Properties

Label 30.0.107...171.1
Degree $30$
Signature $[0, 15]$
Discriminant $-1.075\times 10^{43}$
Root discriminant $27.19$
Ramified primes $11, 31$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{15}\times S_3$ (as 30T15)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 - 4*x^28 + 2*x^27 + 14*x^26 - 18*x^25 - 22*x^24 + 80*x^23 + 2*x^22 - 292*x^21 + 107*x^20 + 966*x^19 - 524*x^18 - 1689*x^17 + 1837*x^16 + 1003*x^15 - 6747*x^14 + 4493*x^13 + 13850*x^12 - 15662*x^11 - 11779*x^10 + 28372*x^9 - 4752*x^8 - 31522*x^7 + 29972*x^6 + 5164*x^5 - 25670*x^4 + 15150*x^3 + 3000*x^2 - 8125*x + 3125)
 
gp: K = bnfinit(x^30 - x^29 - 4*x^28 + 2*x^27 + 14*x^26 - 18*x^25 - 22*x^24 + 80*x^23 + 2*x^22 - 292*x^21 + 107*x^20 + 966*x^19 - 524*x^18 - 1689*x^17 + 1837*x^16 + 1003*x^15 - 6747*x^14 + 4493*x^13 + 13850*x^12 - 15662*x^11 - 11779*x^10 + 28372*x^9 - 4752*x^8 - 31522*x^7 + 29972*x^6 + 5164*x^5 - 25670*x^4 + 15150*x^3 + 3000*x^2 - 8125*x + 3125, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3125, -8125, 3000, 15150, -25670, 5164, 29972, -31522, -4752, 28372, -11779, -15662, 13850, 4493, -6747, 1003, 1837, -1689, -524, 966, 107, -292, 2, 80, -22, -18, 14, 2, -4, -1, 1]);
 

\( x^{30} - x^{29} - 4 x^{28} + 2 x^{27} + 14 x^{26} - 18 x^{25} - 22 x^{24} + 80 x^{23} + 2 x^{22} - 292 x^{21} + 107 x^{20} + 966 x^{19} - 524 x^{18} - 1689 x^{17} + 1837 x^{16} + 1003 x^{15} - 6747 x^{14} + 4493 x^{13} + 13850 x^{12} - 15662 x^{11} - 11779 x^{10} + 28372 x^{9} - 4752 x^{8} - 31522 x^{7} + 29972 x^{6} + 5164 x^{5} - 25670 x^{4} + 15150 x^{3} + 3000 x^{2} - 8125 x + 3125 \)

