Properties

Label 30.0.615...784.1
Degree $30$
Signature $[0, 15]$
Discriminant $-6.152\times 10^{50}$
Root discriminant $49.31$
Ramified primes $2, 31$
Class number $5084$ (GRH)
Class group $[2, 2542]$ (GRH)
Galois group $C_{30}$ (as 30T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 + 29*x^28 + 378*x^26 + 2925*x^24 + 14950*x^22 + 53130*x^20 + 134596*x^18 + 245157*x^16 + 319770*x^14 + 293930*x^12 + 184756*x^10 + 75582*x^8 + 18564*x^6 + 2380*x^4 + 120*x^2 + 1)
 
gp: K = bnfinit(x^30 + 29*x^28 + 378*x^26 + 2925*x^24 + 14950*x^22 + 53130*x^20 + 134596*x^18 + 245157*x^16 + 319770*x^14 + 293930*x^12 + 184756*x^10 + 75582*x^8 + 18564*x^6 + 2380*x^4 + 120*x^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 120, 0, 2380, 0, 18564, 0, 75582, 0, 184756, 0, 293930, 0, 319770, 0, 245157, 0, 134596, 0, 53130, 0, 14950, 0, 2925, 0, 378, 0, 29, 0, 1]);
 

\( x^{30} + 29 x^{28} + 378 x^{26} + 2925 x^{24} + 14950 x^{22} + 53130 x^{20} + 134596 x^{18} + 245157 x^{16} + 319770 x^{14} + 293930 x^{12} + 184756 x^{10} + 75582 x^{8} + 18564 x^{6} + 2380 x^{4} + 120 x^{2} + 1 \)

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:  \(-615215540441622698713738389172402189599059721846784\)\(\medspace = -\,2^{30}\cdot 31^{28}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $49.31$
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:  $2, 31$
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:  \(124=2^{2}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{124}(1,·)$, $\chi_{124}(67,·)$, $\chi_{124}(97,·)$, $\chi_{124}(5,·)$, $\chi_{124}(7,·)$, $\chi_{124}(9,·)$, $\chi_{124}(111,·)$, $\chi_{124}(109,·)$, $\chi_{124}(81,·)$, $\chi_{124}(19,·)$, $\chi_{124}(87,·)$, $\chi_{124}(25,·)$, $\chi_{124}(103,·)$, $\chi_{124}(69,·)$, $\chi_{124}(107,·)$, $\chi_{124}(33,·)$, $\chi_{124}(35,·)$, $\chi_{124}(113,·)$, $\chi_{124}(101,·)$, $\chi_{124}(39,·)$, $\chi_{124}(41,·)$, $\chi_{124}(95,·)$, $\chi_{124}(71,·)$, $\chi_{124}(45,·)$, $\chi_{124}(47,·)$, $\chi_{124}(49,·)$, $\chi_{124}(51,·)$, $\chi_{124}(121,·)$, $\chi_{124}(59,·)$, $\chi_{124}(63,·)$$\rbrace$
This is a CM field.

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

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

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

Class group and class number

$C_{2}\times C_{2542}$, which has order $5084$ (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:  \( a^{29} + 28 a^{27} + 351 a^{25} + 2600 a^{23} + 12650 a^{21} + 42504 a^{19} + 100947 a^{17} + 170544 a^{15} + 203490 a^{13} + 167960 a^{11} + 92378 a^{9} + 31824 a^{7} + 6188 a^{5} + 560 a^{3} + 15 a \) (order $4$)
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

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{15}\cdot 4316173757.895952 \cdot 5084}{4\sqrt{615215540441622698713738389172402189599059721846784}}\approx 0.207696341766930$ (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

\(\Q(\sqrt{-1}) \), 3.3.961.1, 5.5.923521.1, 6.0.59105344.1, 10.0.873360422339584.1, \(\Q(\zeta_{31})^+\)

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 R $30$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{10}$ $30$ $30$ $15^{2}$ $15^{2}$ $30$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{6}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{10}$ $15^{2}$ $30$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{3}$ $15^{2}$ $30$

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
$2$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
31Data not computed