Properties

Label 30.0.141...987.1
Degree $30$
Signature $[0, 15]$
Discriminant $-1.417\times 10^{44}$
Root discriminant $29.63$
Ramified primes $3, 439$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_{30}$ (as 30T14)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 5*x^29 + 14*x^28 - 37*x^27 + 83*x^26 - 152*x^25 + 241*x^24 - 335*x^23 + 465*x^22 - 620*x^21 + 812*x^20 - 1084*x^19 + 1348*x^18 - 1578*x^17 + 1736*x^16 - 1826*x^15 + 1986*x^14 - 2009*x^13 + 2111*x^12 - 2179*x^11 + 1810*x^10 - 1551*x^9 + 1376*x^8 - 959*x^7 + 887*x^6 - 765*x^5 + 462*x^4 - 285*x^3 + 146*x^2 - 13*x + 1)
 
gp: K = bnfinit(x^30 - 5*x^29 + 14*x^28 - 37*x^27 + 83*x^26 - 152*x^25 + 241*x^24 - 335*x^23 + 465*x^22 - 620*x^21 + 812*x^20 - 1084*x^19 + 1348*x^18 - 1578*x^17 + 1736*x^16 - 1826*x^15 + 1986*x^14 - 2009*x^13 + 2111*x^12 - 2179*x^11 + 1810*x^10 - 1551*x^9 + 1376*x^8 - 959*x^7 + 887*x^6 - 765*x^5 + 462*x^4 - 285*x^3 + 146*x^2 - 13*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -13, 146, -285, 462, -765, 887, -959, 1376, -1551, 1810, -2179, 2111, -2009, 1986, -1826, 1736, -1578, 1348, -1084, 812, -620, 465, -335, 241, -152, 83, -37, 14, -5, 1]);
 

\( x^{30} - 5 x^{29} + 14 x^{28} - 37 x^{27} + 83 x^{26} - 152 x^{25} + 241 x^{24} - 335 x^{23} + 465 x^{22} - 620 x^{21} + 812 x^{20} - 1084 x^{19} + 1348 x^{18} - 1578 x^{17} + 1736 x^{16} - 1826 x^{15} + 1986 x^{14} - 2009 x^{13} + 2111 x^{12} - 2179 x^{11} + 1810 x^{10} - 1551 x^{9} + 1376 x^{8} - 959 x^{7} + 887 x^{6} - 765 x^{5} + 462 x^{4} - 285 x^{3} + 146 x^{2} - 13 x + 1 \)

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:  \(-141684422447224323645012862864040611366932987\)\(\medspace = -\,3^{15}\cdot 439^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $29.63$
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:  $3, 439$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{8} + \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{9} + \frac{1}{3} a$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{10} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{19} + \frac{1}{9} a^{17} + \frac{1}{9} a^{16} + \frac{1}{3} a^{15} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{9} a^{11} - \frac{2}{9} a^{9} + \frac{1}{9} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{9} a^{3} + \frac{4}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{20} + \frac{1}{9} a^{18} + \frac{1}{9} a^{17} - \frac{1}{3} a^{15} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{9} a^{12} - \frac{2}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{9} a^{4} + \frac{4}{9} a^{2} + \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{9} a^{21} + \frac{1}{9} a^{18} - \frac{1}{9} a^{17} - \frac{1}{9} a^{16} + \frac{1}{3} a^{15} + \frac{4}{9} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{9} a^{10} - \frac{4}{9} a^{9} - \frac{4}{9} a^{8} - \frac{1}{3} a^{7} - \frac{2}{9} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{9} a^{2} + \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{27} a^{22} - \frac{1}{27} a^{20} - \frac{1}{27} a^{19} + \frac{4}{27} a^{18} + \frac{2}{27} a^{17} - \frac{2}{27} a^{16} + \frac{2}{9} a^{15} + \frac{13}{27} a^{14} + \frac{4}{9} a^{13} + \frac{2}{27} a^{12} + \frac{8}{27} a^{11} + \frac{4}{27} a^{10} - \frac{4}{27} a^{9} - \frac{11}{27} a^{8} - \frac{2}{9} a^{7} + \frac{7}{27} a^{6} - \frac{4}{9} a^{5} - \frac{13}{27} a^{4} - \frac{10}{27} a^{3} - \frac{5}{27} a^{2} - \frac{1}{27} a - \frac{11}{27}$, $\frac{1}{27} a^{23} - \frac{1}{27} a^{21} - \frac{1}{27} a^{20} + \frac{1}{27} a^{19} + \frac{2}{27} a^{18} + \frac{4}{27} a^{17} + \frac{1}{9} a^{16} + \frac{4}{27} a^{15} - \frac{2}{9} a^{14} + \frac{11}{27} a^{13} - \frac{10}{27} a^{12} + \frac{1}{27} a^{11} - \frac{4}{27} a^{10} + \frac{4}{27} a^{9} - \frac{1}{3} a^{8} - \frac{11}{27} a^{7} + \frac{2}{9} a^{6} + \frac{5}{27} a^{5} + \frac{8}{27} a^{4} - \frac{8}{27} a^{3} - \frac{1}{27} a^{2} + \frac{13}{27} a - \frac{1}{9}$, $\frac{1}{27} a^{24} - \frac{1}{27} a^{21} + \frac{1}{27} a^{19} - \frac{1}{27} a^{18} - \frac{4}{27} a^{17} + \frac{2}{27} a^{16} - \frac{1}{9} a^{14} + \frac{2}{27} a^{13} + \frac{1}{9} a^{12} + \frac{4}{27} a^{11} - \frac{1}{27} a^{10} + \frac{5}{27} a^{9} + \frac{5}{27} a^{8} + \frac{4}{9} a^{6} - \frac{4}{27} a^{5} + \frac{2}{9} a^{4} - \frac{11}{27} a^{3} - \frac{1}{27} a^{2} - \frac{13}{27} a - \frac{11}{27}$, $\frac{1}{459} a^{25} - \frac{1}{153} a^{24} + \frac{7}{459} a^{23} - \frac{4}{459} a^{22} - \frac{1}{459} a^{21} + \frac{2}{51} a^{20} + \frac{1}{153} a^{19} - \frac{56}{459} a^{18} + \frac{8}{153} a^{17} - \frac{19}{153} a^{16} + \frac{133}{459} a^{15} + \frac{200}{459} a^{14} - \frac{4}{459} a^{13} + \frac{49}{153} a^{12} + \frac{40}{153} a^{11} - \frac{179}{459} a^{10} + \frac{5}{51} a^{9} + \frac{49}{153} a^{8} - \frac{92}{459} a^{7} + \frac{143}{459} a^{6} - \frac{133}{459} a^{5} + \frac{4}{9} a^{4} + \frac{31}{153} a^{3} + \frac{112}{459} a^{2} + \frac{56}{459} a - \frac{38}{153}$, $\frac{1}{459} a^{26} - \frac{2}{459} a^{24} + \frac{4}{459} a^{22} - \frac{19}{459} a^{21} + \frac{2}{153} a^{20} + \frac{7}{153} a^{19} - \frac{59}{459} a^{18} - \frac{70}{459} a^{17} + \frac{10}{153} a^{16} + \frac{7}{153} a^{15} - \frac{152}{459} a^{14} - \frac{205}{459} a^{13} - \frac{4}{9} a^{12} + \frac{32}{153} a^{11} - \frac{152}{459} a^{10} - \frac{58}{459} a^{9} + \frac{1}{51} a^{8} + \frac{35}{153} a^{7} + \frac{7}{459} a^{6} - \frac{229}{459} a^{5} - \frac{6}{17} a^{4} - \frac{1}{3} a^{3} + \frac{74}{153} a^{2} - \frac{82}{459} a + \frac{134}{459}$, $\frac{1}{18819} a^{27} - \frac{4}{18819} a^{26} + \frac{11}{18819} a^{25} - \frac{65}{18819} a^{24} + \frac{37}{2091} a^{23} + \frac{100}{18819} a^{22} + \frac{40}{6273} a^{21} + \frac{571}{18819} a^{20} + \frac{107}{6273} a^{19} + \frac{317}{6273} a^{18} - \frac{517}{18819} a^{17} + \frac{1727}{18819} a^{16} + \frac{322}{6273} a^{15} + \frac{3343}{18819} a^{14} - \frac{271}{6273} a^{13} - \frac{2192}{18819} a^{12} - \frac{3070}{6273} a^{11} - \frac{137}{2091} a^{10} - \frac{35}{459} a^{9} + \frac{6026}{18819} a^{8} + \frac{1991}{6273} a^{7} - \frac{4025}{18819} a^{6} - \frac{998}{2091} a^{5} - \frac{4163}{18819} a^{4} - \frac{206}{459} a^{3} - \frac{7283}{18819} a^{2} + \frac{14}{369} a - \frac{8699}{18819}$, $\frac{1}{90952227} a^{28} - \frac{50}{3368601} a^{27} - \frac{3953}{90952227} a^{26} - \frac{58331}{90952227} a^{25} + \frac{47369}{30317409} a^{24} - \frac{172267}{10105803} a^{23} + \frac{10163}{2218347} a^{22} - \frac{1638122}{30317409} a^{21} - \frac{4869967}{90952227} a^{20} + \frac{93700}{2218347} a^{19} + \frac{7303033}{90952227} a^{18} + \frac{10881722}{90952227} a^{17} + \frac{5518207}{90952227} a^{16} - \frac{10157443}{30317409} a^{15} - \frac{28940099}{90952227} a^{14} + \frac{9400367}{30317409} a^{13} + \frac{32771510}{90952227} a^{12} + \frac{29485949}{90952227} a^{11} + \frac{18353260}{90952227} a^{10} - \frac{22248364}{90952227} a^{9} - \frac{37284641}{90952227} a^{8} + \frac{9632029}{30317409} a^{7} + \frac{5991811}{90952227} a^{6} - \frac{102539}{1122867} a^{5} + \frac{14884363}{90952227} a^{4} + \frac{26116646}{90952227} a^{3} - \frac{15092483}{30317409} a^{2} + \frac{33734584}{90952227} a + \frac{8100526}{90952227}$, $\frac{1}{8718043814631} a^{29} - \frac{36127}{8718043814631} a^{28} + \frac{106912861}{8718043814631} a^{27} - \frac{21108641}{35876723517} a^{26} - \frac{3015072001}{8718043814631} a^{25} + \frac{21357217930}{2906014604877} a^{24} + \frac{56018246680}{8718043814631} a^{23} - \frac{124499265928}{8718043814631} a^{22} + \frac{169712741441}{8718043814631} a^{21} - \frac{29141350688}{968671534959} a^{20} + \frac{334184994851}{8718043814631} a^{19} - \frac{381887125712}{8718043814631} a^{18} - \frac{5135614850}{212635214991} a^{17} - \frac{430484801305}{8718043814631} a^{16} - \frac{592793941340}{8718043814631} a^{15} + \frac{1666832596355}{8718043814631} a^{14} - \frac{4044430113829}{8718043814631} a^{13} + \frac{900944279843}{2906014604877} a^{12} + \frac{889790626832}{8718043814631} a^{11} - \frac{1312745134775}{8718043814631} a^{10} - \frac{2807199929740}{8718043814631} a^{9} - \frac{405632310460}{8718043814631} a^{8} + \frac{36181679936}{512826106743} a^{7} - \frac{2531012214940}{8718043814631} a^{6} - \frac{102640181165}{8718043814631} a^{5} - \frac{113379010646}{8718043814631} a^{4} + \frac{3184122242}{97955548479} a^{3} - \frac{1066973953016}{8718043814631} a^{2} - \frac{135726440162}{2906014604877} a + \frac{1521117204008}{8718043814631}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -\frac{10493915024}{107630170551} a^{29} + \frac{49355026628}{107630170551} a^{28} - \frac{132614083868}{107630170551} a^{27} + \frac{117342677845}{35876723517} a^{26} - \frac{778750507993}{107630170551} a^{25} + \frac{465876170536}{35876723517} a^{24} - \frac{2195423870762}{107630170551} a^{23} + \frac{3033012597578}{107630170551} a^{22} - \frac{251334944627}{6331186503} a^{21} + \frac{1889209152911}{35876723517} a^{20} - \frac{7406617182097}{107630170551} a^{19} + \frac{9927122893516}{107630170551} a^{18} - \frac{12194172448426}{107630170551} a^{17} + \frac{14243506409576}{107630170551} a^{16} - \frac{921233826571}{6331186503} a^{15} + \frac{16551488080124}{107630170551} a^{14} - \frac{18182842047562}{107630170551} a^{13} + \frac{6001703474113}{35876723517} a^{12} - \frac{19196833721899}{107630170551} a^{11} + \frac{1158491948621}{6331186503} a^{10} - \frac{15767986812016}{107630170551} a^{9} + \frac{14195308636277}{107630170551} a^{8} - \frac{12730208819687}{107630170551} a^{7} + \frac{8286655239413}{107630170551} a^{6} - \frac{8144006299112}{107630170551} a^{5} + \frac{6821780067499}{107630170551} a^{4} - \frac{43866346123}{1209327759} a^{3} + \frac{2551977980950}{107630170551} a^{2} - \frac{469868085199}{35876723517} a + \frac{3070603741}{2625126111} \) (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:  \( 18103283853.50697 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{15}\cdot 18103283853.50697 \cdot 2}{6\sqrt{141684422447224323645012862864040611366932987}}\approx 0.476072482619609$ (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

\(\Q(\sqrt{-3}) \), 3.1.439.1, 5.1.192721.1, 6.0.5203467.1, 10.0.9025356273363.1, 15.1.3142328914862177479.1

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 $30$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{3}$ $15^{2}$ $30$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{10}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{15}$ $30$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{15}$ $30$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{15}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
439Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.1317.2t1.a.a$1$ $ 3 \cdot 439 $ \(\Q(\sqrt{1317}) \) $C_2$ (as 2T1) $1$ $1$
1.439.2t1.a.a$1$ $ 439 $ \(\Q(\sqrt{-439}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.3951.6t3.b.a$2$ $ 3^{2} \cdot 439 $ 6.2.2284322013.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.439.3t2.a.a$2$ $ 439 $ 3.1.439.1 $S_3$ (as 3T2) $1$ $0$
* 2.439.5t2.a.a$2$ $ 439 $ 5.1.192721.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.439.5t2.a.b$2$ $ 439 $ 5.1.192721.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.3951.10t3.a.b$2$ $ 3^{2} \cdot 439 $ 10.2.3962131404006357.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.3951.10t3.a.a$2$ $ 3^{2} \cdot 439 $ 10.2.3962131404006357.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.439.15t2.a.a$2$ $ 439 $ 15.1.3142328914862177479.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.3951.30t14.a.c$2$ $ 3^{2} \cdot 439 $ 30.0.141684422447224323645012862864040611366932987.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.439.15t2.a.c$2$ $ 439 $ 15.1.3142328914862177479.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.3951.30t14.a.a$2$ $ 3^{2} \cdot 439 $ 30.0.141684422447224323645012862864040611366932987.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.439.15t2.a.b$2$ $ 439 $ 15.1.3142328914862177479.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.3951.30t14.a.b$2$ $ 3^{2} \cdot 439 $ 30.0.141684422447224323645012862864040611366932987.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.439.15t2.a.d$2$ $ 439 $ 15.1.3142328914862177479.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.3951.30t14.a.d$2$ $ 3^{2} \cdot 439 $ 30.0.141684422447224323645012862864040611366932987.1 $D_{30}$ (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 *.