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Properties

Label 20.10.2906949957...0656.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{30}\cdot 3^{10}\cdot 71^{9}$
Root discriminant $33.36$
Ramified primes $2, 3, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T647

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -14, 32, 142, -177, -910, 296, 408, 1747, -1364, -740, 1280, -1504, 2006, -1788, 892, -185, -42, 38, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 38*x^18 - 42*x^17 - 185*x^16 + 892*x^15 - 1788*x^14 + 2006*x^13 - 1504*x^12 + 1280*x^11 - 740*x^10 - 1364*x^9 + 1747*x^8 + 408*x^7 + 296*x^6 - 910*x^5 - 177*x^4 + 142*x^3 + 32*x^2 - 14*x + 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 38*x^18 - 42*x^17 - 185*x^16 + 892*x^15 - 1788*x^14 + 2006*x^13 - 1504*x^12 + 1280*x^11 - 740*x^10 - 1364*x^9 + 1747*x^8 + 408*x^7 + 296*x^6 - 910*x^5 - 177*x^4 + 142*x^3 + 32*x^2 - 14*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 38 x^{18} - 42 x^{17} - 185 x^{16} + 892 x^{15} - 1788 x^{14} + 2006 x^{13} - 1504 x^{12} + 1280 x^{11} - 740 x^{10} - 1364 x^{9} + 1747 x^{8} + 408 x^{7} + 296 x^{6} - 910 x^{5} - 177 x^{4} + 142 x^{3} + 32 x^{2} - 14 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2906949957743139152698231750656=-\,2^{30}\cdot 3^{10}\cdot 71^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.36$
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:  $2, 3, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{61} a^{18} - \frac{17}{61} a^{17} + \frac{18}{61} a^{16} - \frac{1}{61} a^{15} + \frac{4}{61} a^{14} + \frac{27}{61} a^{13} + \frac{29}{61} a^{12} + \frac{2}{61} a^{11} + \frac{2}{61} a^{10} + \frac{12}{61} a^{9} - \frac{4}{61} a^{8} - \frac{15}{61} a^{7} + \frac{29}{61} a^{6} - \frac{28}{61} a^{5} - \frac{1}{61} a^{4} + \frac{23}{61} a^{2} - \frac{19}{61} a + \frac{18}{61}$, $\frac{1}{3108729163186258474746727} a^{19} - \frac{576462395225890350074}{3108729163186258474746727} a^{18} + \frac{53404725210364453097607}{3108729163186258474746727} a^{17} + \frac{1228503420205399548160424}{3108729163186258474746727} a^{16} + \frac{1275208971855337328793807}{3108729163186258474746727} a^{15} + \frac{759433653843315546473701}{3108729163186258474746727} a^{14} - \frac{188362098058594536752971}{3108729163186258474746727} a^{13} - \frac{619083052272126922493517}{3108729163186258474746727} a^{12} + \frac{1104505517575787559573970}{3108729163186258474746727} a^{11} - \frac{1081905466274367616166913}{3108729163186258474746727} a^{10} - \frac{646511737495988162515433}{3108729163186258474746727} a^{9} - \frac{1335752139401584111445665}{3108729163186258474746727} a^{8} - \frac{842037658339129893950119}{3108729163186258474746727} a^{7} - \frac{350675841565057057505010}{3108729163186258474746727} a^{6} + \frac{1266413003118819101605482}{3108729163186258474746727} a^{5} - \frac{872090196623470448256621}{3108729163186258474746727} a^{4} - \frac{281265753207327241840534}{3108729163186258474746727} a^{3} - \frac{909802613453061273423244}{3108729163186258474746727} a^{2} + \frac{132334653551778933230673}{3108729163186258474746727} a - \frac{1475021744750169551355995}{3108729163186258474746727}$

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:  \( 80372121.2014 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T647:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 51200
The 152 conjugacy class representatives for t20n647 are not computed
Character table for t20n647 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 10.10.6323239406592.1

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 R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
2Data not computed
3Data not computed
$71$71.10.9.5$x^{10} - 18176$$10$$1$$9$$C_{10}$$[\ ]_{10}$
71.10.0.1$x^{10} - x + 22$$1$$10$$0$$C_{10}$$[\ ]^{10}$