magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -45, -45, 225, -15, -216, 30, 60, -10, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 5*x^9 - 10*x^8 + 60*x^7 + 30*x^6 - 216*x^5 - 15*x^4 + 225*x^3 - 45*x^2 - 45*x + 9)
gp: K = bnfinit(x^10 - 5*x^9 - 10*x^8 + 60*x^7 + 30*x^6 - 216*x^5 - 15*x^4 + 225*x^3 - 45*x^2 - 45*x + 9, 1)
Normalized defining polynomial
\( x^{10} - 5 x^{9} - 10 x^{8} + 60 x^{7} + 30 x^{6} - 216 x^{5} - 15 x^{4} + 225 x^{3} - 45 x^{2} - 45 x + 9 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $10$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[10, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(9563006337890625=3^{6}\cdot 5^{10}\cdot 19^{2}\cdot 61^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $39.63$ | 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: | $3, 5, 19, 61$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{4}$, $\frac{1}{13713} a^{9} + \frac{1933}{13713} a^{8} - \frac{692}{4571} a^{7} - \frac{748}{13713} a^{6} + \frac{1328}{4571} a^{5} + \frac{704}{1959} a^{4} + \frac{2067}{4571} a^{3} + \frac{1725}{4571} a^{2} + \frac{1634}{4571} a - \frac{1026}{4571}$
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: | $9$ | 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: | \( 81512.20144581948 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$D_5^2.C_4$ (as 10T28):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 400 |
| The 13 conjugacy class representatives for $(C_5^2 : C_8):C_2$ |
| Character table for $(C_5^2 : C_8):C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 20 siblings: | data not computed |
| Degree 25 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ | R | R | ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ | ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ | R | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ | ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ | ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ | ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ | ${\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 3.8.6.3 | $x^{8} - 3 x^{4} + 18$ | $4$ | $2$ | $6$ | $C_8:C_2$ | $[\ ]_{4}^{4}$ | |
| $5$ | 5.10.10.15 | $x^{10} + 20 x + 5$ | $10$ | $1$ | $10$ | $(C_5^2 : C_8):C_2$ | $[9/8, 9/8]_{8}^{2}$ |
| $19$ | $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 19.4.0.1 | $x^{4} - 2 x + 10$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 19.4.2.2 | $x^{4} - 19 x^{2} + 722$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| $61$ | $\Q_{61}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 61.2.1.1 | $x^{2} - 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.2.1.1 | $x^{2} - 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.5.0.1 | $x^{5} - x + 6$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |