Properties

Label 30.2.812...137.1
Degree $30$
Signature $[2, 14]$
Discriminant $8.128\times 10^{47}$
Root discriminant $39.54$
Ramified primes $17, 127$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{30}$ (as 30T14)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 9*x^29 + 29*x^28 - 62*x^27 + 140*x^26 - 289*x^25 + 749*x^24 - 2505*x^23 + 6556*x^22 - 13979*x^21 + 24020*x^20 - 33363*x^19 + 38542*x^18 - 41923*x^17 + 47672*x^16 - 60161*x^15 + 70650*x^14 - 64907*x^13 + 32243*x^12 + 11892*x^11 - 39347*x^10 + 39493*x^9 - 24068*x^8 + 9526*x^7 + 164*x^6 - 4686*x^5 + 4431*x^4 - 2269*x^3 + 729*x^2 - 213*x + 43)
 
gp: K = bnfinit(x^30 - 9*x^29 + 29*x^28 - 62*x^27 + 140*x^26 - 289*x^25 + 749*x^24 - 2505*x^23 + 6556*x^22 - 13979*x^21 + 24020*x^20 - 33363*x^19 + 38542*x^18 - 41923*x^17 + 47672*x^16 - 60161*x^15 + 70650*x^14 - 64907*x^13 + 32243*x^12 + 11892*x^11 - 39347*x^10 + 39493*x^9 - 24068*x^8 + 9526*x^7 + 164*x^6 - 4686*x^5 + 4431*x^4 - 2269*x^3 + 729*x^2 - 213*x + 43, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43, -213, 729, -2269, 4431, -4686, 164, 9526, -24068, 39493, -39347, 11892, 32243, -64907, 70650, -60161, 47672, -41923, 38542, -33363, 24020, -13979, 6556, -2505, 749, -289, 140, -62, 29, -9, 1]);
 

\( x^{30} - 9 x^{29} + 29 x^{28} - 62 x^{27} + 140 x^{26} - 289 x^{25} + 749 x^{24} - 2505 x^{23} + 6556 x^{22} - 13979 x^{21} + 24020 x^{20} - 33363 x^{19} + 38542 x^{18} - 41923 x^{17} + 47672 x^{16} - 60161 x^{15} + 70650 x^{14} - 64907 x^{13} + 32243 x^{12} + 11892 x^{11} - 39347 x^{10} + 39493 x^{9} - 24068 x^{8} + 9526 x^{7} + 164 x^{6} - 4686 x^{5} + 4431 x^{4} - 2269 x^{3} + 729 x^{2} - 213 x + 43 \)

