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Properties

Label 30.2.553...125.1
Degree $30$
Signature $[2, 14]$
Discriminant $5.540\times 10^{50}$
Root discriminant $49.14$
Ramified primes $5, 751$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $D_{30}$ (as 30T14)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 5*x^29 + 2*x^28 + 21*x^27 + 40*x^26 - 46*x^25 + 39*x^24 - 296*x^23 - 1612*x^22 - 2561*x^21 + 962*x^20 + 4669*x^19 + 19325*x^18 + 55780*x^17 + 98099*x^16 + 110152*x^15 + 94739*x^14 + 50734*x^13 + 40862*x^12 + 44242*x^11 + 25838*x^10 + 20656*x^9 + 3131*x^8 + 2527*x^7 + 6736*x^6 + 1305*x^5 + 3768*x^4 - 512*x^3 + 661*x^2 - 25*x - 1)
 
gp: K = bnfinit(x^30 - 5*x^29 + 2*x^28 + 21*x^27 + 40*x^26 - 46*x^25 + 39*x^24 - 296*x^23 - 1612*x^22 - 2561*x^21 + 962*x^20 + 4669*x^19 + 19325*x^18 + 55780*x^17 + 98099*x^16 + 110152*x^15 + 94739*x^14 + 50734*x^13 + 40862*x^12 + 44242*x^11 + 25838*x^10 + 20656*x^9 + 3131*x^8 + 2527*x^7 + 6736*x^6 + 1305*x^5 + 3768*x^4 - 512*x^3 + 661*x^2 - 25*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -25, 661, -512, 3768, 1305, 6736, 2527, 3131, 20656, 25838, 44242, 40862, 50734, 94739, 110152, 98099, 55780, 19325, 4669, 962, -2561, -1612, -296, 39, -46, 40, 21, 2, -5, 1]);
 

\( x^{30} - 5 x^{29} + 2 x^{28} + 21 x^{27} + 40 x^{26} - 46 x^{25} + 39 x^{24} - 296 x^{23} - 1612 x^{22} - 2561 x^{21} + 962 x^{20} + 4669 x^{19} + 19325 x^{18} + 55780 x^{17} + 98099 x^{16} + 110152 x^{15} + 94739 x^{14} + 50734 x^{13} + 40862 x^{12} + 44242 x^{11} + 25838 x^{10} + 20656 x^{9} + 3131 x^{8} + 2527 x^{7} + 6736 x^{6} + 1305 x^{5} + 3768 x^{4} - 512 x^{3} + 661 x^{2} - 25 x - 1 \)

