Properties

Label 30.0.266...291.1
Degree $30$
Signature $[0, 15]$
Discriminant $-2.667\times 10^{51}$
Root discriminant $51.78$
Ramified primes $11, 19$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_3\times D_5$ (as 30T4)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 5*x^29 + 23*x^28 - 62*x^27 + 198*x^26 - 367*x^25 + 397*x^24 + 1188*x^23 - 7770*x^22 + 27186*x^21 - 70571*x^20 + 106780*x^19 + 113195*x^18 - 1691169*x^17 + 8127778*x^16 - 29421234*x^15 + 90493414*x^14 - 242325966*x^13 + 570478488*x^12 - 1182396448*x^11 + 2168615679*x^10 - 3480425904*x^9 + 4808261059*x^8 - 5571332106*x^7 + 5293750647*x^6 - 4028704993*x^5 + 2404982206*x^4 - 1088744723*x^3 + 357808383*x^2 - 76833878*x + 8961073)
 
gp: K = bnfinit(x^30 - 5*x^29 + 23*x^28 - 62*x^27 + 198*x^26 - 367*x^25 + 397*x^24 + 1188*x^23 - 7770*x^22 + 27186*x^21 - 70571*x^20 + 106780*x^19 + 113195*x^18 - 1691169*x^17 + 8127778*x^16 - 29421234*x^15 + 90493414*x^14 - 242325966*x^13 + 570478488*x^12 - 1182396448*x^11 + 2168615679*x^10 - 3480425904*x^9 + 4808261059*x^8 - 5571332106*x^7 + 5293750647*x^6 - 4028704993*x^5 + 2404982206*x^4 - 1088744723*x^3 + 357808383*x^2 - 76833878*x + 8961073, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8961073, -76833878, 357808383, -1088744723, 2404982206, -4028704993, 5293750647, -5571332106, 4808261059, -3480425904, 2168615679, -1182396448, 570478488, -242325966, 90493414, -29421234, 8127778, -1691169, 113195, 106780, -70571, 27186, -7770, 1188, 397, -367, 198, -62, 23, -5, 1]);
 

\(x^{30} - 5 x^{29} + 23 x^{28} - 62 x^{27} + 198 x^{26} - 367 x^{25} + 397 x^{24} + 1188 x^{23} - 7770 x^{22} + 27186 x^{21} - 70571 x^{20} + 106780 x^{19} + 113195 x^{18} - 1691169 x^{17} + 8127778 x^{16} - 29421234 x^{15} + 90493414 x^{14} - 242325966 x^{13} + 570478488 x^{12} - 1182396448 x^{11} + 2168615679 x^{10} - 3480425904 x^{9} + 4808261059 x^{8} - 5571332106 x^{7} + 5293750647 x^{6} - 4028704993 x^{5} + 2404982206 x^{4} - 1088744723 x^{3} + 357808383 x^{2} - 76833878 x + 8961073\)  Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 15]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-2666804038012396184155132908419351509268871501674291\)\(\medspace = -\,11^{15}\cdot 19^{28}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $51.78$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $11, 19$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $30$
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{22} a^{18} - \frac{1}{22} a^{17} + \frac{5}{11} a^{16} + \frac{9}{22} a^{15} + \frac{3}{11} a^{14} + \frac{3}{11} a^{13} - \frac{5}{22} a^{12} + \frac{1}{11} a^{11} + \frac{2}{11} a^{10} - \frac{1}{22} a^{9} - \frac{2}{11} a^{8} - \frac{9}{22} a^{7} + \frac{5}{11} a^{6} - \frac{1}{11} a^{5} - \frac{5}{22} a^{4} - \frac{3}{22} a^{3} + \frac{2}{11} a^{2} - \frac{1}{2}$, $\frac{1}{22} a^{19} + \frac{9}{22} a^{17} - \frac{3}{22} a^{16} - \frac{7}{22} a^{15} - \frac{5}{11} a^{14} + \frac{1}{22} a^{13} - \frac{3}{22} a^{12} + \frac{3}{11} a^{11} + \frac{3}{22} a^{10} - \frac{5}{22} a^{9} + \frac{9}{22} a^{8} + \frac{1}{22} a^{7} + \frac{4}{11} a^{6} - \frac{7}{22} a^{5} - \frac{4}{11} a^{4} + \frac{1}{22} a^{3} + \frac{2}{11} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{22} a^{20} + \frac{3}{11} a^{17} - \frac{9}{22} a^{16} - \frac{3}{22} a^{15} - \frac{9}{22} a^{14} + \frac{9}{22} a^{13} + \frac{7}{22} a^{12} + \frac{7}{22} a^{11} + \frac{3}{22} a^{10} - \frac{2}{11} a^{9} - \frac{7}{22} a^{8} + \frac{1}{22} a^{7} - \frac{9}{22} a^{6} + \frac{5}{11} a^{5} + \frac{1}{11} a^{4} + \frac{9}{22} a^{3} - \frac{3}{22} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{22} a^{21} - \frac{3}{22} a^{17} + \frac{3}{22} a^{16} + \frac{3}{22} a^{15} - \frac{5}{22} a^{14} - \frac{7}{22} a^{13} - \frac{7}{22} a^{12} - \frac{9}{22} a^{11} - \frac{3}{11} a^{10} - \frac{1}{22} a^{9} + \frac{3}{22} a^{8} + \frac{1}{22} a^{7} - \frac{3}{11} a^{6} - \frac{4}{11} a^{5} - \frac{5}{22} a^{4} - \frac{7}{22} a^{3} + \frac{9}{22} a^{2} - \frac{1}{2} a$, $\frac{1}{154} a^{22} - \frac{1}{77} a^{21} + \frac{1}{154} a^{20} + \frac{3}{154} a^{19} - \frac{27}{154} a^{17} + \frac{9}{154} a^{16} - \frac{15}{77} a^{15} - \frac{9}{77} a^{14} - \frac{73}{154} a^{13} + \frac{16}{77} a^{12} + \frac{65}{154} a^{11} - \frac{31}{154} a^{10} + \frac{7}{22} a^{9} - \frac{19}{154} a^{8} + \frac{5}{22} a^{7} - \frac{39}{154} a^{6} - \frac{2}{11} a^{5} + \frac{5}{77} a^{4} + \frac{5}{11} a^{3} - \frac{37}{77} a^{2} + \frac{1}{7} a + \frac{1}{14}$, $\frac{1}{154} a^{23} - \frac{3}{154} a^{21} - \frac{1}{77} a^{20} - \frac{1}{154} a^{19} + \frac{1}{154} a^{18} - \frac{12}{77} a^{17} + \frac{2}{7} a^{16} - \frac{32}{77} a^{15} + \frac{19}{77} a^{14} - \frac{8}{77} a^{13} - \frac{39}{154} a^{12} + \frac{32}{77} a^{11} + \frac{57}{154} a^{10} - \frac{20}{77} a^{9} + \frac{25}{154} a^{8} + \frac{73}{154} a^{7} + \frac{27}{154} a^{6} + \frac{31}{154} a^{5} - \frac{4}{77} a^{4} + \frac{3}{7} a^{3} - \frac{3}{22} a^{2} + \frac{5}{14} a + \frac{1}{7}$, $\frac{1}{308} a^{24} - \frac{1}{308} a^{22} + \frac{1}{308} a^{21} - \frac{3}{154} a^{20} - \frac{3}{308} a^{18} + \frac{151}{308} a^{17} - \frac{39}{308} a^{16} + \frac{26}{77} a^{15} + \frac{9}{154} a^{14} - \frac{6}{77} a^{13} + \frac{25}{77} a^{12} - \frac{79}{308} a^{11} + \frac{13}{77} a^{10} + \frac{1}{77} a^{9} + \frac{4}{11} a^{8} - \frac{9}{28} a^{7} - \frac{13}{154} a^{6} + \frac{125}{308} a^{5} - \frac{3}{77} a^{4} + \frac{13}{44} a^{3} - \frac{107}{308} a^{2} + \frac{13}{28} a - \frac{5}{28}$, $\frac{1}{31724} a^{25} + \frac{1}{1133} a^{24} - \frac{1}{2884} a^{23} - \frac{17}{31724} a^{22} - \frac{229}{15862} a^{21} + \frac{130}{7931} a^{20} - \frac{439}{31724} a^{19} - \frac{321}{31724} a^{18} - \frac{1413}{31724} a^{17} + \frac{157}{15862} a^{16} + \frac{1633}{7931} a^{15} - \frac{3730}{7931} a^{14} + \frac{7815}{15862} a^{13} + \frac{14169}{31724} a^{12} + \frac{1781}{15862} a^{11} - \frac{6885}{15862} a^{10} + \frac{1481}{15862} a^{9} - \frac{893}{4532} a^{8} - \frac{487}{1133} a^{7} - \frac{7213}{31724} a^{6} + \frac{325}{2266} a^{5} - \frac{1073}{31724} a^{4} - \frac{9377}{31724} a^{3} + \frac{12689}{31724} a^{2} + \frac{163}{412} a - \frac{705}{1442}$, $\frac{1}{348964} a^{26} - \frac{1}{87241} a^{25} + \frac{5}{87241} a^{24} + \frac{953}{348964} a^{23} - \frac{1047}{348964} a^{22} + \frac{241}{348964} a^{21} + \frac{1873}{348964} a^{20} + \frac{349}{31724} a^{19} - \frac{1931}{87241} a^{18} - \frac{5455}{348964} a^{17} + \frac{2973}{348964} a^{16} - \frac{2215}{7931} a^{15} + \frac{14551}{87241} a^{14} - \frac{41649}{348964} a^{13} - \frac{17650}{87241} a^{12} - \frac{5611}{49852} a^{11} + \frac{85017}{174482} a^{10} + \frac{121651}{348964} a^{9} + \frac{16111}{87241} a^{8} - \frac{6773}{24926} a^{7} + \frac{5750}{12463} a^{6} - \frac{4352}{87241} a^{5} - \frac{76805}{348964} a^{4} - \frac{235}{174482} a^{3} - \frac{4434}{12463} a^{2} - \frac{5889}{31724} a - \frac{1745}{31724}$, $\frac{1}{348964} a^{27} + \frac{1}{87241} a^{25} - \frac{25}{87241} a^{24} + \frac{499}{348964} a^{23} - \frac{137}{87241} a^{22} + \frac{1985}{174482} a^{21} - \frac{971}{49852} a^{20} + \frac{417}{174482} a^{19} - \frac{1747}{174482} a^{18} + \frac{9741}{24926} a^{17} - \frac{168277}{348964} a^{16} + \frac{59649}{174482} a^{15} - \frac{19571}{348964} a^{14} - \frac{25309}{87241} a^{13} + \frac{90735}{348964} a^{12} - \frac{21299}{49852} a^{11} - \frac{170327}{348964} a^{10} - \frac{31055}{87241} a^{9} - \frac{2528}{7931} a^{8} + \frac{172597}{348964} a^{7} - \frac{32269}{174482} a^{6} + \frac{3259}{174482} a^{5} + \frac{243}{174482} a^{4} + \frac{81307}{348964} a^{3} - \frac{26335}{174482} a^{2} + \frac{379}{15862} a + \frac{12281}{31724}$, $\frac{1}{766634475068} a^{28} - \frac{594263}{766634475068} a^{27} - \frac{117503}{383317237534} a^{26} + \frac{639241}{54759605362} a^{25} - \frac{300137163}{766634475068} a^{24} - \frac{778320623}{766634475068} a^{23} - \frac{77962936}{27379802681} a^{22} - \frac{8842599133}{766634475068} a^{21} - \frac{1944296693}{109519210724} a^{20} + \frac{20224901}{27379802681} a^{19} + \frac{2547196952}{191658618767} a^{18} - \frac{252355639149}{766634475068} a^{17} - \frac{44556317137}{109519210724} a^{16} + \frac{116472942301}{766634475068} a^{15} + \frac{365229356535}{766634475068} a^{14} - \frac{34701417409}{69694043188} a^{13} + \frac{1911008370}{27379802681} a^{12} - \frac{47283985773}{383317237534} a^{11} + \frac{381510091899}{766634475068} a^{10} + \frac{24613050991}{383317237534} a^{9} + \frac{307188795357}{766634475068} a^{8} - \frac{177381350007}{766634475068} a^{7} + \frac{183197880741}{383317237534} a^{6} - \frac{50593427}{4978145942} a^{5} + \frac{345457006239}{766634475068} a^{4} - \frac{310543049}{766634475068} a^{3} + \frac{76660817225}{191658618767} a^{2} - \frac{2310629469}{69694043188} a - \frac{19626810321}{69694043188}$, $\frac{1}{5797003220947299463065148326838019680648227080923752497966298336603444741072711080656867894276} a^{29} + \frac{1197182685647357096796919860602271845873000368631043802076669202162528108418877107}{2898501610473649731532574163419009840324113540461876248983149168301722370536355540328433947138} a^{28} - \frac{5707194792910038244813167955721915421711432985780342994605758109847639985398972058637609}{5797003220947299463065148326838019680648227080923752497966298336603444741072711080656867894276} a^{27} - \frac{491540904278734845037853534851952474932422577342897450270050202416540108171033851012391}{527000292813390860278649847894365425513475189174886590724208939691222249188428280059715263116} a^{26} - \frac{1458599391230398747127635192879615761496633705252640894980738773403396446520488985303720}{131750073203347715069662461973591356378368797293721647681052234922805562297107070014928815779} a^{25} + \frac{1246518167052642606069192692394140315823209052248247287223643697275737785923154696443030909}{1449250805236824865766287081709504920162056770230938124491574584150861185268177770164216973569} a^{24} - \frac{719059752788806978573010025648389779778579236795475795291834315962225217721881985286071223}{527000292813390860278649847894365425513475189174886590724208939691222249188428280059715263116} a^{23} + \frac{733857017607994455693489885551228059370946552097653770140065095767482784980669500981449919}{527000292813390860278649847894365425513475189174886590724208939691222249188428280059715263116} a^{22} - \frac{3514277948918833082341724455517884124601047155749223092095881798525113133646806272512787619}{5797003220947299463065148326838019680648227080923752497966298336603444741072711080656867894276} a^{21} + \frac{8432932802020731749