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Properties

Label 30.0.885...327.1
Degree $30$
Signature $[0, 15]$
Discriminant $-8.851\times 10^{51}$
Root discriminant $53.90$
Ramified primes $3, 7, 53$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $D_{30}$ (as 30T14)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 4*x^29 - 32*x^28 + 92*x^27 + 588*x^26 - 752*x^25 - 6717*x^24 - 805*x^23 + 44049*x^22 + 54263*x^21 - 93757*x^20 - 350454*x^19 - 591742*x^18 + 307201*x^17 + 3600412*x^16 + 5669890*x^15 - 632841*x^14 - 17605194*x^13 - 34361042*x^12 - 23087625*x^11 + 45815379*x^10 + 119941341*x^9 + 85868062*x^8 - 30516460*x^7 - 62011703*x^6 + 46150369*x^5 + 155790202*x^4 + 147686157*x^3 + 67768188*x^2 + 10788948*x + 751689)
 
gp: K = bnfinit(x^30 - 4*x^29 - 32*x^28 + 92*x^27 + 588*x^26 - 752*x^25 - 6717*x^24 - 805*x^23 + 44049*x^22 + 54263*x^21 - 93757*x^20 - 350454*x^19 - 591742*x^18 + 307201*x^17 + 3600412*x^16 + 5669890*x^15 - 632841*x^14 - 17605194*x^13 - 34361042*x^12 - 23087625*x^11 + 45815379*x^10 + 119941341*x^9 + 85868062*x^8 - 30516460*x^7 - 62011703*x^6 + 46150369*x^5 + 155790202*x^4 + 147686157*x^3 + 67768188*x^2 + 10788948*x + 751689, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![751689, 10788948, 67768188, 147686157, 155790202, 46150369, -62011703, -30516460, 85868062, 119941341, 45815379, -23087625, -34361042, -17605194, -632841, 5669890, 3600412, 307201, -591742, -350454, -93757, 54263, 44049, -805, -6717, -752, 588, 92, -32, -4, 1]);
 

\( x^{30} - 4 x^{29} - 32 x^{28} + 92 x^{27} + 588 x^{26} - 752 x^{25} - 6717 x^{24} - 805 x^{23} + 44049 x^{22} + 54263 x^{21} - 93757 x^{20} - 350454 x^{19} - 591742 x^{18} + 307201 x^{17} + 3600412 x^{16} + 5669890 x^{15} - 632841 x^{14} - 17605194 x^{13} - 34361042 x^{12} - 23087625 x^{11} + 45815379 x^{10} + 119941341 x^{9} + 85868062 x^{8} - 30516460 x^{7} - 62011703 x^{6} + 46150369 x^{5} + 155790202 x^{4} + 147686157 x^{3} + 67768188 x^{2} + 10788948 x + 751689 \)

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:  \(-8851375005191383462430321349782942769050796640392327\)\(\medspace = -\,3^{14}\cdot 7^{25}\cdot 53^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $53.90$
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:  $3, 7, 53$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{15} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{10} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{19} + \frac{1}{3} a^{11} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{20} + \frac{1}{3} a^{12} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{21} + \frac{1}{3} a^{13} + \frac{1}{3} a^{5}$, $\frac{1}{51} a^{22} + \frac{4}{51} a^{21} + \frac{2}{17} a^{20} + \frac{1}{17} a^{19} + \frac{1}{17} a^{18} - \frac{4}{51} a^{17} - \frac{2}{17} a^{16} - \frac{11}{51} a^{15} - \frac{4}{17} a^{14} + \frac{4}{51} a^{13} - \frac{5}{51} a^{12} - \frac{19}{51} a^{11} - \frac{4}{17} a^{10} + \frac{2}{17} a^{9} + \frac{2}{51} a^{8} - \frac{20}{51} a^{7} + \frac{1}{51} a^{6} + \frac{8}{17} a^{5} - \frac{5}{51} a^{4} + \frac{8}{17} a^{3} - \frac{22}{51} a^{2} - \frac{2}{17} a$, $\frac{1}{8109} a^{23} + \frac{23}{8109} a^{22} - \frac{139}{8109} a^{21} + \frac{47}{901} a^{20} + \frac{485}{8109} a^{19} - \frac{355}{8109} a^{18} - \frac{407}{2703} a^{17} - \frac{992}{8109} a^{16} - \frac{26}{53} a^{15} + \frac{3839}{8109} a^{14} - \frac{968}{2703} a^{13} - \frac{23}{153} a^{12} - \frac{1648}{8109} a^{11} + \frac{691}{2703} a^{10} - \frac{2893}{8109} a^{9} + \frac{103}{8109} a^{8} - \frac{2198}{8109} a^{7} - \frac{2456}{8109} a^{6} + \frac{700}{2703} a^{5} + \frac{283}{901} a^{4} - \frac{722}{8109} a^{3} + \frac{427}{901} a^{2} + \frac{64}{2703} a - \frac{21}{53}$, $\frac{1}{1694781} a^{24} + \frac{5}{1694781} a^{23} - \frac{10570}{1694781} a^{22} + \frac{92135}{564927} a^{21} + \frac{105761}{1694781} a^{20} - \frac{1633}{89199} a^{19} + \frac{3751}{51357} a^{18} - \frac{68690}{1694781} a^{17} - \frac{17909}{188309} a^{16} + \frac{177521}{1694781} a^{15} + \frac{7358}{51357} a^{14} + \frac{332642}{1694781} a^{13} - \frac{462112}{1694781} a^{12} + \frac{15821}{51357} a^{11} - \frac{136243}{1694781} a^{10} + \frac{632686}{1694781} a^{9} + \frac{657070}{1694781} a^{8} - \frac{613997}{1694781} a^{7} - \frac{114944}{564927} a^{6} - \frac{122521}{564927} a^{5} - \frac{374903}{1694781} a^{4} + \frac{21110}{188309} a^{3} + \frac{83801}{564927} a^{2} + \frac{61799}{188309} a - \frac{2378}{11077}$, $\frac{1}{1694781} a^{25} + \frac{64}{1694781} a^{23} + \frac{9485}{1694781} a^{22} + \frac{6070}{154071} a^{21} - \frac{5564}{1694781} a^{20} - \frac{200737}{1694781} a^{19} + \frac{47866}{1694781} a^{18} + \frac{160951}{1694781} a^{17} + \frac{13457}{1694781} a^{16} + \frac{453086}{1694781} a^{15} - \frac{71344}{1694781} a^{14} - \frac{313292}{1694781} a^{13} + \frac{30155}{89199} a^{12} + \frac{589559}{1694781} a^{11} + \frac{82667}{564927} a^{10} - \frac{577081}{1694781} a^{9} - \frac{541762}{1694781} a^{8} + \frac{198970}{1694781} a^{7} + \frac{18176}{51357} a^{6} + \frac{10898}{154071} a^{5} + \frac{401701}{1694781} a^{4} + \frac{520}{11077} a^{3} - \frac{137677}{564927} a^{2} + \frac{66280}{188309} a - \frac{1486}{11077}$, $\frac{1}{1694781} a^{26} - \frac{31}{1694781} a^{23} + \frac{46}{1694781} a^{22} + \frac{98567}{1694781} a^{21} + \frac{24550}{564927} a^{20} - \frac{1315}{33231} a^{19} + \frac{122528}{1694781} a^{18} + \frac{251980}{1694781} a^{17} - \frac{7339}{154071} a^{16} + \frac{54622}{188309} a^{15} + \frac{417680}{1694781} a^{14} - \frac{219053}{564927} a^{13} + \frac{533752}{1694781} a^{12} + \frac{565186}{1694781} a^{11} + \frac{181286}{564927} a^{10} + \frac{457850}{1694781} a^{9} + \frac{200216}{1694781} a^{8} + \frac{126469}{1694781} a^{7} - \frac{7577}{33231} a^{6} + \frac{327334}{1694781} a^{5} - \frac{644869}{1694781} a^{4} - \frac{50992}{1694781} a^{3} - \frac{80128}{188309} a^{2} - \frac{167806}{564927} a + \frac{1921}{11077}$, $\frac{1}{5084343} a^{27} - \frac{1}{5084343} a^{26} + \frac{1}{5084343} a^{24} - \frac{181}{5084343} a^{23} + \frac{16582}{1694781} a^{22} - \frac{94225}{5084343} a^{21} - \frac{787973}{5084343} a^{20} - \frac{108764}{5084343} a^{19} + \frac{17264}{1694781} a^{18} + \frac{172835}{5084343} a^{17} + \frac{14642}{299079} a^{16} + \frac{385018}{1694781} a^{15} - \frac{11366}{299079} a^{14} + \frac{7204}{299079} a^{13} - \frac{488974}{5084343} a^{12} - \frac{32741}{89199} a^{11} + \frac{26019}{188309} a^{10} - \frac{56647}{1694781} a^{9} - \frac{363161}{1694781} a^{8} - \frac{15118}{51357} a^{7} + \frac{34087}{99693} a^{6} - \frac{1914707}{5084343} a^{5} - \frac{969739}{5084343} a^{4} + \frac{530224}{1694781} a^{3} + \frac{402124}{1694781} a^{2} - \frac{8411}{188309} a - \frac{2777}{33231}$, $\frac{1}{2506581099} a^{28} - \frac{157}{2506581099} a^{27} + \frac{131}{835527033} a^{26} - \frac{707}{2506581099} a^{25} + \frac{401}{2506581099} a^{24} - \frac{1933}{43975107} a^{23} - \frac{7616461}{2506581099} a^{22} - \frac{247049072}{2506581099} a^{21} + \frac{250713010}{2506581099} a^{20} + \frac{31436755}{278509011} a^{19} + \frac{315730934}{2506581099} a^{18} + \frac{431920}{2506581099} a^{17} - \frac{651784}{92836337} a^{16} + \frac{1044972518}{2506581099} a^{15} - \frac{941542927}{2506581099} a^{14} + \frac{718809554}{2506581099} a^{13} - \frac{236056894}{835527033} a^{12} + \frac{2417695}{5254887} a^{11} + \frac{204527228}{835527033} a^{10} + \frac{18497759}{835527033} a^{9} - \frac{121706132}{835527033} a^{8} + \frac{10692005}{49148649} a^{7} + \frac{317576617}{2506581099} a^{6} + \frac{1143896495}{2506581099} a^{5} - \frac{241321297}{835527033} a^{4} - \frac{5258386}{278509011} a^{3} - \frac{32497403}{278509011} a^{2} - \frac{103801}{862257} a - \frac{25055}{321233}$, $\frac{1}{100522270381252986846117061108478613973472491442074696302603150542356542787146918867} a^{29} + \frac{28988331187906842321876952695097548878767045664252057021502882521674171}{33507423460417662282039020369492871324490830480691565434201050180785514262382306289} a^{28} - \frac{190870950223088284207922787060053596813592704953290656030249766701093152923}{3046129405492514752912638215408442847680984589153778675836459107344137660216573299} a^{27} + \frac{149698896550460803485854533951578435445973549148622432356853589456623624457}{3723047051157518031337668929943652369387870053410173937133450020087279362486922921} a^{26} + \frac{2814342695324367637287930689761971378145451345928408577518931461996791930752}{11169141153472554094013006789830957108163610160230521811400350060261838087460768763} a^{25} - \frac{8642103325008859058758605429244064405156898964288126206911113175624890631239}{33507423460417662282039020369492871324490830480691565434201050180785514262382306289} a^{24} + \frac{3054460247404546985922120050621906026145421710791949537804881081754517191947804}{100522270381252986846117061108478613973472491442074696302603150542356542787146918867} a^{23} - \frac{114173270238134663480421841644122038349112326741092112414746377154080137773322185}{33507423460417662282039020369492871324490830480691565434201050180785514262382306289} a^{22} + \frac{123068449036757822801907104038002541258796896897559611685029002149604479726956199}{1896646610967037487662586058650539886291933800793862194388738689478425335606545639} a^{21} - \frac{12654392952077889191110741320800830727555766874913165701363706837589073108753593406}{100522270381252986846117061108478613973472491442074696302603150542356542787146918867} a^{20} + \frac{5143071052547948398716111808235707010882466294759665774548935805979116970137832503}{33507423460417662282039020369492871324490830480691565434201050180785514262382306289} a^{19} - \frac{216253848153983554592383735151191456932706372370421909917410503709269049290423059}{3723047051157518031337668929943652369387870053410173937133450020087279362486922921} a^{18} - \frac{460737614373010527508592880907409421221100684274288023602800257179425358172642654}{3723047051157518031337668929943652369387870053410173937133450020087279362486922921} a^{17} - \frac{126214180502846720396496311428878616743650781721434052801471633631717579152797815}{1155428395186815940759966219637685218085890706230743635662105178647776353875251941} a^{16} + \frac{526518998602693377761712570126713664326433689026436451304585770548798015365424210}{3466285185560447822279898658913055654257672118692230906986315535943329061625755823} a^{15} - \frac{358844333786848059758011609617297575181676200433817509904324948851701079127286865}{11169141153472554094013006789830957108163610160230521811400350060261838087460768763} a^{14} + \frac{6594132777637477890811421473955076481786415901522506334730777541633201713625338063}{100522270381252986846117061108478613973472491442074696302603150542356542787146918867} a^{13} - \frac{15934666237937150147343079037292021616028285355851892744492089605426097280121718239}{100522270381252986846117061108478613973472491442074696302603150542356542787146918867} a^{12} - \frac{12872950802270582631538033122142281872143874259042144572636714238008002989516142949}{33507423460417662282039020369492871324490830480691565434201050180785514262382306289} a^{11} - \frac{6309100189059383269000863659953234041524020115607737364807679908166223979249294548}{33507423460417662282039020369492871324490830480691565434201050180785514262382306289} a^{10} - \frac{230866178605831669047388269038830919541722113821276189955416829114034615898791060}{11169141153472554094013006789830957108163610160230521811400350060261838087460768763} a^{9} + \frac{103441735402759308376510527863527220010594221250480738751548073954923885259732156}{657008303145444358471353340578291594597859421190030694788255885897755181615339339} a^{8} - \frac{43508226510466319161584035845020274170236615956870536327271939497813180899258941004}{100522270381252986846117061108478613973472491442074696302603150542356542787146918867} a^{7} - \frac{2844598624605909829526372213221171908829230949116907101574565156944929388602623206}{11169141153472554094013006789830957108163610160230521811400350060261838087460768763} a^{6} + \frac{4688221283667227994675970430783934045648308197878346396488644344055222933215934840}{33507423460417662282039020369492871324490830480691565434201050180785514262382306289} a^{5} + \frac{428068567769707122447168369815507373801753113077579581635214699080190157694546513}{5290645809539630886637740058340979682814341654846036647505428975913502251955100993} a^{4} + \frac{3649251369576561721203201873313764048186945370468506453132291450471537575637864584}{33507423460417662282039020369492871324490830480691565434201050180785514262382306289} a^{3} + \frac{704862693573237244864774695955088123311057847948590879102440804295817960396475278}{1971024909436333075414060021734874783793578263570092084364767657693265544846018017} a^{2} - \frac{18545517547735499931315862274658345534189943949852717702252168485426006932331726}{38647547243849668145373725916370093799874083599413570281662110935162069506784667} a + \frac{673457886300274473058897116378634141786075143023130437411303904504140811137842}{2273385131991156949727866230374711399992593152906680604803653584421298206281451}$