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:  \(553999258054841955537532525076904905639678955078125\)\(\medspace = 5^{15}\cdot 751^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $49.14$
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:  $5, 751$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{14} + \frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{4}{9} a^{7} + \frac{4}{9} a^{6} - \frac{4}{9} a^{5} + \frac{4}{9} a^{4} - \frac{4}{9} a^{3} + \frac{4}{9} a^{2} - \frac{4}{9} a + \frac{4}{9}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{19} + \frac{1}{9} a^{11} - \frac{2}{9} a^{3}$, $\frac{1}{27} a^{20} + \frac{1}{27} a^{19} - \frac{1}{27} a^{18} + \frac{1}{27} a^{17} - \frac{1}{27} a^{15} - \frac{2}{27} a^{14} + \frac{2}{27} a^{13} - \frac{4}{27} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{27} a^{8} + \frac{4}{27} a^{7} + \frac{8}{27} a^{6} + \frac{10}{27} a^{5} + \frac{1}{3} a^{4} - \frac{1}{27} a^{3} + \frac{4}{27} a^{2} - \frac{13}{27} a + \frac{5}{27}$, $\frac{1}{27} a^{21} + \frac{1}{27} a^{19} - \frac{1}{27} a^{18} - \frac{1}{27} a^{17} - \frac{1}{27} a^{16} - \frac{1}{27} a^{15} + \frac{4}{27} a^{14} + \frac{1}{9} a^{13} - \frac{2}{27} a^{12} + \frac{1}{9} a^{11} + \frac{4}{27} a^{9} + \frac{1}{9} a^{8} + \frac{4}{27} a^{7} + \frac{2}{27} a^{6} - \frac{10}{27} a^{5} - \frac{1}{27} a^{4} - \frac{1}{27} a^{3} + \frac{7}{27} a^{2} - \frac{1}{3} a - \frac{5}{27}$, $\frac{1}{27} a^{22} + \frac{1}{27} a^{19} + \frac{1}{27} a^{17} - \frac{1}{27} a^{16} - \frac{1}{27} a^{15} + \frac{2}{27} a^{14} - \frac{1}{27} a^{13} + \frac{4}{27} a^{12} + \frac{1}{9} a^{11} - \frac{2}{27} a^{10} + \frac{1}{9} a^{9} - \frac{5}{27} a^{7} - \frac{2}{9} a^{6} + \frac{4}{27} a^{5} + \frac{2}{27} a^{4} - \frac{10}{27} a^{3} - \frac{1}{27} a^{2} - \frac{1}{27} a + \frac{7}{27}$, $\frac{1}{27} a^{23} - \frac{1}{27} a^{19} - \frac{1}{27} a^{18} + \frac{1}{27} a^{17} - \frac{1}{27} a^{16} + \frac{4}{27} a^{14} - \frac{1}{27} a^{13} + \frac{1}{27} a^{12} + \frac{1}{27} a^{11} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{2}{27} a^{7} + \frac{11}{27} a^{6} + \frac{4}{27} a^{5} + \frac{5}{27} a^{4} + \frac{1}{9} a^{3} - \frac{11}{27} a^{2} - \frac{1}{27} a + \frac{10}{27}$, $\frac{1}{5481} a^{24} + \frac{34}{5481} a^{23} - \frac{16}{5481} a^{22} + \frac{44}{5481} a^{21} + \frac{2}{5481} a^{20} - \frac{235}{5481} a^{19} + \frac{115}{5481} a^{18} + \frac{17}{1827} a^{17} - \frac{92}{5481} a^{16} + \frac{20}{609} a^{15} - \frac{260}{1827} a^{14} - \frac{128}{783} a^{13} + \frac{296}{1827} a^{12} - \frac{563}{5481} a^{11} - \frac{76}{5481} a^{10} - \frac{796}{5481} a^{9} + \frac{311}{5481} a^{8} - \frac{2263}{5481} a^{7} + \frac{19}{5481} a^{6} - \frac{20}{87} a^{5} - \frac{1298}{5481} a^{4} + \frac{215}{609} a^{3} + \frac{583}{1827} a^{2} - \frac{449}{5481} a - \frac{2140}{5481}$, $\frac{1}{16443} a^{25} - \frac{1}{16443} a^{24} - \frac{191}{16443} a^{23} + \frac{22}{1827} a^{22} + \frac{86}{16443} a^{21} + \frac{101}{16443} a^{20} + \frac{47}{1827} a^{19} + \frac{289}{16443} a^{18} - \frac{659}{16443} a^{17} - \frac{17}{5481} a^{16} - \frac{787}{16443} a^{15} + \frac{2}{2349} a^{14} - \frac{26}{189} a^{13} + \frac{634}{16443} a^{12} + \frac{1156}{16443} a^{11} + \frac{18}{203} a^{10} + \frac{1172}{16443} a^{9} - \frac{1780}{16443} a^{8} + \frac{1642}{5481} a^{7} - \frac{710}{2349} a^{6} - \frac{5918}{16443} a^{5} - \frac{1399}{5481} a^{4} - \frac{2437}{16443} a^{3} - \frac{2185}{16443} a^{2} - \frac{3883}{16443} a - \frac{175}{2349}$, $\frac{1}{49329} a^{26} - \frac{1}{16443} a^{24} - \frac{38}{7047} a^{23} - \frac{913}{49329} a^{22} + \frac{586}{49329} a^{21} + \frac{293}{49329} a^{20} + \frac{5}{1701} a^{19} + \frac{1877}{49329} a^{18} - \frac{1424}{49329} a^{17} + \frac{1871}{49329} a^{16} + \frac{1579}{49329} a^{15} - \frac{463}{49329} a^{14} + \frac{6247}{49329} a^{13} + \frac{7628}{49329} a^{12} + \frac{6436}{49329} a^{11} + \frac{2882}{49329} a^{10} - \frac{20}{49329} a^{9} + \frac{4679}{49329} a^{8} + \frac{14383}{49329} a^{7} + \frac{13409}{49329} a^{6} + \frac{2728}{7047} a^{5} + \frac{170}{49329} a^{4} + \frac{18835}{49329} a^{3} + \frac{24256}{49329} a^{2} - \frac{24197}{49329} a + \frac{74}{7047}$, $\frac{1}{49329} a^{27} + \frac{1}{49329} a^{24} + \frac{386}{49329} a^{23} + \frac{514}{49329} a^{22} - \frac{358}{49329} a^{21} - \frac{839}{49329} a^{20} + \frac{1814}{49329} a^{19} - \frac{2393}{49329} a^{18} - \frac{136}{7047} a^{17} - \frac{1490}{49329} a^{16} + \frac{1928}{49329} a^{15} - \frac{5168}{49329} a^{14} - \frac{5395}{49329} a^{13} - \frac{2201}{49329} a^{12} + \frac{5981}{49329} a^{11} + \frac{3931}{49329} a^{10} + \frac{5207}{49329} a^{9} + \frac{3490}{49329} a^{8} - \frac{10972}{49329} a^{7} - \frac{23570}{49329} a^{6} + \frac{3176}{7047} a^{5} - \frac{4394}{49329} a^{4} + \frac{7738}{49329} a^{3} - \frac{17099}{49329} a^{2} - \frac{22741}{49329} a + \frac{5927}{16443}$, $\frac{1}{1698525676071} a^{28} - \frac{2137156}{242646525153} a^{27} + \frac{267763}{58569850899} a^{26} - \frac{367088}{54791150841} a^{25} + \frac{1186888}{242646525153} a^{24} + \frac{11888307520}{1698525676071} a^{23} - \frac{3093067303}{1698525676071} a^{22} - \frac{4169648075}{242646525153} a^{21} - \frac{25503223408}{1698525676071} a^{20} - \frac{3114117326}{1698525676071} a^{19} + \frac{29428150796}{1698525676071} a^{18} + \frac{973512686}{58569850899} a^{17} + \frac{69276765407}{1698525676071} a^{16} + \frac{26119636432}{1698525676071} a^{15} - \frac{229960298653}{1698525676071} a^{14} + \frac{35976612691}{1698525676071} a^{13} + \frac{152462260997}{1698525676071} a^{12} + \frac{24182799313}{242646525153} a^{11} - \frac{125374199056}{1698525676071} a^{10} - \frac{259367916014}{1698525676071} a^{9} + \frac{101168133212}{1698525676071} a^{8} - \frac{322141988129}{1698525676071} a^{7} - \frac{204137227345}{1698525676071} a^{6} + \frac{1776112561}{242646525153} a^{5} - \frac{11353595342}{54791150841} a^{4} - \frac{602903428150}{1698525676071} a^{3} + \frac{33877139299}{188725075119} a^{2} + \frac{8980159702}{33304425021} a + \frac{118981838386}{1698525676071}$, $\frac{1}{4634583293229377106287478276191459511} a^{29} + \frac{103794522720642869416004}{662083327604196729469639753741637073} a^{28} + \frac{19269935420306996845505940747818}{4634583293229377106287478276191459511} a^{27} + \frac{10720167178107747165913144312096}{4634583293229377106287478276191459511} a^{26} - \frac{25630557569499600611140075427852}{4634583293229377106287478276191459511} a^{25} + \frac{5450790303852013395347160031948}{149502686878367003428628331490047081} a^{24} - \frac{35718617576885291949375400009759198}{4634583293229377106287478276191459511} a^{23} + \frac{84790019335751630171660777538861532}{4634583293229377106287478276191459511} a^{22} - \frac{38185647942947650490515623684983368}{4634583293229377106287478276191459511} a^{21} - \frac{51142856836829079524070843927958835}{4634583293229377106287478276191459511} a^{20} - \frac{118654380862941898052031305275325272}{4634583293229377106287478276191459511} a^{19} - \frac{40195456973738356593185026272764861}{4634583293229377106287478276191459511} a^{18} - \frac{856119732594203484031168103910220}{94583332514885247067091393391662439} a^{17} + \frac{164078576019005937998052417195651694}{4634583293229377106287478276191459511} a^{16} - \frac{23982430859315560696257601279172398}{662083327604196729469639753741637073} a^{15} + \frac{11081897274248611311816981403542229}{662083327604196729469639753741637073} a^{14} - \frac{2588849376923463702873144072534172}{4634583293229377106287478276191459511} a^{13} + \frac{255611601803071193772036367713129535}{4634583293229377106287478276191459511} a^{12} + \frac{547307846818859750206996728950850506}{4634583293229377106287478276191459511} a^{11} + \frac{454496060944340210956519553573998081}{4634583293229377106287478276191459511} a^{10} + \frac{598378329756934360300317104324118632}{4634583293229377106287478276191459511} a^{9} - \frac{608045335128255467065083455694497783}{4634583293229377106287478276191459511} a^{8} - \frac{18291730736113101157617960990464674}{94583332514885247067091393391662439} a^{7} + \frac{39237889473688585637586433477142656}{662083327604196729469639753741637073} a^{6} - \frac{764002448284593116469521898813378881}{4634583293229377106287478276191459511} a^{5} - \frac{1633650018686413898506570823357740882}{4634583293229377106287478276191459511} a^{4} - \frac{184280233010641738613290988954404673}{1544861097743125702095826092063819837} a^{3} - \frac{195379467646771073955678973821125344}{514953699247708567365275364021273279} a^{2} - \frac{23368459629184273723744433241418820}{662083327604196729469639753741637073} a - \frac{659105124440624055827120945664922784}{1544861097743125702095826092063819837}$

