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Properties

Label 30.0.10495827164...1875.1
Degree $30$
Signature $[0, 15]$
Discriminant $-\,3^{45}\cdot 5^{48}$
Root discriminant $68.24$
Ramified primes $3, 5$
Class number $3641$ (GRH)
Class group $[3641]$ (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, 0, 0, -820, 0, 0, 670505, 0, 0, -1555150, 0, 0, 4103510, 0, 0, 1208974, 0, 0, 361380, 0, 0, 13165, 0, 0, 850, 0, 0, -15, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 15*x^27 + 850*x^24 + 13165*x^21 + 361380*x^18 + 1208974*x^15 + 4103510*x^12 - 1555150*x^9 + 670505*x^6 - 820*x^3 + 1)
 
gp: K = bnfinit(x^30 - 15*x^27 + 850*x^24 + 13165*x^21 + 361380*x^18 + 1208974*x^15 + 4103510*x^12 - 1555150*x^9 + 670505*x^6 - 820*x^3 + 1, 1)
 

Normalized defining polynomial

\( x^{30} - 15 x^{27} + 850 x^{24} + 13165 x^{21} + 361380 x^{18} + 1208974 x^{15} + 4103510 x^{12} - 1555150 x^{9} + 670505 x^{6} - 820 x^{3} + 1 \)

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:  \(-10495827164017277150673379537693108431994915008544921875=-\,3^{45}\cdot 5^{48}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.24$
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:  $3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(225=3^{2}\cdot 5^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{225}(1,·)$, $\chi_{225}(131,·)$, $\chi_{225}(196,·)$, $\chi_{225}(71,·)$, $\chi_{225}(136,·)$, $\chi_{225}(11,·)$, $\chi_{225}(76,·)$, $\chi_{225}(206,·)$, $\chi_{225}(16,·)$, $\chi_{225}(146,·)$, $\chi_{225}(211,·)$, $\chi_{225}(86,·)$, $\chi_{225}(151,·)$, $\chi_{225}(26,·)$, $\chi_{225}(91,·)$, $\chi_{225}(221,·)$, $\chi_{225}(31,·)$, $\chi_{225}(161,·)$, $\chi_{225}(101,·)$, $\chi_{225}(166,·)$, $\chi_{225}(41,·)$, $\chi_{225}(106,·)$, $\chi_{225}(46,·)$, $\chi_{225}(176,·)$, $\chi_{225}(116,·)$, $\chi_{225}(181,·)$, $\chi_{225}(56,·)$, $\chi_{225}(121,·)$, $\chi_{225}(61,·)$, $\chi_{225}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{6} + \frac{1}{7}$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{7} + \frac{1}{7} a$, $\frac{1}{7} a^{14} + \frac{1}{7} a^{8} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{15} + \frac{1}{7} a^{9} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{16} + \frac{1}{7} a^{10} + \frac{1}{7} a^{4}$, $\frac{1}{7} a^{17} + \frac{1}{7} a^{11} + \frac{1}{7} a^{5}$, $\frac{1}{7} a^{18} - \frac{1}{7}$, $\frac{1}{7} a^{19} - \frac{1}{7} a$, $\frac{1}{7} a^{20} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{21} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{22} - \frac{1}{7} a^{4}$, $\frac{1}{7} a^{23} - \frac{1}{7} a^{5}$, $\frac{1}{1530907} a^{24} + \frac{2994}{218701} a^{21} - \frac{95044}{1530907} a^{18} - \frac{4303}{218701} a^{15} + \frac{554}{9751} a^{12} - \frac{33990}{218701} a^{9} + \frac{25286}{1530907} a^{6} + \frac{26492}{218701} a^{3} + \frac{476708}{1530907}$, $\frac{1}{1530907} a^{25} + \frac{2994}{218701} a^{22} - \frac{95044}{1530907} a^{19} - \frac{4303}{218701} a^{16} + \frac{554}{9751} a^{13} - \frac{33990}{218701} a^{10} + \frac{25286}{1530907} a^{7} + \frac{26492}{218701} a^{4} + \frac{476708}{1530907} a$, $\frac{1}{1530907} a^{26} + \frac{2994}{218701} a^{23} - \frac{95044}{1530907} a^{20} - \frac{4303}{218701} a^{17} + \frac{554}{9751} a^{14} - \frac{33990}{218701} a^{11} + \frac{25286}{1530907} a^{8} + \frac{26492}{218701} a^{5} + \frac{476708}{1530907} a^{2}$, $\frac{1}{1386296242742438692504366093} a^{27} + \frac{25839207799662200753}{198042320391776956072052299} a^{24} - \frac{76800525431980296212847306}{1386296242742438692504366093} a^{21} + \frac{9874110840781584924434926}{198042320391776956072052299} a^{18} + \frac{4322164493317961997116203}{1386296242742438692504366093} a^{15} + \frac{11947669440043090324057698}{198042320391776956072052299} a^{12} + \frac{86618278913860980725668385}{1386296242742438692504366093} a^{9} + \frac{22492269422920273528587549}{198042320391776956072052299} a^{6} + \frac{129351915236563585730714404}{1386296242742438692504366093} a^{3} + \frac{25049776656422499942044942}{198042320391776956072052299}$, $\frac{1}{1386296242742438692504366093} a^{28} + \frac{25839207799662200753}{198042320391776956072052299} a^{25} - \frac{76800525431980296212847306}{1386296242742438692504366093} a^{22} + \frac{9874110840781584924434926}{198042320391776956072052299} a^{19} + \frac{4322164493317961997116203}{1386296242742438692504366093} a^{16} + \frac{11947669440043090324057698}{198042320391776956072052299} a^{13} + \frac{86618278913860980725668385}{1386296242742438692504366093} a^{10} + \frac{22492269422920273528587549}{198042320391776956072052299} a^{7} + \frac{129351915236563585730714404}{1386296242742438692504366093} a^{4} + \frac{25049776656422499942044942}{198042320391776956072052299} a$, $\frac{1}{1386296242742438692504366093} a^{29} + \frac{25839207799662200753}{198042320391776956072052299} a^{26} - \frac{76800525431980296212847306}{1386296242742438692504366093} a^{23} + \frac{9874110840781584924434926}{198042320391776956072052299} a^{20} + \frac{4322164493317961997116203}{1386296242742438692504366093} a^{17} + \frac{11947669440043090324057698}{198042320391776956072052299} a^{14} + \frac{86618278913860980725668385}{1386296242742438692504366093} a^{11} + \frac{22492269422920273528587549}{198042320391776956072052299} a^{8} + \frac{129351915236563585730714404}{1386296242742438692504366093} a^{5} + \frac{25049776656422499942044942}{198042320391776956072052299} a^{2}$