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

Class group and class number

$C_{3}$, which has order $3$ (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 $2$)
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:  \( 44958053496730.96 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{15}\cdot 44958053496730.96 \cdot 3}{2\sqrt{8851375005191383462430321349782942769050796640392327}}\approx 0.673118324687278$ (assuming GRH)

Galois group

$D_{30}$ (as 30T14):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 60
The 18 conjugacy class representatives for $D_{30}$
Character table for $D_{30}$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.1.7791.1, 5.1.25281.1, 6.0.424897767.1, 10.0.10741840447527.1, 15.1.725705259120295972581231.1

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

Sibling fields

Degree 30 sibling: 30.2.1407368625825429970526421094615487900279076665822379993.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $15^{2}$ R $30$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $30$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{15}$ $15^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{6}$ $30$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{15}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{15}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7Data not computed
$53$$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.7.2t1.a.a$1$ $ 7 $ \(\Q(\sqrt{-7}) \) $C_2$ (as 2T1) $1$ $-1$
1.159.2t1.a.a$1$ $ 3 \cdot 53 $ \(\Q(\sqrt{-159}) \) $C_2$ (as 2T1) $1$ $-1$
1.1113.2t1.a.a$1$ $ 3 \cdot 7 \cdot 53 $ \(\Q(\sqrt{1113}) \) $C_2$ (as 2T1) $1$ $1$
* 2.7791.6t3.a.a$2$ $ 3 \cdot 7^{2} \cdot 53 $ 6.0.424897767.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.7791.3t2.a.a$2$ $ 3 \cdot 7^{2} \cdot 53 $ 3.1.7791.1 $S_3$ (as 3T2) $1$ $0$
* 2.159.5t2.a.b$2$ $ 3 \cdot 53 $ 5.1.25281.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.159.5t2.a.a$2$ $ 3 \cdot 53 $ 5.1.25281.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.7791.10t3.b.b$2$ $ 3 \cdot 7^{2} \cdot 53 $ 10.0.10741840447527.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.7791.10t3.b.a$2$ $ 3 \cdot 7^{2} \cdot 53 $ 10.0.10741840447527.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.7791.15t2.a.b$2$ $ 3 \cdot 7^{2} \cdot 53 $ 15.1.725705259120295972581231.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.7791.30t14.b.c$2$ $ 3 \cdot 7^{2} \cdot 53 $ 30.0.8851375005191383462430321349782942769050796640392327.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.7791.15t2.a.a$2$ $ 3 \cdot 7^{2} \cdot 53 $ 15.1.725705259120295972581231.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.7791.30t14.b.d$2$ $ 3 \cdot 7^{2} \cdot 53 $ 30.0.8851375005191383462430321349782942769050796640392327.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.7791.30t14.b.a$2$ $ 3 \cdot 7^{2} \cdot 53 $ 30.0.8851375005191383462430321349782942769050796640392327.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.7791.30t14.b.b$2$ $ 3 \cdot 7^{2} \cdot 53 $ 30.0.8851375005191383462430321349782942769050796640392327.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.7791.15t2.a.d$2$ $ 3 \cdot 7^{2} \cdot 53 $ 15.1.725705259120295972581231.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.7791.15t2.a.c$2$ $ 3 \cdot 7^{2} \cdot 53 $ 15.1.725705259120295972581231.1 $D_{15}$ (as 15T2) $1$ $0$

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 *.