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

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(812804153180660145912426894477473567856769041137\)\(\medspace = 17^{15}\cdot 127^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $39.54$
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:  $17, 127$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{17} a^{23} - \frac{4}{17} a^{22} + \frac{1}{17} a^{21} + \frac{6}{17} a^{20} - \frac{4}{17} a^{19} - \frac{8}{17} a^{18} - \frac{8}{17} a^{17} - \frac{7}{17} a^{16} - \frac{3}{17} a^{15} + \frac{8}{17} a^{14} + \frac{6}{17} a^{13} + \frac{1}{17} a^{12} - \frac{5}{17} a^{11} + \frac{5}{17} a^{10} + \frac{3}{17} a^{9} + \frac{8}{17} a^{8} - \frac{4}{17} a^{7} + \frac{1}{17} a^{6} + \frac{3}{17} a^{5} - \frac{4}{17} a^{4} - \frac{2}{17} a^{3} - \frac{2}{17} a^{2} - \frac{5}{17} a - \frac{4}{17}$, $\frac{1}{17} a^{24} + \frac{2}{17} a^{22} - \frac{7}{17} a^{21} + \frac{3}{17} a^{20} - \frac{7}{17} a^{19} - \frac{6}{17} a^{18} - \frac{5}{17} a^{17} + \frac{3}{17} a^{16} - \frac{4}{17} a^{15} + \frac{4}{17} a^{14} + \frac{8}{17} a^{13} - \frac{1}{17} a^{12} + \frac{2}{17} a^{11} + \frac{6}{17} a^{10} + \frac{3}{17} a^{9} - \frac{6}{17} a^{8} + \frac{2}{17} a^{7} + \frac{7}{17} a^{6} + \frac{8}{17} a^{5} - \frac{1}{17} a^{4} + \frac{7}{17} a^{3} + \frac{4}{17} a^{2} - \frac{7}{17} a + \frac{1}{17}$, $\frac{1}{17} a^{25} + \frac{1}{17} a^{22} + \frac{1}{17} a^{21} - \frac{2}{17} a^{20} + \frac{2}{17} a^{19} - \frac{6}{17} a^{18} + \frac{2}{17} a^{17} - \frac{7}{17} a^{16} - \frac{7}{17} a^{15} - \frac{8}{17} a^{14} + \frac{4}{17} a^{13} - \frac{1}{17} a^{11} - \frac{7}{17} a^{10} + \frac{5}{17} a^{9} + \frac{3}{17} a^{8} - \frac{2}{17} a^{7} + \frac{6}{17} a^{6} - \frac{7}{17} a^{5} - \frac{2}{17} a^{4} + \frac{8}{17} a^{3} - \frac{3}{17} a^{2} - \frac{6}{17} a + \frac{8}{17}$, $\frac{1}{17} a^{26} + \frac{5}{17} a^{22} - \frac{3}{17} a^{21} - \frac{4}{17} a^{20} - \frac{2}{17} a^{19} - \frac{7}{17} a^{18} + \frac{1}{17} a^{17} - \frac{5}{17} a^{15} - \frac{4}{17} a^{14} - \frac{6}{17} a^{13} - \frac{2}{17} a^{12} - \frac{2}{17} a^{11} + \frac{7}{17} a^{8} - \frac{7}{17} a^{7} - \frac{8}{17} a^{6} - \frac{5}{17} a^{5} - \frac{5}{17} a^{4} - \frac{1}{17} a^{3} - \frac{4}{17} a^{2} - \frac{4}{17} a + \frac{4}{17}$, $\frac{1}{731} a^{27} - \frac{16}{731} a^{26} - \frac{9}{731} a^{25} + \frac{16}{731} a^{24} + \frac{21}{731} a^{23} + \frac{182}{731} a^{22} + \frac{262}{731} a^{21} + \frac{105}{731} a^{20} + \frac{205}{731} a^{19} + \frac{232}{731} a^{18} + \frac{81}{731} a^{17} - \frac{74}{731} a^{16} - \frac{262}{731} a^{15} + \frac{50}{731} a^{14} - \frac{177}{731} a^{13} - \frac{208}{731} a^{12} - \frac{126}{731} a^{11} + \frac{256}{731} a^{10} - \frac{282}{731} a^{9} + \frac{260}{731} a^{8} + \frac{141}{731} a^{7} - \frac{143}{731} a^{6} + \frac{76}{731} a^{5} - \frac{4}{43} a^{4} - \frac{31}{731} a^{3} + \frac{15}{43} a^{2} + \frac{185}{731} a + \frac{6}{17}$, $\frac{1}{13211363} a^{28} - \frac{4584}{13211363} a^{27} + \frac{230029}{13211363} a^{26} - \frac{224053}{13211363} a^{25} + \frac{150533}{13211363} a^{24} + \frac{321612}{13211363} a^{23} - \frac{587863}{13211363} a^{22} + \frac{1078806}{13211363} a^{21} + \frac{5447513}{13211363} a^{20} - \frac{22515}{777139} a^{19} - \frac{3744787}{13211363} a^{18} + \frac{855246}{13211363} a^{17} - \frac{218220}{13211363} a^{16} + \frac{753493}{13211363} a^{15} + \frac{6070708}{13211363} a^{14} + \frac{3632482}{13211363} a^{13} + \frac{5541902}{13211363} a^{12} - \frac{5428309}{13211363} a^{11} - \frac{159675}{1201033} a^{10} - \frac{6234543}{13211363} a^{9} + \frac{4886985}{13211363} a^{8} + \frac{2383270}{13211363} a^{7} - \frac{2830001}{13211363} a^{6} - \frac{1389857}{13211363} a^{5} + \frac{1658084}{13211363} a^{4} + \frac{4053788}{13211363} a^{3} - \frac{152407}{307241} a^{2} + \frac{5228240}{13211363} a + \frac{92439}{307241}$, $\frac{1}{77528033104658552131978365575348266293887288354096441119} a^{29} - \frac{43493096884567073061339517291680381938346542343}{2500904293698662971999302115333815041738299624325691649} a^{28} - \frac{41134854391956217930969611178637895686954856832984989}{77528033104658552131978365575348266293887288354096441119} a^{27} + \frac{797547298324061351668122840606341776974281280216595}{1802977514061826793766938734310424797532262519862707933} a^{26} - \frac{36648611719654118166731074765920001190227220395584473}{77528033104658552131978365575348266293887288354096441119} a^{25} + \frac{1899133090779078746246544108984613759714078280935516619}{77528033104658552131978365575348266293887288354096441119} a^{24} - \frac{1618406002595107547078174620457730853509635823342555337}{77528033104658552131978365575348266293887288354096441119} a^{23} + \frac{2726303914587532719598230957542448917718412229897243321}{7048003009514413830179851415940751481262480759463312829} a^{22} - \frac{36905329668401280365456527356745650442702171911602212023}{77528033104658552131978365575348266293887288354096441119} a^{21} + \frac{29996873564654728446933241932285047055936219736537147760}{77528033104658552131978365575348266293887288354096441119} a^{20} + \frac{27993945681019497999200822435787272516559497203352392897}{77528033104658552131978365575348266293887288354096441119} a^{19} + \frac{36450863037083159156345470060212555214215192338191759986}{77528033104658552131978365575348266293887288354096441119} a^{18} + \frac{85125456085411578070375924129814173036273505649502087}{1314034459400992409016582467378784174472665904306719341} a^{17} + \frac{25770375848887801259346889396059970287197846610811553490}{77528033104658552131978365575348266293887288354096441119} a^{16} + \frac{27819217288121362210762267422093128437471863224128615789}{77528033104658552131978365575348266293887288354096441119} a^{15} - \frac{33562278556180081184992297071568123246058733233907577000}{77528033104658552131978365575348266293887288354096441119} a^{14} - \frac{11533454954500265494527105383516796965428449537488567401}{77528033104658552131978365575348266293887288354096441119} a^{13} + \frac{6044505540441010888142767597022347169733274111587643038}{77528033104658552131978365575348266293887288354096441119} a^{12} + \frac{23787397549972769853964484827451727318451052814357447196}{77528033104658552131978365575348266293887288354096441119} a^{11} - \frac{16055280724523740901824533134369716843999031679896770475}{77528033104658552131978365575348266293887288354096441119} a^{10} + \frac{35557818671259391126026403653622764254671652274617392519}{77528033104658552131978365575348266293887288354096441119} a^{9} - \frac{1727528027393125187907593849518118716750811578338244474}{7048003009514413830179851415940751481262480759463312829} a^{8} - \frac{353013786473038556918403678786055313026477962359385152}{4560472535568150125410492092667545076111016962005673007} a^{7} + \frac{31573958510583113212824552235310745801830274409425668771}{77528033104658552131978365575348266293887288354096441119} a^{6} + \frac{2180604334324365311201276261612627444898660620422279006}{7048003009514413830179851415940751481262480759463312829} a^{5} - \frac{19491739444676403404551890971521553039709444259918397459}{77528033104658552131978365575348266293887288354096441119} a^{4} - \frac{21232902572475854477221806084462012814758667648264808283}{77528033104658552131978365575348266293887288354096441119} a^{3} - \frac{22423380878538661216874801276731877951705853128581680043}{77528033104658552131978365575348266293887288354096441119} a^{2} + \frac{19025311579055495145257179482013036808677158847391881903}{77528033104658552131978365575348266293887288354096441119} a - \frac{477813529240900248330406819468169912470486240776024240}{1802977514061826793766938734310424797532262519862707933}$

