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Properties

Label 20.4.16447502024...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{28}\cdot 5^{10}\cdot 89^{4}$
Root discriminant $14.48$
Ramified primes $2, 5, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1013

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 43, -110, 199, -272, 250, -58, -273, 598, -763, 716, -521, 300, -138, 46, 1, -18, 15, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 15*x^18 - 18*x^17 + x^16 + 46*x^15 - 138*x^14 + 300*x^13 - 521*x^12 + 716*x^11 - 763*x^10 + 598*x^9 - 273*x^8 - 58*x^7 + 250*x^6 - 272*x^5 + 199*x^4 - 110*x^3 + 43*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^20 - 6*x^19 + 15*x^18 - 18*x^17 + x^16 + 46*x^15 - 138*x^14 + 300*x^13 - 521*x^12 + 716*x^11 - 763*x^10 + 598*x^9 - 273*x^8 - 58*x^7 + 250*x^6 - 272*x^5 + 199*x^4 - 110*x^3 + 43*x^2 - 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 15 x^{18} - 18 x^{17} + x^{16} + 46 x^{15} - 138 x^{14} + 300 x^{13} - 521 x^{12} + 716 x^{11} - 763 x^{10} + 598 x^{9} - 273 x^{8} - 58 x^{7} + 250 x^{6} - 272 x^{5} + 199 x^{4} - 110 x^{3} + 43 x^{2} - 10 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(164475020247040000000000=2^{28}\cdot 5^{10}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.48$
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:  $2, 5, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2888902} a^{19} - \frac{419709}{2888902} a^{18} - \frac{280955}{1444451} a^{17} - \frac{101029}{1444451} a^{16} + \frac{630565}{2888902} a^{15} - \frac{598831}{2888902} a^{14} - \frac{418043}{2888902} a^{13} - \frac{672539}{2888902} a^{12} - \frac{756769}{2888902} a^{11} + \frac{778835}{2888902} a^{10} + \frac{1318983}{2888902} a^{9} - \frac{22415}{61466} a^{8} + \frac{601017}{1444451} a^{7} + \frac{173503}{1444451} a^{6} + \frac{202009}{2888902} a^{5} - \frac{287703}{2888902} a^{4} - \frac{256694}{1444451} a^{3} - \frac{580459}{1444451} a^{2} + \frac{264}{1444451} a + \frac{421130}{1444451}$

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:  \( 5058.99177989 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1013:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3686400
The 114 conjugacy class representatives for t20n1013 are not computed
Character table for t20n1013 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.2.25347200000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: data not computed
Degree 32 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$2.8.16.16$x^{8} + 2 x^{6} + 4 x^{5} + 6 x^{4} + 8 x^{3} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
2.12.12.28$x^{12} - x^{10} + 2 x^{8} - x^{6} - 2 x^{4} + 3 x^{2} + 1$$6$$2$$12$$S_4$$[4/3, 4/3]_{3}^{2}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.3.0.1$x^{3} - x + 7$$1$$3$$0$$C_3$$[\ ]^{3}$
89.3.0.1$x^{3} - x + 7$$1$$3$$0$$C_3$$[\ ]^{3}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.6.4.1$x^{6} + 1513 x^{3} + 1710936$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$