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Properties

Label 30.0.47644328292...4375.1
Degree $30$
Signature $[0, 15]$
Discriminant $-\,5^{48}\cdot 7^{25}$
Root discriminant $66.47$
Ramified primes $5, 7$
Class number $9616$ (GRH)
Class group $[2, 2, 2, 1202]$ (GRH)
Galois group $C_{30}$ (as 30T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 95, -890, 8325, -77851, 727995, -555375, 1136155, -1313050, 1973826, -2539455, 2950295, 1796430, 1745550, 1091474, 861265, 435185, 460135, 89825, 61451, 14795, 7860, 1800, 825, -101, 90, -5, 10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^30 + 10*x^28 - 5*x^27 + 90*x^26 - 101*x^25 + 825*x^24 + 1800*x^23 + 7860*x^22 + 14795*x^21 + 61451*x^20 + 89825*x^19 + 460135*x^18 + 435185*x^17 + 861265*x^16 + 1091474*x^15 + 1745550*x^14 + 1796430*x^13 + 2950295*x^12 - 2539455*x^11 + 1973826*x^10 - 1313050*x^9 + 1136155*x^8 - 555375*x^7 + 727995*x^6 - 77851*x^5 + 8325*x^4 - 890*x^3 + 95*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^30 + 10*x^28 - 5*x^27 + 90*x^26 - 101*x^25 + 825*x^24 + 1800*x^23 + 7860*x^22 + 14795*x^21 + 61451*x^20 + 89825*x^19 + 460135*x^18 + 435185*x^17 + 861265*x^16 + 1091474*x^15 + 1745550*x^14 + 1796430*x^13 + 2950295*x^12 - 2539455*x^11 + 1973826*x^10 - 1313050*x^9 + 1136155*x^8 - 555375*x^7 + 727995*x^6 - 77851*x^5 + 8325*x^4 - 890*x^3 + 95*x^2 - 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{30} + 10 x^{28} - 5 x^{27} + 90 x^{26} - 101 x^{25} + 825 x^{24} + 1800 x^{23} + 7860 x^{22} + 14795 x^{21} + 61451 x^{20} + 89825 x^{19} + 460135 x^{18} + 435185 x^{17} + 861265 x^{16} + 1091474 x^{15} + 1745550 x^{14} + 1796430 x^{13} + 2950295 x^{12} - 2539455 x^{11} + 1973826 x^{10} - 1313050 x^{9} + 1136155 x^{8} - 555375 x^{7} + 727995 x^{6} - 77851 x^{5} + 8325 x^{4} - 890 x^{3} + 95 x^{2} - 10 x + 1 \)

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

Invariants

Degree:  $30$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 15]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-4764432829290274543179606325793429277837276458740234375=-\,5^{48}\cdot 7^{25}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.47$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(175=5^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{175}(1,·)$, $\chi_{175}(66,·)$, $\chi_{175}(131,·)$, $\chi_{175}(6,·)$, $\chi_{175}(71,·)$, $\chi_{175}(136,·)$, $\chi_{175}(11,·)$, $\chi_{175}(76,·)$, $\chi_{175}(141,·)$, $\chi_{175}(16,·)$, $\chi_{175}(81,·)$, $\chi_{175}(146,·)$, $\chi_{175}(86,·)$, $\chi_{175}(151,·)$, $\chi_{175}(26,·)$, $\chi_{175}(156,·)$, $\chi_{175}(31,·)$, $\chi_{175}(96,·)$, $\chi_{175}(36,·)$, $\chi_{175}(101,·)$, $\chi_{175}(166,·)$, $\chi_{175}(41,·)$, $\chi_{175}(106,·)$, $\chi_{175}(171,·)$, $\chi_{175}(46,·)$, $\chi_{175}(111,·)$, $\chi_{175}(51,·)$, $\chi_{175}(116,·)$, $\chi_{175}(121,·)$, $\chi_{175}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{43} a^{19} + \frac{12}{43} a^{18} + \frac{19}{43} a^{17} + \frac{18}{43} a^{16} - \frac{9}{43} a^{15} + \frac{7}{43} a^{14} + \frac{5}{43} a^{13} + \frac{9}{43} a^{12} - \frac{1}{43} a^{11} - \frac{19}{43} a^{10} - \frac{17}{43} a^{9} + \frac{21}{43} a^{8} + \frac{12}{43} a^{7} + \frac{13}{43} a^{6} - \frac{4}{43} a^{5} + \frac{4}{43} a^{4} - \frac{11}{43} a^{3} + \frac{13}{43} a^{2} - \frac{17}{43} a + \frac{20}{43}$, $\frac{1}{43} a^{20} + \frac{4}{43} a^{18} + \frac{5}{43} a^{17} - \frac{10}{43} a^{16} - \frac{14}{43} a^{15} + \frac{7}{43} a^{14} - \frac{8}{43} a^{13} + \frac{20}{43} a^{12} - \frac{7}{43} a^{11} - \frac{4}{43} a^{10} + \frac{10}{43} a^{9} + \frac{18}{43} a^{8} - \frac{2}{43} a^{7} + \frac{12}{43} a^{6} + \frac{9}{43} a^{5} - \frac{16}{43} a^{4} + \frac{16}{43} a^{3} - \frac{1}{43} a^{2} + \frac{9}{43} a + \frac{18}{43}$, $\frac{1}{43} a^{21} + \frac{7}{43} a^{14} + \frac{7}{43} a^{7} + \frac{6}{43}$, $\frac{1}{43} a^{22} + \frac{7}{43} a^{15} + \frac{7}{43} a^{8} + \frac{6}{43} a$, $\frac{1}{43} a^{23} + \frac{7}{43} a^{16} + \frac{7}{43} a^{9} + \frac{6}{43} a^{2}$, $\frac{1}{301} a^{24} - \frac{3}{301} a^{23} + \frac{3}{301} a^{22} + \frac{1}{7} a^{18} + \frac{1}{43} a^{17} + \frac{22}{301} a^{16} + \frac{107}{301} a^{15} + \frac{1}{7} a^{12} + \frac{1}{43} a^{10} - \frac{107}{301} a^{9} - \frac{108}{301} a^{8} + \frac{1}{7} a^{6} + \frac{7}{43} a^{3} + \frac{68}{301} a^{2} + \frac{104}{301} a + \frac{1}{7}$, $\frac{1}{3672672871922190857} a^{25} + \frac{3050578242858763}{3672672871922190857} a^{24} + \frac{39407524389523820}{3672672871922190857} a^{23} + \frac{30505782428587625}{3672672871922190857} a^{22} + \frac{5311194942165597}{524667553131741551} a^{21} - \frac{161154634934220}{74952507590248793} a^{20} - \frac{7049094286550700}{3672672871922190857} a^{19} - \frac{765797680135852486}{3672672871922190857} a^{18} + \frac{1267225674278429069}{3672672871922190857} a^{17} - \frac{645062494977534488}{3672672871922190857} a^{16} - \frac{1783731696170117385}{3672672871922190857} a^{15} - \frac{23984228325504189}{524667553131741551} a^{14} - \frac{1407630425156727708}{3672672871922190857} a^{13} - \frac{1071775755826305736}{3672672871922190857} a^{12} - \frac{139090072734657575}{524667553131741551} a^{11} + \frac{1785441945505934280}{3672672871922190857} a^{10} - \frac{1567335597800027857}{3672672871922190857} a^{9} + \frac{1793904136892425280}{3672672871922190857} a^{8} - \frac{1493301287189346928}{3672672871922190857} a^{7} + \frac{951707689013146503}{3672672871922190857} a^{6} - \frac{210502831673803070}{524667553131741551} a^{5} - \frac{433951721474767}{1743081571866251} a^{4} - \frac{766896369006854963}{3672672871922190857} a^{3} + \frac{950546013126674263}{3672672871922190857} a^{2} - \frac{331810391945383459}{3672672871922190857} a - \frac{129076590556876693}{3672672871922190857}$, $\frac{1}{157924933492654206851} a^{26} + \frac{12}{157924933492654206851} a^{25} + \frac{144220853087294734}{157924933492654206851} a^{24} + \frac{252779540374690662}{157924933492654206851} a^{23} + \frac{1442208530872947270}{157924933492654206851} a^{22} + \frac{258098734044347547}{22560704784664886693} a^{21} + \frac{697960460216499467}{157924933492654206851} a^{20} + \frac{425798999629781254}{157924933492654206851} a^{19} - \frac{43717245911199209395}{157924933492654206851} a^{18} - \frac{15316350892699211548}{157924933492654206851} a^{17} - \frac{26494753161267833172}{157924933492654206851} a^{16} + \frac{60466770172489619488}{157924933492654206851} a^{15} - \frac{67468984022128374986}{157924933492654206851} a^{14} + \frac{67271424595789643363}{157924933492654206851} a^{13} + \frac{62816037010286531904}{157924933492654206851} a^{12} + \frac{60282336393580994453}{157924933492654206851} a^{11} + \frac{10387079755763006620}{157924933492654206851} a^{10} - \frac{7655362900204271704}{22560704784664886693} a^{9} + \frac{54576921846995372638}{157924933492654206851} a^{8} + \frac{9949265202023125994}{157924933492654206851} a^{7} - \frac{59272078255090660034}{157924933492654206851} a^{6} - \frac{9462216178995548358}{22560704784664886693} a^{5} - \frac{6576950692909535136}{157924933492654206851} a^{4} + \frac{18783274742751855697}{157924933492654206851} a^{3} - \frac{59882560244001618176}{157924933492654206851} a^{2} - \frac{5204889279079963362}{157924933492654206851} a + \frac{70962777998905557025}{157924933492654206851}$, $\frac{1}{157924933492654206851} a^{27} + \frac{4}{157924933492654206851} a^{25} + \frac{14823058133082715}{22560704784664886693} a^{24} + \frac{1403206634138957020}{157924933492654206851} a^{23} + \frac{148230581330827147}{22560704784664886693} a^{22} - \frac{1177432180957088293}{157924933492654206851} a^{21} - \frac{1762887147206093339}{157924933492654206851} a^{20} - \frac{1716836929093455424}{157924933492654206851} a^{19} - \frac{61444213115129082954}{157924933492654206851} a^{18} + \frac{11807265276437907026}{157924933492654206851} a^{17} + \frac{16926381505534261928}{157924933492654206851} a^{16} - \frac{29549825699797487944}{157924933492654206851} a^{15} - \frac{8148117630900802286}{157924933492654206851} a^{14} + \frac{32676084531891581847}{157924933492654206851} a^{13} - \frac{67058380806812628305}{157924933492654206851} a^{12} + \frac{77140974341514702498}{157924933492654206851} a^{11} + \frac{2297137639584932227}{157924933492654206851} a^{10} - \frac{41693956400166301617}{157924933492654206851} a^{9} - \frac{40675638760797902719}{157924933492654206851} a^{8} + \frac{27788012719490236832}{157924933492654206851} a^{7} - \frac{8474627366742661198}{22560704784664886693} a^{6} - \frac{3904106259966794107}{157924933492654206851} a^{5} + \frac{7961139030464078828}{22560704784664886693} a^{4} - \frac{56030151192552210388}{157924933492654206851} a^{3} + \frac{492352149460651609}{22560704784664886693} a^{2} + \frac{49046743638372740851}{157924933492654206851} a - \frac{225602069149718901}{3222957826380698099}$, $\frac{1}{157924933492654206851} a^{28} - \frac{412707728431227618}{157924933492654206851} a^{21} + \frac{21368598332280473679}{157924933492654206851} a^{14} - \frac{51064592104540162216}{157924933492654206851} a^{7} - \frac{8223011677419768031}{157924933492654206851}$, $\frac{1}{157924933492654206851} a^{29} - \frac{412707728431227618}{157924933492654206851} a^{22} + \frac{21368598332280473679}{157924933492654206851} a^{15} - \frac{51064592104540162216}{157924933492654206851} a^{8} - \frac{8223011677419768031}{157924933492654206851} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{1202}$, which has order $9616$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $14$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{2412612091064194800}{22560704784664886693} a^{29} - \frac{24126123104910695585}{22560704784664886693} a^{27} + \frac{12063060455320974000}{22560704784664886693} a^{26} - \frac{217135088195777532000}{22560704784664886693} a^{25} + \frac{243673821197483674800}{22560704784664886693} a^{24} - \frac{1990404975127960710000}{22560704784664886693} a^{23} - \frac{4342701763915550640000}{22560704784664886693} a^{22} - \frac{18963131035764571128000}{22560704784664886693} a^{21} - \frac{35694602951979507415218}{22560704784664886693} a^{20} - \frac{148257425607985834654800}{22560704784664886693} a^{19} - \frac{216712881079841297910000}{22560704784664886693} a^{18} - \frac{1110127264521823274298000}{22560704784664886693} a^{17} - \frac{1049932592849771614038000}{22560704784664886693} a^{16} - \frac{2077898352610403734422000}{22560704784664886693} a^{15} - \frac{2633303369482200955135200}{22560704784664886693} a^{14} - \frac{4211329192401012206918270}{22560704784664886693} a^{13} - \frac{4334088738750451464564000}{22560704784664886693} a^{12} - \frac{7117917389206238597466000}{22560704784664886693} a^{11} + \frac{6126719837713424805834000}{22560704784664886693} a^{10} - \frac{4762076473256875365304800}{22560704784664886693} a^{9} + \frac{3167880306171840982140000}{22560704784664886693} a^{8} - \frac{2741101290323040242994000}{22560704784664886693} a^{7} + \frac{1339773657277972324101055}{22560704784664886693} a^{6} - \frac{1756369539234278493426000}{22560704784664886693} a^{5} + \frac{187824263901438629374800}{22560704784664886693} a^{4} - \frac{20084995658109421710000}{22560704784664886693} a^{3} + \frac{2147224761047133372000}{22560704784664886693} a^{2} - \frac{229198148651098506000}{22560704784664886693} a + \frac{24126120910641948000}{22560704784664886693} \) (order $14$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1429364684924.111 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{30}$ (as 30T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 30
The 30 conjugacy class representatives for $C_{30}$
Character table for $C_{30}$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), 5.5.390625.1, \(\Q(\zeta_{7})\), 10.0.2564544677734375.1, 15.15.16836836874485015869140625.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $15^{2}$ $30$ R R $15^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{3}$ $30$ $30$ $15^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{6}$ $30$ $15^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{30}$ $30$ $15^{2}$ $30$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$