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:  \(-10745322081507339108891258824483901499337171\)\(\medspace = -\,11^{27}\cdot 31^{10}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $27.19$
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:  $11, 31$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $15$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{19} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{19} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{10} a^{26} - \frac{1}{10} a^{25} + \frac{1}{10} a^{24} + \frac{1}{5} a^{23} - \frac{1}{10} a^{22} + \frac{1}{5} a^{21} - \frac{1}{5} a^{20} - \frac{1}{2} a^{19} - \frac{3}{10} a^{18} + \frac{3}{10} a^{17} + \frac{1}{5} a^{16} + \frac{1}{10} a^{15} - \frac{2}{5} a^{14} + \frac{1}{10} a^{13} + \frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{3}{10} a^{10} + \frac{3}{10} a^{9} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{3}{10} a^{5} + \frac{3}{10} a^{4} - \frac{1}{5} a^{3} - \frac{3}{10} a^{2} - \frac{1}{10} a$, $\frac{1}{17702350} a^{27} + \frac{326932}{8851175} a^{26} + \frac{1385153}{8851175} a^{25} - \frac{1802454}{8851175} a^{24} - \frac{1355128}{8851175} a^{23} + \frac{1515571}{8851175} a^{22} - \frac{4103717}{17702350} a^{21} + \frac{112357}{708094} a^{20} - \frac{699223}{17702350} a^{19} + \frac{3769319}{8851175} a^{18} + \frac{2831026}{8851175} a^{17} + \frac{1521221}{17702350} a^{16} + \frac{1104908}{8851175} a^{15} - \frac{3731887}{8851175} a^{14} + \frac{4149977}{17702350} a^{13} + \frac{4383054}{8851175} a^{12} - \frac{3034576}{8851175} a^{11} + \frac{3336613}{17702350} a^{10} - \frac{1010771}{3540470} a^{9} + \frac{7875663}{17702350} a^{8} + \frac{636683}{8851175} a^{7} - \frac{113344}{8851175} a^{6} + \frac{4099914}{8851175} a^{5} - \frac{2495226}{8851175} a^{4} - \frac{3485804}{8851175} a^{3} + \frac{744719}{17702350} a^{2} + \frac{170669}{1770235} a - \frac{24324}{354047}$, $\frac{1}{2412153355264986649644427250} a^{28} + \frac{48470023020454467259}{2412153355264986649644427250} a^{27} + \frac{22437346896253688275773011}{2412153355264986649644427250} a^{26} + \frac{205049187562398650647362906}{1206076677632493324822213625} a^{25} - \frac{203958037198621847326608783}{1206076677632493324822213625} a^{24} - \frac{186508829112048576130778914}{1206076677632493324822213625} a^{23} + \frac{178705967503599080054026574}{1206076677632493324822213625} a^{22} - \frac{55353338203558159410660884}{241215335526498664964442725} a^{21} - \frac{869150926057627358787336}{11064923648004525915800125} a^{20} + \frac{342172110289600705461290364}{1206076677632493324822213625} a^{19} - \frac{366540890402262968460150263}{2412153355264986649644427250} a^{18} - \frac{562468340183566599415966007}{1206076677632493324822213625} a^{17} - \frac{556546403288786091233755407}{1206076677632493324822213625} a^{16} + \frac{796431851400196811294432971}{2412153355264986649644427250} a^{15} + \frac{1169700017813607810023354347}{2412153355264986649644427250} a^{14} + \frac{562138044394015767229828824}{1206076677632493324822213625} a^{13} - \frac{930424358493083587097111667}{2412153355264986649644427250} a^{12} + \frac{789084147117568033307048473}{2412153355264986649644427250} a^{11} - \frac{26506994772187620805087527}{241215335526498664964442725} a^{10} + \frac{396089052077804498481566894}{1206076677632493324822213625} a^{9} - \frac{351891545471917578662726137}{1206076677632493324822213625} a^{8} - \frac{391178891117511845044173093}{2412153355264986649644427250} a^{7} + \frac{1021974212227205870788333243}{2412153355264986649644427250} a^{6} - \frac{652692147877814465694322867}{2412153355264986649644427250} a^{5} - \frac{366557999177366132201793399}{1206076677632493324822213625} a^{4} - \frac{306462975093411394097899541}{2412153355264986649644427250} a^{3} + \frac{19583906590276167159689677}{241215335526498664964442725} a^{2} - \frac{25922295074336226036454873}{96486134210599465985777090} a + \frac{3738133685389718212048163}{19297226842119893197155418}$, $\frac{1}{12060766776324933248222136250} a^{29} - \frac{1}{12060766776324933248222136250} a^{28} + \frac{247954445345637526521}{12060766776324933248222136250} a^{27} + \frac{100463592335237439899879176}{6030383388162466624111068125} a^{26} - \frac{194194590998218396563448461}{12060766776324933248222136250} a^{25} + \frac{1133656230549493326897478016}{6030383388162466624111068125} a^{24} + \frac{765835080339170977833633664}{6030383388162466624111068125} a^{23} + \frac{17323575184232723443618588}{1206076677632493324822213625} a^{22} - \frac{967109531978101074666570024}{6030383388162466624111068125} a^{21} - \frac{1292240380304505854095046521}{6030383388162466624111068125} a^{20} + \frac{2168368603258769928556085016}{6030383388162466624111068125} a^{19} - \frac{3533930708351200782508053959}{12060766776324933248222136250} a^{18} + \frac{2083566271763687786263109151}{12060766776324933248222136250} a^{17} + \frac{5937793301166225877743334961}{12060766776324933248222136250} a^{16} - \frac{3650175578405713948323464263}{12060766776324933248222136250} a^{15} + \frac{5854101628220921627002435153}{12060766776324933248222136250} a^{14} + \frac{2546351965988796749030709553}{12060766776324933248222136250} a^{13} + \frac{3570811151084219318507695693}{12060766776324933248222136250} a^{12} - \frac{156275502341174967691746603}{482430671052997329928885450} a^{11} - \frac{1452236177797861357251721481}{6030383388162466624111068125} a^{10} - \frac{2359744339799791181861874827}{6030383388162466624111068125} a^{9} - \frac{1419183665280358453415483089}{6030383388162466624111068125} a^{8} + \frac{1340227771075772716799142199}{6030383388162466624111068125} a^{7} - \frac{130860033950219285364301236}{6030383388162466624111068125} a^{6} - \frac{1040872772559400045040200414}{6030383388162466624111068125} a^{5} - \frac{1491401418570319670207998761}{12060766776324933248222136250} a^{4} + \frac{123376617645251754596634563}{1206076677632493324822213625} a^{3} + \frac{6473588291441416846712069}{19297226842119893197155418} a^{2} - \frac{1979029828432511845225539}{96486134210599465985777090} a + \frac{5932803778160384000641597}{19297226842119893197155418}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( \frac{9727516702630088716870914}{6030383388162466624111068125} a^{29} - \frac{2294911267042609066533144}{6030383388162466624111068125} a^{28} - \frac{39442099053684818060510601}{6030383388162466624111068125} a^{27} - \frac{108818645489696764161878}{55324618240022629579000625} a^{26} + \frac{121847934667376751086920911}{6030383388162466624111068125} a^{25} - \frac{80957803195482482152910022}{6030383388162466624111068125} a^{24} - \frac{256250868607174873205034543}{6030383388162466624111068125} a^{23} + \frac{113127532972992218921628141}{1206076677632493324822213625} a^{22} + \frac{419945592867374599325626653}{6030383388162466624111068125} a^{21} - \frac{2433186759714864829092541098}{6030383388162466624111068125} a^{20} - \frac{791567335824299188016029392}{6030383388162466624111068125} a^{19} + \frac{8440163662706098374989925414}{6030383388162466624111068125} a^{18} + \frac{1377580667579010033297816909}{6030383388162466624111068125} a^{17} - \frac{14105180527410398348374560651}{6030383388162466624111068125} a^{16} + \frac{6762372258392856416757180438}{6030383388162466624111068125} a^{15} + \frac{12594408990981787506055509057}{6030383388162466624111068125} a^{14} - \frac{54905549335169999056671421923}{6030383388162466624111068125} a^{13} + \frac{3680472257945871555141933762}{6030383388162466624111068125} a^{12} + \frac{26082899368030849007934349122}{1206076677632493324822213625} a^{11} - \frac{48743617508046519389628876468}{6030383388162466624111068125} a^{10} - \frac{133146732544870383695712037371}{6030383388162466624111068125} a^{9} + \frac{160791392265447600848846811153}{6030383388162466624111068125} a^{8} + \frac{55130958267088567882745538687}{6030383388162466624111068125} a^{7} - \frac{241468051137964941137737808748}{6030383388162466624111068125} a^{6} + \frac{113249275720342073531652311943}{6030383388162466624111068125} a^{5} + \frac{103291464352691597261385297186}{6030383388162466624111068125} a^{4} - \frac{5870674656092294547247421253}{241215335526498664964442725} a^{3} + \frac{324697725096020850405750948}{48243067105299732992888545} a^{2} + \frac{76808417406570367319664774}{9648613421059946598577709} a - \frac{56349166079551001431890390}{9648613421059946598577709} \) (order $22$)
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:  \( 10842704727.78492 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{15}\cdot 10842704727.78492 \cdot 1}{22\sqrt{10745322081507339108891258824483901499337171}}\approx 0.141189778474009$ (assuming GRH)

