Properties

Label 30.0.822...187.1
Degree $30$
Signature $[0, 15]$
Discriminant $-8.221\times 10^{48}$
Root discriminant $42.71$
Ramified primes $3, 31$
Class number $755$ (GRH)
Class group $[755]$ (GRH)
Galois group $C_{30}$ (as 30T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 + 15*x^28 - 12*x^27 + 131*x^26 - 92*x^25 + 755*x^24 - 449*x^23 + 3202*x^22 - 1648*x^21 + 10173*x^20 - 4368*x^19 + 24856*x^18 - 8980*x^17 + 46255*x^16 - 13276*x^15 + 65515*x^14 - 15154*x^13 + 68312*x^12 - 11198*x^11 + 51310*x^10 - 6592*x^9 + 25719*x^8 - 1518*x^7 + 8148*x^6 - 588*x^5 + 1330*x^4 + 56*x^3 + 92*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^30 - x^29 + 15*x^28 - 12*x^27 + 131*x^26 - 92*x^25 + 755*x^24 - 449*x^23 + 3202*x^22 - 1648*x^21 + 10173*x^20 - 4368*x^19 + 24856*x^18 - 8980*x^17 + 46255*x^16 - 13276*x^15 + 65515*x^14 - 15154*x^13 + 68312*x^12 - 11198*x^11 + 51310*x^10 - 6592*x^9 + 25719*x^8 - 1518*x^7 + 8148*x^6 - 588*x^5 + 1330*x^4 + 56*x^3 + 92*x^2 - 8*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 92, 56, 1330, -588, 8148, -1518, 25719, -6592, 51310, -11198, 68312, -15154, 65515, -13276, 46255, -8980, 24856, -4368, 10173, -1648, 3202, -449, 755, -92, 131, -12, 15, -1, 1]);
 

\( x^{30} - x^{29} + 15 x^{28} - 12 x^{27} + 131 x^{26} - 92 x^{25} + 755 x^{24} - 449 x^{23} + 3202 x^{22} - 1648 x^{21} + 10173 x^{20} - 4368 x^{19} + 24856 x^{18} - 8980 x^{17} + 46255 x^{16} - 13276 x^{15} + 65515 x^{14} - 15154 x^{13} + 68312 x^{12} - 11198 x^{11} + 51310 x^{10} - 6592 x^{9} + 25719 x^{8} - 1518 x^{7} + 8148 x^{6} - 588 x^{5} + 1330 x^{4} + 56 x^{3} + 92 x^{2} - 8 x + 1 \)

