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Properties

Label 20.12.2066353575...3125.1
Degree $20$
Signature $[12, 4]$
Discriminant $3^{6}\cdot 5^{15}\cdot 23^{6}\cdot 89^{4}$
Root discriminant $29.23$
Ramified primes $3, 5, 23, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T802

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-179, 1625, -5595, 8715, -3390, -8921, 14195, -5810, -5215, 7465, -2884, -925, 1475, -705, 125, 84, -95, 40, 0, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 40*x^17 - 95*x^16 + 84*x^15 + 125*x^14 - 705*x^13 + 1475*x^12 - 925*x^11 - 2884*x^10 + 7465*x^9 - 5215*x^8 - 5810*x^7 + 14195*x^6 - 8921*x^5 - 3390*x^4 + 8715*x^3 - 5595*x^2 + 1625*x - 179)
 
gp: K = bnfinit(x^20 - 5*x^19 + 40*x^17 - 95*x^16 + 84*x^15 + 125*x^14 - 705*x^13 + 1475*x^12 - 925*x^11 - 2884*x^10 + 7465*x^9 - 5215*x^8 - 5810*x^7 + 14195*x^6 - 8921*x^5 - 3390*x^4 + 8715*x^3 - 5595*x^2 + 1625*x - 179, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 40 x^{17} - 95 x^{16} + 84 x^{15} + 125 x^{14} - 705 x^{13} + 1475 x^{12} - 925 x^{11} - 2884 x^{10} + 7465 x^{9} - 5215 x^{8} - 5810 x^{7} + 14195 x^{6} - 8921 x^{5} - 3390 x^{4} + 8715 x^{3} - 5595 x^{2} + 1625 x - 179 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(206635357553265518829345703125=3^{6}\cdot 5^{15}\cdot 23^{6}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.23$
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:  $3, 5, 23, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} + \frac{2}{5} a^{8} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{3} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{25} a^{14} - \frac{1}{25} a^{13} + \frac{1}{25} a^{12} - \frac{1}{25} a^{11} + \frac{1}{25} a^{10} + \frac{12}{25} a^{9} + \frac{3}{25} a^{8} - \frac{8}{25} a^{7} - \frac{12}{25} a^{6} - \frac{3}{25} a^{5} - \frac{4}{25} a^{4} - \frac{6}{25} a^{3} - \frac{4}{25} a^{2} - \frac{6}{25} a + \frac{1}{25}$, $\frac{1}{25} a^{15} - \frac{12}{25} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{7}{25} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{25}$, $\frac{1}{125} a^{16} + \frac{1}{125} a^{15} - \frac{37}{125} a^{11} + \frac{28}{125} a^{10} + \frac{12}{25} a^{9} + \frac{2}{5} a^{8} + \frac{8}{25} a^{7} + \frac{3}{125} a^{6} + \frac{33}{125} a^{5} + \frac{11}{25} a^{4} + \frac{1}{25} a^{3} + \frac{7}{25} a^{2} + \frac{21}{125} a - \frac{49}{125}$, $\frac{1}{125} a^{17} - \frac{1}{125} a^{15} - \frac{12}{125} a^{12} - \frac{2}{25} a^{11} + \frac{57}{125} a^{10} - \frac{2}{25} a^{9} - \frac{2}{25} a^{8} + \frac{13}{125} a^{7} + \frac{1}{25} a^{6} - \frac{53}{125} a^{5} - \frac{2}{5} a^{4} + \frac{6}{25} a^{3} + \frac{11}{125} a^{2} - \frac{4}{25} a - \frac{51}{125}$, $\frac{1}{625} a^{18} - \frac{2}{625} a^{17} - \frac{2}{625} a^{16} + \frac{1}{625} a^{15} - \frac{37}{625} a^{13} + \frac{14}{625} a^{12} - \frac{186}{625} a^{11} + \frac{23}{625} a^{10} - \frac{2}{25} a^{9} + \frac{183}{625} a^{8} - \frac{311}{625} a^{7} - \frac{166}{625} a^{6} - \frac{127}{625} a^{5} + \frac{8}{25} a^{4} + \frac{46}{625} a^{3} - \frac{77}{625} a^{2} + \frac{293}{625} a + \frac{76}{625}$, $\frac{1}{197085892431875} a^{19} + \frac{15463177969}{197085892431875} a^{18} + \frac{568199185651}{197085892431875} a^{17} - \frac{560504781901}{197085892431875} a^{16} + \frac{190070223991}{197085892431875} a^{15} - \frac{2287738387812}{197085892431875} a^{14} + \frac{1097388934237}{197085892431875} a^{13} + \frac{15898639431368}{197085892431875} a^{12} + \frac{83897122995012}{197085892431875} a^{11} - \frac{61242990007982}{197085892431875} a^{10} - \frac{39574608605967}{197085892431875} a^{9} - \frac{56272354646518}{197085892431875} a^{8} - \frac{32781292216737}{197085892431875} a^{7} - \frac{45891039853043}{197085892431875} a^{6} - \frac{4920044219907}{197085892431875} a^{5} + \frac{1695799552671}{197085892431875} a^{4} + \frac{10633127370039}{197085892431875} a^{3} + \frac{81162062657071}{197085892431875} a^{2} + \frac{94534391994419}{197085892431875} a - \frac{80684233589834}{197085892431875}$

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:  $15$
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:  \( 31103723.1775 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T802:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 138 conjugacy class representatives for t20n802 are not computed
Character table for t20n802 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.767625.1, 10.10.2946240703125.1

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 $20$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.12.0.1$x^{12} - x^{4} - x^{3} - x^{2} + x - 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
5Data not computed
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$89$89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{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}$