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

Class group and class number

$C_{3641}$, which has order $3641$ (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:  \( -\frac{148248976519146198942551125}{1386296242742438692504366093} a^{29} + \frac{317675954960175014841890835}{198042320391776956072052299} a^{26} - \frac{126011586872750921645367842390}{1386296242742438692504366093} a^{23} - \frac{278814325074580350522364281451}{198042320391776956072052299} a^{20} - \frac{53574255225438895825289073358465}{1386296242742438692504366093} a^{17} - \frac{25604320847243540531889400997250}{198042320391776956072052299} a^{14} - \frac{608345203718322207803827803644090}{1386296242742438692504366093} a^{11} + \frac{32933660986169431170986149895110}{198042320391776956072052299} a^{8} - \frac{99402335352361051203938763690093}{1386296242742438692504366093} a^{5} + \frac{17366423173342244910048040640}{198042320391776956072052299} a^{2} \) (order $18$)
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:  \( 1819602443243.4526 \) (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

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), 5.5.390625.1, \(\Q(\zeta_{9})\), 10.0.37078857421875.1, 15.15.207828545629978179931640625.1

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 $30$ R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{10}$ $30$ $15^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{6}$ $30$ $30$ $15^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{6}$ $30$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{10}$ $30$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{3}$ $30$

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
3Data not computed
5Data not computed