$ |
| * | 2.1089.30t15.a.g | $2$ | $ 3^{2} \cdot 11^{2}$ | 30.0.45711723244122563996456104229882472771.1 | $S_3\times C_{15}$ (as 30T15) | $0$ | $0$ |
| * | 2.1089.30t15.a.h | $2$ | $ 3^{2} \cdot 11^{2}$ | 30.0.45711723244122563996456104229882472771.1 | $S_3\times C_{15}$ (as 30T15) | $0$ | $0$ |
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 *.