magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-31, 0, 1240, 0, -14756, 0, 82212, 0, -260338, 0, 520676, 0, -700910, 0, 660858, 0, -447051, 0, 219604, 0, -78430, 0, 20150, 0, -3627, 0, 434, 0, -31, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 31*x^28 + 434*x^26 - 3627*x^24 + 20150*x^22 - 78430*x^20 + 219604*x^18 - 447051*x^16 + 660858*x^14 - 700910*x^12 + 520676*x^10 - 260338*x^8 + 82212*x^6 - 14756*x^4 + 1240*x^2 - 31)
gp: K = bnfinit(x^30 - 31*x^28 + 434*x^26 - 3627*x^24 + 20150*x^22 - 78430*x^20 + 219604*x^18 - 447051*x^16 + 660858*x^14 - 700910*x^12 + 520676*x^10 - 260338*x^8 + 82212*x^6 - 14756*x^4 + 1240*x^2 - 31, 1)
Normalized defining polynomial
\( x^{30} - 31 x^{28} + 434 x^{26} - 3627 x^{24} + 20150 x^{22} - 78430 x^{20} + 219604 x^{18} - 447051 x^{16} + 660858 x^{14} - 700910 x^{12} + 520676 x^{10} - 260338 x^{8} + 82212 x^{6} - 14756 x^{4} + 1240 x^{2} - 31 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $30$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[30, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(19071681753690303660125890064344467877570851377250304=2^{30}\cdot 31^{29}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $55.29$ | 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: | $2, 31$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(124=2^{2}\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{124}(123,·)$, $\chi_{124}(1,·)$, $\chi_{124}(3,·)$, $\chi_{124}(5,·)$, $\chi_{124}(97,·)$, $\chi_{124}(9,·)$, $\chi_{124}(79,·)$, $\chi_{124}(11,·)$, $\chi_{124}(109,·)$, $\chi_{124}(15,·)$, $\chi_{124}(81,·)$, $\chi_{124}(75,·)$, $\chi_{124}(83,·)$, $\chi_{124}(23,·)$, $\chi_{124}(25,·)$, $\chi_{124}(27,·)$, $\chi_{124}(69,·)$, $\chi_{124}(33,·)$, $\chi_{124}(91,·)$, $\chi_{124}(101,·)$, $\chi_{124}(113,·)$, $\chi_{124}(41,·)$, $\chi_{124}(43,·)$, $\chi_{124}(45,·)$, $\chi_{124}(99,·)$, $\chi_{124}(49,·)$, $\chi_{124}(115,·)$, $\chi_{124}(55,·)$, $\chi_{124}(121,·)$, $\chi_{124}(119,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
Trivial group, which has order $1$ (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: | $29$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | 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: | \( 31320406865860856 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
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
| \(\Q(\sqrt{31}) \), 3.3.961.1, 5.5.923521.1, 6.6.1832265664.1, 10.10.27074173092527104.1, \(\Q(\zeta_{31})^+\) |
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 | $15^{2}$ | ${\href{/LocalNumberField/5.3.0.1}{3} }^{10}$ | $30$ | $15^{2}$ | $30$ | $30$ | $30$ | ${\href{/LocalNumberField/23.5.0.1}{5} }^{6}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{3}$ | R | ${\href{/LocalNumberField/37.6.0.1}{6} }^{5}$ | $15^{2}$ | $15^{2}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{3}$ | $30$ | $30$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | 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 | $x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ | |
| 31 | Data not computed | ||||||