magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, 0, 211, 0, 475, 0, 188, 0, 25, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^10 + 25*x^8 + 188*x^6 + 475*x^4 + 211*x^2 + 25)
gp: K = bnfinit(x^10 + 25*x^8 + 188*x^6 + 475*x^4 + 211*x^2 + 25, 1)
Normalized defining polynomial
\( x^{10} + 25 x^{8} + 188 x^{6} + 475 x^{4} + 211 x^{2} + 25 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $10$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 5]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-873360422339584=-\,2^{10}\cdot 31^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $31.20$ | 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$$\chi_{124}(1,·)$, $\chi_{124}(35,·)$, $\chi_{124}(97,·)$, $\chi_{124}(101,·)$, $\chi_{124}(33,·)$, $\chi_{124}(39,·)$, $\chi_{124}(109,·)$, $\chi_{124}(47,·)$, $\chi_{124}(63,·)$, $\chi_{124}(95,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{335} a^{8} + \frac{32}{335} a^{6} + \frac{144}{335} a^{4} + \frac{143}{335} a^{2} + \frac{28}{67}$, $\frac{1}{335} a^{9} + \frac{32}{335} a^{7} + \frac{2}{67} a^{5} + \frac{143}{335} a^{3} - \frac{61}{335} a$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{41}$, which has order $41$
magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp
Unit group
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | $4$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{37}{335} a^{9} - \frac{916}{335} a^{7} - \frac{1347}{67} a^{5} - \frac{15944}{335} a^{3} - \frac{3773}{335} a \) (order $4$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | \( \frac{22}{335} a^{8} + \frac{114}{67} a^{6} + \frac{4508}{335} a^{4} + \frac{2331}{67} a^{2} + \frac{482}{67} \), \( \frac{128}{335} a^{9} + \frac{3158}{335} a^{7} + \frac{4611}{67} a^{5} + \frac{53747}{335} a^{3} + \frac{11287}{335} a \), \( \frac{112}{335} a^{9} + \frac{556}{67} a^{7} + \frac{4110}{67} a^{5} + \frac{9863}{67} a^{3} + \frac{13603}{335} a \), \( \frac{2}{335} a^{8} + \frac{64}{335} a^{6} + \frac{623}{335} a^{4} + \frac{1626}{335} a^{2} + \frac{56}{67} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 485.913224212 \) | 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 10 |
| The 10 conjugacy class representatives for $C_{10}$ |
| Character table for $C_{10}$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 5.5.923521.1 |
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 | R | ${\href{/LocalNumberField/3.10.0.1}{10} }$ | ${\href{/LocalNumberField/5.1.0.1}{1} }^{10}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }$ | ${\href{/LocalNumberField/11.10.0.1}{10} }$ | ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }$ | ${\href{/LocalNumberField/23.10.0.1}{10} }$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ | R | ${\href{/LocalNumberField/37.1.0.1}{1} }^{10}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }$ | ${\href{/LocalNumberField/47.10.0.1}{10} }$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }$ |
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 | $x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ |
| $31$ | 31.10.8.1 | $x^{10} - 20491 x^{5} + 239127552$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |