Properties

Label 30.30.254...797.1
Degree $30$
Signature $[30, 0]$
Discriminant $2.549\times 10^{50}$
Root discriminant $47.89$
Ramified primes $3, 31$
Class number $1$ (GRH)
Class group trivial (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 - 30*x^28 + 30*x^27 + 404*x^26 - 404*x^25 - 3223*x^24 + 3223*x^23 + 16927*x^22 - 16927*x^21 - 61503*x^20 + 61503*x^19 + 158101*x^18 - 158101*x^17 - 288950*x^16 + 288950*x^15 + 371908*x^14 - 371908*x^13 - 329002*x^12 + 329002*x^11 + 191674*x^10 - 191674*x^9 - 68664*x^8 + 68664*x^7 + 13548*x^6 - 13548*x^5 - 1208*x^4 + 1208*x^3 + 32*x^2 - 32*x + 1)
 
gp: K = bnfinit(x^30 - x^29 - 30*x^28 + 30*x^27 + 404*x^26 - 404*x^25 - 3223*x^24 + 3223*x^23 + 16927*x^22 - 16927*x^21 - 61503*x^20 + 61503*x^19 + 158101*x^18 - 158101*x^17 - 288950*x^16 + 288950*x^15 + 371908*x^14 - 371908*x^13 - 329002*x^12 + 329002*x^11 + 191674*x^10 - 191674*x^9 - 68664*x^8 + 68664*x^7 + 13548*x^6 - 13548*x^5 - 1208*x^4 + 1208*x^3 + 32*x^2 - 32*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -32, 32, 1208, -1208, -13548, 13548, 68664, -68664, -191674, 191674, 329002, -329002, -371908, 371908, 288950, -288950, -158101, 158101, 61503, -61503, -16927, 16927, 3223, -3223, -404, 404, 30, -30, -1, 1]);
 

\( x^{30} - x^{29} - 30 x^{28} + 30 x^{27} + 404 x^{26} - 404 x^{25} - 3223 x^{24} + 3223 x^{23} + 16927 x^{22} - 16927 x^{21} - 61503 x^{20} + 61503 x^{19} + 158101 x^{18} - 158101 x^{17} - 288950 x^{16} + 288950 x^{15} + 371908 x^{14} - 371908 x^{13} - 329002 x^{12} + 329002 x^{11} + 191674 x^{10} - 191674 x^{9} - 68664 x^{8} + 68664 x^{7} + 13548 x^{6} - 13548 x^{5} - 1208 x^{4} + 1208 x^{3} + 32 x^{2} - 32 x + 1 \)

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

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[30, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(254863675513583304379979153001217903140700917991797\)\(\medspace = 3^{15}\cdot 31^{29}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $47.89$
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$$\chi_{93}(64,·)$, $\chi_{93}(1,·)$, $\chi_{93}(67,·)$, $\chi_{93}(4,·)$, $\chi_{93}(68,·)$, $\chi_{93}(7,·)$, $\chi_{93}(10,·)$, $\chi_{93}(11,·)$, $\chi_{93}(76,·)$, $\chi_{93}(77,·)$, $\chi_{93}(16,·)$, $\chi_{93}(17,·)$, $\chi_{93}(82,·)$, $\chi_{93}(19,·)$, $\chi_{93}(86,·)$, $\chi_{93}(23,·)$, $\chi_{93}(25,·)$, $\chi_{93}(26,·)$, $\chi_{93}(28,·)$, $\chi_{93}(29,·)$, $\chi_{93}(70,·)$, $\chi_{93}(65,·)$, $\chi_{93}(40,·)$, $\chi_{93}(92,·)$, $\chi_{93}(44,·)$, $\chi_{93}(49,·)$, $\chi_{93}(83,·)$, $\chi_{93}(53,·)$, $\chi_{93}(89,·)$, $\chi_{93}(74,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $29$
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:  \( 3165460564789336.0 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{30}\cdot(2\pi)^{0}\cdot 3165460564789336.0 \cdot 1}{2\sqrt{254863675513583304379979153001217903140700917991797}}\approx 0.106451751314542$ (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

\(\Q(\sqrt{93}) \), 3.3.961.1, 5.5.923521.1, 6.6.772987077.1, 10.10.6424828185043053.1, \(\Q(\zeta_{31})^+\)

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{5}$ $15^{2}$ $15^{2}$ $30$ $15^{2}$ $15^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{6}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{6}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{5}$ $30$ $30$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{3}$ $15^{2}$ $30$

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
31Data not computed