magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![439, 0, 3365, 0, 10523, 0, 17865, 0, 18387, 0, 12084, 0, 5173, 0, 1433, 0, 247, 0, 24, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 24*x^18 + 247*x^16 + 1433*x^14 + 5173*x^12 + 12084*x^10 + 18387*x^8 + 17865*x^6 + 10523*x^4 + 3365*x^2 + 439)
gp: K = bnfinit(x^20 + 24*x^18 + 247*x^16 + 1433*x^14 + 5173*x^12 + 12084*x^10 + 18387*x^8 + 17865*x^6 + 10523*x^4 + 3365*x^2 + 439, 1)
Normalized defining polynomial
\( x^{20} + 24 x^{18} + 247 x^{16} + 1433 x^{14} + 5173 x^{12} + 12084 x^{10} + 18387 x^{8} + 17865 x^{6} + 10523 x^{4} + 3365 x^{2} + 439 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 10]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(175563371587677622582606371487744=2^{20}\cdot 11^{16}\cdot 43^{2}\cdot 67^{2}\cdot 439\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $40.95$ | 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: | $2, 11, 43, 67, 439$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{2}\times C_{200}$, which has order $400$ (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: | \( 128474.772693 \) (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 163840 |
| The 649 conjugacy class representatives for t20n846 are not computed |
| Character table for t20n846 is not computed |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.10.617567936161.1 |
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 |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ | $20$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | $20$ | $20$ | R | ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.10.12 | $x^{10} - 11 x^{8} + 54 x^{6} - 10 x^{4} + 9 x^{2} - 11$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2, 2]^{5}$ |
| 2.10.10.4 | $x^{10} - 5 x^{8} + 14 x^{6} - 22 x^{4} + 17 x^{2} - 37$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2]^{10}$ | |
| 11 | Data not computed | ||||||
| 43 | Data not computed | ||||||
| $67$ | $\Q_{67}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{67}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 67.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.2.1.2 | $x^{2} + 268$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 268$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 439 | Data not computed | ||||||