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

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 15]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-8221408887534945302579972677458642036796803806187\)\(\medspace = -\,3^{15}\cdot 31^{28}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $42.71$
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:  $3, 31$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $30$
This field is Galois and abelian over $\Q$.
Conductor:  \(93=3\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{93}(64,·)$, $\chi_{93}(1,·)$, $\chi_{93}(2,·)$, $\chi_{93}(67,·)$, $\chi_{93}(4,·)$, $\chi_{93}(5,·)$, $\chi_{93}(70,·)$, $\chi_{93}(7,·)$, $\chi_{93}(8,·)$, $\chi_{93}(10,·)$, $\chi_{93}(76,·)$, $\chi_{93}(14,·)$, $\chi_{93}(16,·)$, $\chi_{93}(82,·)$, $\chi_{93}(19,·)$, $\chi_{93}(20,·)$, $\chi_{93}(25,·)$, $\chi_{93}(28,·)$, $\chi_{93}(32,·)$, $\chi_{93}(80,·)$, $\chi_{93}(35,·)$, $\chi_{93}(38,·)$, $\chi_{93}(40,·)$, $\chi_{93}(41,·)$, $\chi_{93}(71,·)$, $\chi_{93}(47,·)$, $\chi_{93}(49,·)$, $\chi_{93}(50,·)$, $\chi_{93}(56,·)$, $\chi_{93}(59,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{61} a^{26} + \frac{2}{61} a^{25} - \frac{26}{61} a^{24} - \frac{9}{61} a^{23} - \frac{16}{61} a^{22} - \frac{26}{61} a^{21} - \frac{13}{61} a^{20} - \frac{14}{61} a^{19} + \frac{2}{61} a^{18} - \frac{15}{61} a^{17} - \frac{5}{61} a^{16} - \frac{14}{61} a^{15} + \frac{8}{61} a^{14} - \frac{19}{61} a^{13} - \frac{18}{61} a^{12} + \frac{24}{61} a^{11} - \frac{21}{61} a^{10} + \frac{26}{61} a^{9} + \frac{28}{61} a^{8} - \frac{11}{61} a^{7} + \frac{7}{61} a^{6} - \frac{6}{61} a^{5} - \frac{17}{61} a^{4} + \frac{6}{61} a^{3} - \frac{25}{61} a^{2} + \frac{10}{61} a + \frac{19}{61}$, $\frac{1}{61} a^{27} - \frac{30}{61} a^{25} - \frac{18}{61} a^{24} + \frac{2}{61} a^{23} + \frac{6}{61} a^{22} - \frac{22}{61} a^{21} + \frac{12}{61} a^{20} + \frac{30}{61} a^{19} - \frac{19}{61} a^{18} + \frac{25}{61} a^{17} - \frac{4}{61} a^{16} - \frac{25}{61} a^{15} + \frac{26}{61} a^{14} + \frac{20}{61} a^{13} - \frac{1}{61} a^{12} - \frac{8}{61} a^{11} + \frac{7}{61} a^{10} - \frac{24}{61} a^{9} - \frac{6}{61} a^{8} + \frac{29}{61} a^{7} - \frac{20}{61} a^{6} - \frac{5}{61} a^{5} - \frac{21}{61} a^{4} + \frac{24}{61} a^{3} - \frac{1}{61} a^{2} - \frac{1}{61} a + \frac{23}{61}$, $\frac{1}{61} a^{28} - \frac{19}{61} a^{25} + \frac{15}{61} a^{24} - \frac{20}{61} a^{23} - \frac{14}{61} a^{22} + \frac{25}{61} a^{21} + \frac{6}{61} a^{20} - \frac{12}{61} a^{19} + \frac{24}{61} a^{18} - \frac{27}{61} a^{17} + \frac{8}{61} a^{16} - \frac{28}{61} a^{15} + \frac{16}{61} a^{14} - \frac{22}{61} a^{13} + \frac{1}{61} a^{12} - \frac{5}{61} a^{11} + \frac{17}{61} a^{10} - \frac{19}{61} a^{9} + \frac{15}{61} a^{8} + \frac{16}{61} a^{7} + \frac{22}{61} a^{6} - \frac{18}{61} a^{5} + \frac{2}{61} a^{4} - \frac{4}{61} a^{3} - \frac{19}{61} a^{2} + \frac{18}{61} a + \frac{21}{61}$, $\frac{1}{759857395299017198901947312187494172568579} a^{29} + \frac{382537929751015274923584050846642532999}{759857395299017198901947312187494172568579} a^{28} + \frac{1628461332882284069461254487091239772659}{759857395299017198901947312187494172568579} a^{27} - \frac{2871376645011751900594497534628628111924}{759857395299017198901947312187494172568579} a^{26} + \frac{99208657811287623413634573143622423548501}{759857395299017198901947312187494172568579} a^{25} + \frac{137012052609914680594455204274176221245633}{759857395299017198901947312187494172568579} a^{24} - \frac{6493567038016191194488863924640592432951}{759857395299017198901947312187494172568579} a^{23} - \frac{222631448709651541975144865488535238635827}{759857395299017198901947312187494172568579} a^{22} - \frac{195829220022871981857145039086114086208194}{759857395299017198901947312187494172568579} a^{21} - \frac{59714710054479090680246086839093842309067}{759857395299017198901947312187494172568579} a^{20} + \frac{69371438469294370275633351673329216594289}{759857395299017198901947312187494172568579} a^{19} + \frac{313610599936918413488653716385093977431140}{759857395299017198901947312187494172568579} a^{18} + \frac{267722332273016129808876739491423034665240}{759857395299017198901947312187494172568579} a^{17} + \frac{237096968983990367223007723145671279751767}{759857395299017198901947312187494172568579} a^{16} - \frac{29221299497428756152929353075231917093607}{759857395299017198901947312187494172568579} a^{15} + \frac{174845000123532114716036529260582643826719}{759857395299017198901947312187494172568579} a^{14} - \frac{148771268564280077829826873740780524387873}{759857395299017198901947312187494172568579} a^{13} + \frac{317246057099262261231771996031575902404479}{759857395299017198901947312187494172568579} a^{12} + \frac{60804438258481746059513602421523331315239}{759857395299017198901947312187494172568579} a^{11} - \frac{28249149912399991694200130193474122119686}{759857395299017198901947312187494172568579} a^{10} - \frac{41715476879566479894050717669780974503653}{759857395299017198901947312187494172568579} a^{9} - \frac{283577482431879124771515940131379727413328}{759857395299017198901947312187494172568579} a^{8} + \frac{166372333210082420516515082373051122389943}{759857395299017198901947312187494172568579} a^{7} - \frac{110491332289702908310684698041128199399950}{759857395299017198901947312187494172568579} a^{6} - \frac{166797466432777793784594369713839650354513}{759857395299017198901947312187494172568579} a^{5} + \frac{88360322103650162125741991638255406724512}{759857395299017198901947312187494172568579} a^{4} + \frac{297278530957790207870104939778236946340538}{759857395299017198901947312187494172568579} a^{3} + \frac{191327361531845951744645710046080100353655}{759857395299017198901947312187494172568579} a^{2} - \frac{189061387450638488334608358130250986503676}{759857395299017198901947312187494172568579} a + \frac{168070571443457787073506044856200544776385}{759857395299017198901947312187494172568579}$

