magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3625, -576520, 62118, 36743, -2233, -609, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - 609*x^5 - 2233*x^4 + 36743*x^3 + 62118*x^2 - 576520*x + 3625)
gp: K = bnfinit(x^7 - 609*x^5 - 2233*x^4 + 36743*x^3 + 62118*x^2 - 576520*x + 3625, 1)
Normalized defining polynomial
\( x^{7} - 609 x^{5} - 2233 x^{4} + 36743 x^{3} + 62118 x^{2} - 576520 x + 3625 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $7$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[7, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(8233120419813614521=7^{12}\cdot 29^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $503.76$ | 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: | $7, 29$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1421=7^{2}\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1421}(1,·)$, $\chi_{1421}(1051,·)$, $\chi_{1421}(1156,·)$, $\chi_{1421}(596,·)$, $\chi_{1421}(1387,·)$, $\chi_{1421}(1212,·)$, $\chi_{1421}(484,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{215} a^{5} + \frac{19}{215} a^{4} - \frac{17}{215} a^{3} - \frac{59}{215} a^{2} - \frac{39}{215} a - \frac{5}{43}$, $\frac{1}{24321428875} a^{6} + \frac{8416536}{24321428875} a^{5} + \frac{11875818}{223132375} a^{4} - \frac{10840722801}{24321428875} a^{3} - \frac{765479643}{24321428875} a^{2} - \frac{694599251}{4864285775} a - \frac{24660061}{194571431}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{7}$, which has order $7$
magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp
Unit group
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | $6$ | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 5753883.56452 \) | 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 7 |
| The 7 conjugacy class representatives for $C_7$ |
| Character table for $C_7$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.7.0.1}{7} }$ | ${\href{/LocalNumberField/3.7.0.1}{7} }$ | ${\href{/LocalNumberField/5.1.0.1}{1} }^{7}$ | R | ${\href{/LocalNumberField/11.7.0.1}{7} }$ | ${\href{/LocalNumberField/13.7.0.1}{7} }$ | ${\href{/LocalNumberField/17.7.0.1}{7} }$ | ${\href{/LocalNumberField/19.7.0.1}{7} }$ | ${\href{/LocalNumberField/23.7.0.1}{7} }$ | R | ${\href{/LocalNumberField/31.7.0.1}{7} }$ | ${\href{/LocalNumberField/37.7.0.1}{7} }$ | ${\href{/LocalNumberField/41.7.0.1}{7} }$ | ${\href{/LocalNumberField/43.1.0.1}{1} }^{7}$ | ${\href{/LocalNumberField/47.7.0.1}{7} }$ | ${\href{/LocalNumberField/53.7.0.1}{7} }$ | ${\href{/LocalNumberField/59.7.0.1}{7} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $7$ | 7.7.12.3 | $x^{7} - 7 x^{6} + 105$ | $7$ | $1$ | $12$ | $C_7$ | $[2]$ |
| $29$ | 29.7.6.1 | $x^{7} + 232$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |