Properties

Label 30.30.595...341.1
Degree $30$
Signature $[30, 0]$
Discriminant $5.951\times 10^{51}$
Root discriminant $53.19$
Ramified prime $61$
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 - 29*x^28 + 28*x^27 + 378*x^26 - 351*x^25 - 2925*x^24 + 2600*x^23 + 14950*x^22 - 12650*x^21 - 53130*x^20 + 42504*x^19 + 134596*x^18 - 100947*x^17 - 245157*x^16 + 170544*x^15 + 319770*x^14 - 203490*x^13 - 293930*x^12 + 167960*x^11 + 184756*x^10 - 92378*x^9 - 75582*x^8 + 31824*x^7 + 18564*x^6 - 6188*x^5 - 2380*x^4 + 560*x^3 + 120*x^2 - 15*x - 1)
 
gp: K = bnfinit(x^30 - x^29 - 29*x^28 + 28*x^27 + 378*x^26 - 351*x^25 - 2925*x^24 + 2600*x^23 + 14950*x^22 - 12650*x^21 - 53130*x^20 + 42504*x^19 + 134596*x^18 - 100947*x^17 - 245157*x^16 + 170544*x^15 + 319770*x^14 - 203490*x^13 - 293930*x^12 + 167960*x^11 + 184756*x^10 - 92378*x^9 - 75582*x^8 + 31824*x^7 + 18564*x^6 - 6188*x^5 - 2380*x^4 + 560*x^3 + 120*x^2 - 15*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -15, 120, 560, -2380, -6188, 18564, 31824, -75582, -92378, 184756, 167960, -293930, -203490, 319770, 170544, -245157, -100947, 134596, 42504, -53130, -12650, 14950, 2600, -2925, -351, 378, 28, -29, -1, 1]);
 

\( x^{30} - x^{29} - 29 x^{28} + 28 x^{27} + 378 x^{26} - 351 x^{25} - 2925 x^{24} + 2600 x^{23} + 14950 x^{22} - 12650 x^{21} - 53130 x^{20} + 42504 x^{19} + 134596 x^{18} - 100947 x^{17} - 245157 x^{16} + 170544 x^{15} + 319770 x^{14} - 203490 x^{13} - 293930 x^{12} + 167960 x^{11} + 184756 x^{10} - 92378 x^{9} - 75582 x^{8} + 31824 x^{7} + 18564 x^{6} - 6188 x^{5} - 2380 x^{4} + 560 x^{3} + 120 x^{2} - 15 x - 1 \)

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:  \(5950661074415937716058277355262049126611998411687341\)\(\medspace = 61^{29}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $53.19$
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:  $61$
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:  \(61\)
Dirichlet character group:    $\lbrace$$\chi_{61}(1,·)$, $\chi_{61}(3,·)$, $\chi_{61}(4,·)$, $\chi_{61}(5,·)$, $\chi_{61}(9,·)$, $\chi_{61}(12,·)$, $\chi_{61}(13,·)$, $\chi_{61}(14,·)$, $\chi_{61}(15,·)$, $\chi_{61}(16,·)$, $\chi_{61}(19,·)$, $\chi_{61}(20,·)$, $\chi_{61}(22,·)$, $\chi_{61}(25,·)$, $\chi_{61}(27,·)$, $\chi_{61}(34,·)$, $\chi_{61}(36,·)$, $\chi_{61}(39,·)$, $\chi_{61}(41,·)$, $\chi_{61}(42,·)$, $\chi_{61}(45,·)$, $\chi_{61}(46,·)$, $\chi_{61}(47,·)$, $\chi_{61}(48,·)$, $\chi_{61}(49,·)$, $\chi_{61}(52,·)$, $\chi_{61}(56,·)$, $\chi_{61}(57,·)$, $\chi_{61}(58,·)$, $\chi_{61}(60,·)$$\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:  \( 17190292874679612 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{30}\cdot(2\pi)^{0}\cdot 17190292874679612 \cdot 1}{2\sqrt{5950661074415937716058277355262049126611998411687341}}\approx 0.119638385183071$ (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

\(\Q(\sqrt{61}) \), 3.3.3721.1, 5.5.13845841.1, 6.6.844596301.1, 10.10.11694146092834141.1, 15.15.9876832533361318095112441.1

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 $30$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{6}$ $15^{2}$ $30$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{10}$ $30$ $15^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{5}$ $30$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{6}$ $30$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{3}$ $30$

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
61Data not computed