Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 + x^28 - x^27 + x^26 - x^25 + x^24 - x^23 + x^22 - x^21 + x^20 - x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)
gp: K = bnfinit(x^30 - x^29 + x^28 - x^27 + x^26 - x^25 + x^24 - x^23 + x^22 - x^21 + x^20 - x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]);
\( x^{30} - x^{29} + x^{28} - x^{27} + x^{26} - x^{25} + x^{24} - x^{23} + x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \)
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: | \(-17761887753093897979823770061456102763834271\)\(\medspace = -\,31^{29}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $27.65$ | 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: | $31$ | 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: | \(31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{31}(1,·)$, $\chi_{31}(2,·)$, $\chi_{31}(3,·)$, $\chi_{31}(4,·)$, $\chi_{31}(5,·)$, $\chi_{31}(6,·)$, $\chi_{31}(7,·)$, $\chi_{31}(8,·)$, $\chi_{31}(9,·)$, $\chi_{31}(10,·)$, $\chi_{31}(11,·)$, $\chi_{31}(12,·)$, $\chi_{31}(13,·)$, $\chi_{31}(14,·)$, $\chi_{31}(15,·)$, $\chi_{31}(16,·)$, $\chi_{31}(17,·)$, $\chi_{31}(18,·)$, $\chi_{31}(19,·)$, $\chi_{31}(20,·)$, $\chi_{31}(21,·)$, $\chi_{31}(22,·)$, $\chi_{31}(23,·)$, $\chi_{31}(24,·)$, $\chi_{31}(25,·)$, $\chi_{31}(26,·)$, $\chi_{31}(27,·)$, $\chi_{31}(28,·)$, $\chi_{31}(29,·)$, $\chi_{31}(30,·)$$\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
$C_{9}$, which has order $9$ (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: | \( a \) (order $62$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| Fundamental units: | \( a - 1 \), \( a^{2} + 1 \), \( a^{3} - 1 \), \( a^{2} - a + 1 \), \( a^{4} + 1 \), \( a^{27} + a^{23} - a^{18} - a^{14} + a^{9} + a^{5} - 1 \), \( a^{22} - a^{11} + 1 \), \( a^{21} + a^{11} + a \), \( a^{6} + 1 \), \( a^{6} - a^{5} + a^{4} - a^{3} + a^{2} - a + 1 \), \( a^{25} + a^{19} + a^{13} \), \( a^{4} - a^{3} + a^{2} - a + 1 \), \( a^{17} + a^{3} \), \( a^{16} - a \) (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
Galois group
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
| \(\Q(\sqrt{-31}) \), 3.3.961.1, 5.5.923521.1, 6.0.28629151.1, 10.0.26439622160671.1, \(\Q(\zeta_{31})^+\) |
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 | ${\href{/LocalNumberField/2.5.0.1}{5} }^{6}$ | $30$ | ${\href{/LocalNumberField/5.3.0.1}{3} }^{10}$ | $15^{2}$ | $30$ | $30$ | $30$ | $15^{2}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{3}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{3}$ | R | ${\href{/LocalNumberField/37.6.0.1}{6} }^{5}$ | $15^{2}$ | $30$ | ${\href{/LocalNumberField/47.5.0.1}{5} }^{6}$ | $30$ | $15^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 31 | Data not computed | ||||||