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:  $15$
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:  \( 1232398776399.036 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{14}\cdot 1232398776399.036 \cdot 1}{2\sqrt{812804153180660145912426894477473567856769041137}}\approx 0.408607613133020$ (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

\(\Q(\sqrt{17}) \), 3.1.2159.1, 5.1.4661281.1, 6.2.79241777.1, 10.2.369368189536337.2, 15.1.218659573334046061397519.1

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

Sibling fields

Degree 30 sibling: 30.0.6072125144349637560639895035214067242224098130847.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $15^{2}$ $30$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{6}$ R $15^{2}$ $30$ $30$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
$127$$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.127.2t1.a.a$1$ $ 127 $ \(\Q(\sqrt{-127}) \) $C_2$ (as 2T1) $1$ $-1$
1.2159.2t1.a.a$1$ $ 17 \cdot 127 $ \(\Q(\sqrt{-2159}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.17.2t1.a.a$1$ $ 17 $ \(\Q(\sqrt{17}) \) $C_2$ (as 2T1) $1$ $1$
* 2.2159.6t3.a.a$2$ $ 17 \cdot 127 $ 6.0.591982687.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.2159.3t2.a.a$2$ $ 17 \cdot 127 $ 3.1.2159.1 $S_3$ (as 3T2) $1$ $0$
* 2.2159.5t2.a.b$2$ $ 17 \cdot 127 $ 5.1.4661281.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.2159.5t2.a.a$2$ $ 17 \cdot 127 $ 5.1.4661281.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.2159.10t3.b.b$2$ $ 17 \cdot 127 $ 10.0.2759397651242047.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.2159.10t3.b.a$2$ $ 17 \cdot 127 $ 10.0.2759397651242047.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.2159.15t2.a.d$2$ $ 17 \cdot 127 $ 15.1.218659573334046061397519.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.2159.30t14.b.a$2$ $ 17 \cdot 127 $ 30.2.812804153180660145912426894477473567856769041137.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.2159.15t2.a.b$2$ $ 17 \cdot 127 $ 15.1.218659573334046061397519.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.2159.30t14.b.c$2$ $ 17 \cdot 127 $ 30.2.812804153180660145912426894477473567856769041137.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.2159.15t2.a.a$2$ $ 17 \cdot 127 $ 15.1.218659573334046061397519.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.2159.30t14.b.b$2$ $ 17 \cdot 127 $ 30.2.812804153180660145912426894477473567856769041137.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.2159.15t2.a.c$2$ $ 17 \cdot 127 $ 15.1.218659573334046061397519.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.2159.30t14.b.d$2$ $ 17 \cdot 127 $ 30.2.812804153180660145912426894477473567856769041137.1 $D_{30}$ (as 30T14) $1$ $0$

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 *.