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Properties

Label 20.6.59313783170...0256.3
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{20}\cdot 3^{19}\cdot 13^{5}\cdot 107^{4}$
Root discriminant $27.46$
Ramified primes $2, 3, 13, 107$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1023

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -2, -10, 23, -28, 23, 32, -22, 38, -25, 38, -22, 32, 23, -28, 23, -10, -2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 2*x^18 - 10*x^17 + 23*x^16 - 28*x^15 + 23*x^14 + 32*x^13 - 22*x^12 + 38*x^11 - 25*x^10 + 38*x^9 - 22*x^8 + 32*x^7 + 23*x^6 - 28*x^5 + 23*x^4 - 10*x^3 - 2*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - 2*x^18 - 10*x^17 + 23*x^16 - 28*x^15 + 23*x^14 + 32*x^13 - 22*x^12 + 38*x^11 - 25*x^10 + 38*x^9 - 22*x^8 + 32*x^7 + 23*x^6 - 28*x^5 + 23*x^4 - 10*x^3 - 2*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 2 x^{18} - 10 x^{17} + 23 x^{16} - 28 x^{15} + 23 x^{14} + 32 x^{13} - 22 x^{12} + 38 x^{11} - 25 x^{10} + 38 x^{9} - 22 x^{8} + 32 x^{7} + 23 x^{6} - 28 x^{5} + 23 x^{4} - 10 x^{3} - 2 x^{2} - 2 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-59313783170950766384309600256=-\,2^{20}\cdot 3^{19}\cdot 13^{5}\cdot 107^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.46$
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:  $2, 3, 13, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{45} a^{16} + \frac{1}{9} a^{15} - \frac{1}{15} a^{14} + \frac{2}{15} a^{13} - \frac{2}{5} a^{12} - \frac{7}{15} a^{11} - \frac{16}{45} a^{10} + \frac{4}{45} a^{9} + \frac{1}{15} a^{8} + \frac{19}{45} a^{7} - \frac{1}{45} a^{6} - \frac{7}{15} a^{5} - \frac{2}{5} a^{4} + \frac{2}{15} a^{3} - \frac{1}{15} a^{2} - \frac{2}{9} a - \frac{14}{45}$, $\frac{1}{135} a^{17} + \frac{1}{135} a^{16} + \frac{22}{135} a^{15} + \frac{2}{15} a^{14} - \frac{14}{45} a^{13} - \frac{13}{45} a^{12} - \frac{67}{135} a^{11} - \frac{22}{135} a^{10} - \frac{58}{135} a^{9} + \frac{7}{135} a^{8} + \frac{58}{135} a^{7} - \frac{62}{135} a^{6} - \frac{8}{45} a^{5} + \frac{11}{45} a^{4} + \frac{2}{15} a^{3} + \frac{2}{135} a^{2} + \frac{26}{135} a - \frac{34}{135}$, $\frac{1}{2868345} a^{18} - \frac{584}{573669} a^{17} - \frac{20737}{2868345} a^{16} + \frac{65761}{2868345} a^{15} - \frac{31472}{956115} a^{14} + \frac{19697}{63741} a^{13} + \frac{737336}{2868345} a^{12} - \frac{1237772}{2868345} a^{11} - \frac{547463}{2868345} a^{10} - \frac{49416}{106235} a^{9} - \frac{334993}{2868345} a^{8} + \frac{568223}{2868345} a^{7} + \frac{206161}{2868345} a^{6} - \frac{1550}{63741} a^{5} - \frac{456412}{956115} a^{4} + \frac{596936}{2868345} a^{3} + \frac{85498}{2868345} a^{2} - \frac{43078}{573669} a + \frac{212471}{2868345}$, $\frac{1}{8605035} a^{19} - \frac{1}{8605035} a^{18} + \frac{2036}{956115} a^{17} - \frac{81568}{8605035} a^{16} + \frac{132038}{1721007} a^{15} - \frac{15364}{191223} a^{14} + \frac{3721667}{8605035} a^{13} - \frac{1997486}{8605035} a^{12} + \frac{736843}{2868345} a^{11} + \frac{1901426}{8605035} a^{10} - \frac{340607}{8605035} a^{9} + \frac{1324619}{2868345} a^{8} - \frac{3622417}{8605035} a^{7} - \frac{2975411}{8605035} a^{6} - \frac{17833}{106235} a^{5} + \frac{3162356}{8605035} a^{4} + \frac{1838713}{8605035} a^{3} - \frac{82661}{191223} a^{2} - \frac{492113}{8605035} a + \frac{1320233}{8605035}$

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:  $12$
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:  \( 15118132.1444 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1023:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 324 conjugacy class representatives for t20n1023 are not computed
Character table for t20n1023 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 10.6.38998285028352.1

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 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$3$3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.12.13.6$x^{12} + 3 x^{11} + 3 x^{7} + 3 x^{6} + 3 x^{4} + 3 x^{2} - 3$$12$$1$$13$12T35$[5/4, 5/4]_{4}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.5.1$x^{6} - 52$$6$$1$$5$$C_6$$[\ ]_{6}$
$107$$\Q_{107}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{107}$$x + 3$$1$$1$$0$Trivial$[\ ]$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
107.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
107.6.4.1$x^{6} + 1498 x^{3} + 1431125$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$