magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -11, 40, -28, -152, 314, 15, -500, 293, 218, -226, -7, 54, -9, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 - 9*x^13 + 54*x^12 - 7*x^11 - 226*x^10 + 218*x^9 + 293*x^8 - 500*x^7 + 15*x^6 + 314*x^5 - 152*x^4 - 28*x^3 + 40*x^2 - 11*x + 1)
gp: K = bnfinit(x^15 - 4*x^14 - 9*x^13 + 54*x^12 - 7*x^11 - 226*x^10 + 218*x^9 + 293*x^8 - 500*x^7 + 15*x^6 + 314*x^5 - 152*x^4 - 28*x^3 + 40*x^2 - 11*x + 1, 1)
Normalized defining polynomial
\( x^{15} - 4 x^{14} - 9 x^{13} + 54 x^{12} - 7 x^{11} - 226 x^{10} + 218 x^{9} + 293 x^{8} - 500 x^{7} + 15 x^{6} + 314 x^{5} - 152 x^{4} - 28 x^{3} + 40 x^{2} - 11 x + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $15$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[15, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(8489248153826893370401=11^{12}\cdot 52009^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $28.97$ | 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: | $11, 52009$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$
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: | $14$ | 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: | \( 1056561.55278 \) (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 solvable group of order 1215 |
| The 63 conjugacy class representatives for [3^5]5=3wr5 are not computed |
| Character table for [3^5]5=3wr5 is not computed |
Intermediate fields
| \(\Q(\zeta_{11})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ | $15$ | ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ | ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ | R | $15$ | $15$ | $15$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ | $15$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ | $15$ | $15$ |
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 |
|---|---|---|---|---|---|---|---|
| 11 | Data not computed | ||||||
| 52009 | Data not computed | ||||||