173775300007172695711128302428565194870588957014239706165750447676155697}{414071658639092818790367737631287120046301934351696606997592738328817481505193648618347706734} a^{20} - \frac{53242078429660147866825538161271360926324612957421390331834952784506512150197776111449097967}{2898501610473649731532574163419009840324113540461876248983149168301722370536355540328433947138} a^{19} + \frac{674154972472616536742081138111433071578547421618115197654664436124824596784491959411236185}{207035829319546409395183868815643560023150967175848303498796369164408740752596824309173853367} a^{18} - \frac{559455728622544072523908621797074675964611132802434335467049001383862385539349859607055081621}{1449250805236824865766287081709504920162056770230938124491574584150861185268177770164216973569} a^{17} - \frac{546502978526413589790274029095748282179346134239309863862281278413745305749800849317748304409}{5797003220947299463065148326838019680648227080923752497966298336603444741072711080656867894276} a^{16} - \frac{1288967143525290851170451590431120089003208592463787077274567850775978781594776465333050427617}{2898501610473649731532574163419009840324113540461876248983149168301722370536355540328433947138} a^{15} + \frac{12893774234807423200346715429615859922204206003160883634214502487464374159327236578603098733}{29576547045649487056454838402234794289021566739406900499828052737772677250370974901310550481} a^{14} + \frac{308120996289528874413706700276567177732012429253447716875686516898319815644342534607738617643}{2898501610473649731532574163419009840324113540461876248983149168301722370536355540328433947138} a^{13} + \frac{2230261136384740995605591756496479213754597747988647514786601675850909681135452106862089927907}{5797003220947299463065148326838019680648227080923752497966298336603444741072711080656867894276} a^{12} + \frac{234353935542594724780424287574785063123586227285650508505548543871423060500326140805816473621}{1449250805236824865766287081709504920162056770230938124491574584150861185268177770164216973569} a^{11} - \frac{837449044814516694692748240286886142191018351469652351764919082737540866369795091282894232327}{5797003220947299463065148326838019680648227080923752497966298336603444741072711080656867894276} a^{10} + \frac{36075399285062070822208130387853652577246500590061126726702078765351892785295243721024955225}{414071658639092818790367737631287120046301934351696606997592738328817481505193648618347706734} a^{9} + \frac{441228327940303565054759739214889844843508161979847049222982249779196399463783035661730245935}{5797003220947299463065148326838019680648227080923752497966298336603444741072711080656867894276} a^{8} - \frac{2661783026635735074308544746226778154608111964281181133787628051476853295383370903301999664595}{5797003220947299463065148326838019680648227080923752497966298336603444741072711080656867894276} a^{7} - \frac{1034663388862094673907224362962141609655575903770868678545541328072039672283290180273016622685}{5797003220947299463065148326838019680648227080923752497966298336603444741072711080656867894276} a^{6} + \frac{1120633958307525891873394177653391693971452743152845696375997729887183669283304167726266853507}{5797003220947299463065148326838019680648227080923752497966298336603444741072711080656867894276} a^{5} + \frac{288103137968576389719293314215663274044451913715503052163292793966230068558857090429223181500}{1449250805236824865766287081709504920162056770230938124491574584150861185268177770164216973569} a^{4} + \frac{258577877825413308660690704468194739585976426602621642060194344008665093659614688464900623085}{1449250805236824865766287081709504920162056770230938124491574584150861185268177770164216973569} a^{3} + \frac{692579099342807454921311235682568305551731059973209186674031156029618925588192040809126404691}{2898501610473649731532574163419009840324113540461876248983149168301722370536355540328433947138} a^{2} - \frac{5908366636029139261449233742401236584260989498530109157304459177927855783713440567897229257}{131750073203347715069662461973591356378368797293721647681052234922805562297107070014928815779} a + \frac{6045802421109204789202350401045875409019660409569360122580307500550528747174759845181906422}{131750073203347715069662461973591356378368797293721647681052234922805562297107070014928815779}$  Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 22790934759037.23 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{15}\cdot 22790934759037.23 \cdot 4}{2\sqrt{2666804038012396184155132908419351509268871501674291}}\approx 0.828885623106817$ (assuming GRH)