Galois group

$C_{15}\times S_3$ (as 30T15):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 90
The 45 conjugacy class representatives for $C_{15}\times S_3$
Character table for $C_{15}\times S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\zeta_{11})^+\), 6.0.1279091.2, \(\Q(\zeta_{11})\)

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

Sibling fields

Degree 45 sibling: data not computed

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }^{3}$ $15^{2}$ $15{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ $30$ R $30$ $30$ $30$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{3}$ R $15{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ $30$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{5}$ $15^{2}$ $15{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ $15{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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

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
11Data not computed
$31$31.15.10.2$x^{15} - 923521 x^{3} + 286291510$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$
31.15.0.1$x^{15} + x^{2} + 9$$1$$15$$0$$C_{15}$$[\ ]^{15}$

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) $1$ $-1$
1.341.6t1.b.a$1$ $ 11 \cdot 31 $ 6.0.1229206451.2 $C_6$ (as 6T1) $0$ $-1$
1.341.6t1.b.b$1$ $ 11 \cdot 31 $ 6.0.1229206451.2 $C_6$ (as 6T1) $0$ $-1$
1.31.3t1.a.a$1$ $ 31 $ 3.3.961.1 $C_3$ (as 3T1) $0$ $1$
1.31.3t1.a.b$1$ $ 31 $ 3.3.961.1 $C_3$ (as 3T1) $0$ $1$
* 1.11.10t1.a.a$1$ $ 11 $ \(\Q(\zeta_{11})\) $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.10t1.a.b$1$ $ 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.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) $0$ $1$
* 1.11.10t1.a.d$1$ $ 11 $ \(\Q(\zeta_{11})\) $C_{10}$ (as 10T1) $0$ $-1$
* 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) $0$ $1$
1.341.30t1.a.a$1$ $ 11 \cdot 31 $ 30.0.8807169930722835293105789536033770566807190677270302653971.1 $C_{30}$ (as 30T1) $0$ $-1$
1.341.15t1.a.a$1$ $ 11 \cdot 31 $ 15.15.2572344674223769220522333521.1 $C_{15}$ (as 15T1) $0$ $1$
1.341.30t1.a.b$1$ $ 11 \cdot 31 $ 30.0.8807169930722835293105789536033770566807190677270302653971.1 $C_{30}$ (as 30T1) $0$ $-1$
1.341.15t1.a.b$1$ $ 11 \cdot 31 $ 15.15.2572344674223769220522333521.1 $C_{15}$ (as 15T1) $0$ $1$
1.341.15t1.a.c$1$ $ 11 \cdot 31 $ 15.15.2572344674223769220522333521.1 $C_{15}$ (as 15T1) $0$ $1$
1.341.30t1.a.c$1$ $ 11 \cdot 31 $ 30.0.8807169930722835293105789536033770566807190677270302653971.1 $C_{30}$ (as 30T1) $0$ $-1$
1.341.30t1.a.d$1$ $ 11 \cdot 31 $ 30.0.8807169930722835293105789536033770566807190677270302653971.1 $C_{30}$ (as 30T1) $0$ $-1$
1.341.30t1.a.e$1$ $ 11 \cdot 31 $ 30.0.8807169930722835293105789536033770566807190677270302653971.1 $C_{30}$ (as 30T1) $0$ $-1$