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

Class group and class number

$C_{4}$, 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:  $15$
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:  \( 26934148162576.97 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{14}\cdot 26934148162576.97 \cdot 4}{2\sqrt{553999258054841955537532525076904905639678955078125}}\approx 1.36822271826315$ (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

\(\Q(\sqrt{5}) \), 3.1.751.1, 5.1.564001.1, 6.2.70500125.1, 10.2.994053525003125.2, 15.1.134734730815558692751.1

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

Sibling fields

Degree 30 sibling: Deg 30

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $30$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{15}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $30$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{10}$ $30$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $30$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{3}$ $30$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{5}$ $15^{2}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
751Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3755.2t1.a.a$1$ $ 5 \cdot 751 $ \(\Q(\sqrt{-3755}) \) $C_2$ (as 2T1) $1$ $-1$
1.751.2t1.a.a$1$ $ 751 $ \(\Q(\sqrt{-751}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 2.18775.6t3.a.a$2$ $ 5^{2} \cdot 751 $ 6.0.52945593875.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.751.3t2.a.a$2$ $ 751 $ 3.1.751.1 $S_3$ (as 3T2) $1$ $0$
* 2.751.5t2.a.b$2$ $ 751 $ 5.1.564001.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.751.5t2.a.a$2$ $ 751 $ 5.1.564001.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.18775.10t3.a.b$2$ $ 5^{2} \cdot 751 $ 10.0.746534197277346875.2 $D_{10}$ (as 10T3) $1$ $0$
* 2.18775.10t3.a.a$2$ $ 5^{2} \cdot 751 $ 10.0.746534197277346875.2 $D_{10}$ (as 10T3) $1$ $0$
* 2.751.15t2.a.a$2$ $ 751 $ 15.1.134734730815558692751.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.751.15t2.a.d$2$ $ 751 $ 15.1.134734730815558692751.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.751.15t2.a.c$2$ $ 751 $ 15.1.134734730815558692751.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.751.15t2.a.b$2$ $ 751 $ 15.1.134734730815558692751.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.18775.30t14.a.d$2$ $ 5^{2} \cdot 751 $ 30.2.553999258054841955537532525076904905639678955078125.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.18775.30t14.a.c$2$ $ 5^{2} \cdot 751 $ 30.2.553999258054841955537532525076904905639678955078125.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.18775.30t14.a.a$2$ $ 5^{2} \cdot 751 $ 30.2.553999258054841955537532525076904905639678955078125.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.18775.30t14.a.b$2$ $ 5^{2} \cdot 751 $ 30.2.553999258054841955537532525076904905639678955078125.1 $D_{30}$ (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 *.