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Properties

Label 20.4.32473988377...0464.2
Degree $20$
Signature $[4, 8]$
Discriminant $2^{10}\cdot 11^{17}\cdot 89^{4}$
Root discriminant $26.64$
Ramified primes $2, 11, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T427

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1451, 3362, 1761, -8443, -1455, 14292, -3468, -10080, 1339, 7462, -1079, -3420, 270, 1201, 10, -360, 24, 56, -6, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 6*x^18 + 56*x^17 + 24*x^16 - 360*x^15 + 10*x^14 + 1201*x^13 + 270*x^12 - 3420*x^11 - 1079*x^10 + 7462*x^9 + 1339*x^8 - 10080*x^7 - 3468*x^6 + 14292*x^5 - 1455*x^4 - 8443*x^3 + 1761*x^2 + 3362*x - 1451)
 
gp: K = bnfinit(x^20 - 5*x^19 - 6*x^18 + 56*x^17 + 24*x^16 - 360*x^15 + 10*x^14 + 1201*x^13 + 270*x^12 - 3420*x^11 - 1079*x^10 + 7462*x^9 + 1339*x^8 - 10080*x^7 - 3468*x^6 + 14292*x^5 - 1455*x^4 - 8443*x^3 + 1761*x^2 + 3362*x - 1451, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 6 x^{18} + 56 x^{17} + 24 x^{16} - 360 x^{15} + 10 x^{14} + 1201 x^{13} + 270 x^{12} - 3420 x^{11} - 1079 x^{10} + 7462 x^{9} + 1339 x^{8} - 10080 x^{7} - 3468 x^{6} + 14292 x^{5} - 1455 x^{4} - 8443 x^{3} + 1761 x^{2} + 3362 x - 1451 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(32473988377432635504517950464=2^{10}\cdot 11^{17}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.64$
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:  $2, 11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{43} a^{18} + \frac{8}{43} a^{17} + \frac{16}{43} a^{16} - \frac{5}{43} a^{15} - \frac{20}{43} a^{14} + \frac{5}{43} a^{13} - \frac{5}{43} a^{12} - \frac{5}{43} a^{11} + \frac{13}{43} a^{10} - \frac{3}{43} a^{9} + \frac{9}{43} a^{8} - \frac{1}{43} a^{7} - \frac{14}{43} a^{6} + \frac{11}{43} a^{5} + \frac{16}{43} a^{4} + \frac{10}{43} a^{3} - \frac{14}{43} a^{2} + \frac{15}{43} a + \frac{8}{43}$, $\frac{1}{17768337073979999301686737} a^{19} - \frac{42778831023699905506419}{17768337073979999301686737} a^{18} + \frac{3728861438509234005081898}{17768337073979999301686737} a^{17} + \frac{8391359833553463795439086}{17768337073979999301686737} a^{16} + \frac{6710110013638192362972596}{17768337073979999301686737} a^{15} - \frac{1620582451367955114710277}{17768337073979999301686737} a^{14} - \frac{5927501020158376598361424}{17768337073979999301686737} a^{13} - \frac{7819877742272027439018890}{17768337073979999301686737} a^{12} - \frac{3166577295225414796161538}{17768337073979999301686737} a^{11} + \frac{8842360165546240006426750}{17768337073979999301686737} a^{10} + \frac{216474385980844428283024}{17768337073979999301686737} a^{9} + \frac{2532484377851562338560234}{17768337073979999301686737} a^{8} + \frac{2512852882215134236526158}{17768337073979999301686737} a^{7} + \frac{6848018278144847840162503}{17768337073979999301686737} a^{6} + \frac{316396201828706327717138}{17768337073979999301686737} a^{5} + \frac{7040335360452345533190421}{17768337073979999301686737} a^{4} - \frac{8305641173878004002671950}{17768337073979999301686737} a^{3} + \frac{4864194174002364637306625}{17768337073979999301686737} a^{2} + \frac{37801174237409366599428}{413217141255348820969459} a + \frac{5003740052224945850895343}{17768337073979999301686737}$

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:  $11$
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:  \( 2582998.13537 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T427:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n427 are not computed
Character table for t20n427 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.2.19077940409.1

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
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 R $20$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$89$89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$