1.341.15t1.a.d$1$ $ 11 \cdot 31 $ 15.15.2572344674223769220522333521.1 $C_{15}$ (as 15T1) $0$ $1$
1.341.30t1.a.f$1$ $ 11 \cdot 31 $ 30.0.8807169930722835293105789536033770566807190677270302653971.1 $C_{30}$ (as 30T1) $0$ $-1$
1.341.30t1.a.g$1$ $ 11 \cdot 31 $ 30.0.8807169930722835293105789536033770566807190677270302653971.1 $C_{30}$ (as 30T1) $0$ $-1$
1.341.15t1.a.e$1$ $ 11 \cdot 31 $ 15.15.2572344674223769220522333521.1 $C_{15}$ (as 15T1) $0$ $1$
1.341.15t1.a.f$1$ $ 11 \cdot 31 $ 15.15.2572344674223769220522333521.1 $C_{15}$ (as 15T1) $0$ $1$
1.341.15t1.a.g$1$ $ 11 \cdot 31 $ 15.15.2572344674223769220522333521.1 $C_{15}$ (as 15T1) $0$ $1$
1.341.30t1.a.h$1$ $ 11 \cdot 31 $ 30.0.8807169930722835293105789536033770566807190677270302653971.1 $C_{30}$ (as 30T1) $0$ $-1$
1.341.15t1.a.h$1$ $ 11 \cdot 31 $ 15.15.2572344674223769220522333521.1 $C_{15}$ (as 15T1) $0$ $1$
2.10571.3t2.a.a$2$ $ 11 \cdot 31^{2}$ 3.1.10571.1 $S_3$ (as 3T2) $1$ $0$
* 2.341.6t5.b.a$2$ $ 11 \cdot 31 $ 6.0.1279091.2 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.341.6t5.b.b$2$ $ 11 \cdot 31 $ 6.0.1279091.2 $S_3\times C_3$ (as 6T5) $0$ $0$
2.116281.15t4.a.a$2$ $ 11^{2} \cdot 31^{2}$ 15.5.28295791416461461425745668731.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
2.116281.15t4.a.b$2$ $ 11^{2} \cdot 31^{2}$ 15.5.28295791416461461425745668731.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
2.116281.15t4.a.c$2$ $ 11^{2} \cdot 31^{2}$ 15.5.28295791416461461425745668731.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
2.116281.15t4.a.d$2$ $ 11^{2} \cdot 31^{2}$ 15.5.28295791416461461425745668731.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
* 2.3751.30t15.a.a$2$ $ 11^{2} \cdot 31 $ 30.0.10745322081507339108891258824483901499337171.1 $C_{15}\times S_3$ (as 30T15) $0$ $0$
* 2.3751.30t15.a.b$2$ $ 11^{2} \cdot 31 $ 30.0.10745322081507339108891258824483901499337171.1 $C_{15}\times S_3$ (as 30T15) $0$ $0$
* 2.3751.30t15.a.c$2$ $ 11^{2} \cdot 31 $ 30.0.10745322081507339108891258824483901499337171.1 $C_{15}\times S_3$ (as 30T15) $0$ $0$
* 2.3751.30t15.a.d$2$ $ 11^{2} \cdot 31 $ 30.0.10745322081507339108891258824483901499337171.1 $C_{15}\times S_3$ (as 30T15) $0$ $0$
* 2.3751.30t15.a.e$2$ $ 11^{2} \cdot 31 $ 30.0.10745322081507339108891258824483901499337171.1 $C_{15}\times S_3$ (as 30T15) $0$ $0$
* 2.3751.30t15.a.f$2$ $ 11^{2} \cdot 31 $ 30.0.10745322081507339108891258824483901499337171.1 $C_{15}\times S_3$ (as 30T15) $0$ $0$
* 2.3751.30t15.a.g$2$ $ 11^{2} \cdot 31 $ 30.0.10745322081507339108891258824483901499337171.1 $C_{15}\times S_3$ (as 30T15) $0$ $0$
* 2.3751.30t15.a.h$2$ $ 11^{2} \cdot 31 $ 30.0.10745322081507339108891258824483901499337171.1 $C_{15}\times S_3$ (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 *.