Galois group

$C_3\times D_5$ (as 30T4):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 30
The 12 conjugacy class representatives for $C_3\times D_5$
Character table for $C_3\times D_5$

Intermediate fields

\(\Q(\sqrt{-11}) \), 3.3.361.1, 5.1.15768841.1 x5, 6.0.173457251.1, 10.0.2735219811316091.1, 15.3.1415489083272211976282881.1 x5

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

Sibling fields

Degree 15 sibling: 15.3.1415489083272211976282881.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{5}$ $15^{2}$ $15^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{15}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{5}$ R $15^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{6}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{5}$ $15^{2}$ $15^{2}$ $15^{2}$

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

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
19Data not computed

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) $1$ $-1$
* 1.19.3t1.a.b$1$ $ 19 $ 3.3.361.1 $C_3$ (as 3T1) $0$ $1$
* 1.19.3t1.a.a$1$ $ 19 $ 3.3.361.1 $C_3$ (as 3T1) $0$ $1$
* 1.209.6t1.b.b$1$ $ 11 \cdot 19 $ 6.0.173457251.1 $C_6$ (as 6T1) $0$ $-1$
* 1.209.6t1.b.a$1$ $ 11 \cdot 19 $ 6.0.173457251.1 $C_6$ (as 6T1) $0$ $-1$
*2 2.3971.5t2.a.b$2$ $ 11 \cdot 19^{2}$ 5.1.15768841.1 $D_{5}$ (as 5T2) $1$ $0$
*2 2.3971.5t2.a.a$2$ $ 11 \cdot 19^{2}$ 5.1.15768841.1 $D_{5}$ (as 5T2) $1$ $0$
*2 2.3971.15t3.b.c$2$ $ 11 \cdot 19^{2}$ 30.0.2666804038012396184155132908419351509268871501674291.1 $C_3\times D_5$ (as 30T4) $0$ $0$
*2 2.3971.15t3.b.d$2$ $ 11 \cdot 19^{2}$ 30.0.2666804038012396184155132908419351509268871501674291.1 $C_3\times D_5$ (as 30T4) $0$ $0$
*2 2.3971.15t3.b.b$2$ $ 11 \cdot 19^{2}$ 30.0.2666804038012396184155132908419351509268871501674291.1 $C_3\times D_5$ (as 30T4) $0$ $0$
*2 2.3971.15t3.b.a$2$ $ 11 \cdot 19^{2}$ 30.0.2666804038012396184155132908419351509268871501674291.1 $C_3\times D_5$ (as 30T4) $0$ $0$

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 *.