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

Class group and class number

$C_{755}$, which has order $755$ (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{86409833652564845976273692487227615687646}{759857395299017198901947312187494172568579} a^{29} - \frac{82574480569149263065235702464437698149667}{759857395299017198901947312187494172568579} a^{28} + \frac{1289242792797114731662210662196013738864235}{759857395299017198901947312187494172568579} a^{27} - \frac{975959730486929796303617740834842984509661}{759857395299017198901947312187494172568579} a^{26} + \frac{11227817719854210221781892368379742320571227}{759857395299017198901947312187494172568579} a^{25} - \frac{7405655364015080290185948193029535056503476}{759857395299017198901947312187494172568579} a^{24} + \frac{64488312366088500034207267118783999154367215}{759857395299017198901947312187494172568579} a^{23} - \frac{35579964932960055873303138038456490057169180}{759857395299017198901947312187494172568579} a^{22} + \frac{272681806318500060650710760296943459936259983}{759857395299017198901947312187494172568579} a^{21} - \frac{128527015465108970998968081323293025252202179}{759857395299017198901947312187494172568579} a^{20} + \frac{863133371743457917451303038012858927068139458}{759857395299017198901947312187494172568579} a^{19} - \frac{5450349727930869607186974483743214067927800}{12456678611459298342654873970286789714239} a^{18} + \frac{2100881444527627256797385340560824579584156001}{759857395299017198901947312187494172568579} a^{17} - \frac{664507312482231198495660546590816940924667853}{759857395299017198901947312187494172568579} a^{16} + \frac{3889695929589655881885223559300164749421735283}{759857395299017198901947312187494172568579} a^{15} - \frac{935727872405118979223393009665961877220478435}{759857395299017198901947312187494172568579} a^{14} + \frac{5477165077124753102710070952364176587977252712}{759857395299017198901947312187494172568579} a^{13} - \frac{1005766733569688630409898385879848726826205902}{759857395299017198901947312187494172568579} a^{12} + \frac{5660624444817725809786301395201312657642511780}{759857395299017198901947312187494172568579} a^{11} - \frac{642635714791618432704284052464634742269537202}{759857395299017198901947312187494172568579} a^{10} + \frac{4205057008370573931080003032469558772195969137}{759857395299017198901947312187494172568579} a^{9} - \frac{321722807628743239892897530953725927796698927}{759857395299017198901947312187494172568579} a^{8} + \frac{2064296366113094533493901476879156686464846528}{759857395299017198901947312187494172568579} a^{7} + \frac{1030802241192079374449019734668133815375224}{759857395299017198901947312187494172568579} a^{6} + \frac{636637846260630871684161521375070431432417986}{759857395299017198901947312187494172568579} a^{5} - \frac{8452321548655737296924077656642752742586170}{759857395299017198901947312187494172568579} a^{4} + \frac{95620100167841271437018175714447563656730360}{759857395299017198901947312187494172568579} a^{3} + \frac{15084490256522422850953561898836223518653628}{759857395299017198901947312187494172568579} a^{2} + \frac{6029203372858148817500324479651534387343827}{759857395299017198901947312187494172568579} a + \frac{247389804989818164360839378287490495109261}{759857395299017198901947312187494172568579} \) (order $6$)
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:  \( 4316173757.895952 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{15}\cdot 4316173757.895952 \cdot 755}{6\sqrt{8221408887534945302579972677458642036796803806187}}\approx 0.177876887881771$ (assuming GRH)

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.961.1, 5.5.923521.1, 6.0.24935067.1, 10.0.207252522098163.1, \(\Q(\zeta_{31})^+\)

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{5}$ $15^{2}$ $30$ $15^{2}$ $30$ $15^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{3}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{10}$ $30$ $15^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{3}$ $30$ $30$

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
3Data not computed
$31$31.15.14.1$x^{15} - 31$$15$$1$$14$$C_{15}$$[\ ]_{15}$
31.15.14.1$x^{15} - 31$$15$$1$$14$$C_{15}$$[\ ]_{15}$