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Properties

Label 20.0.16494274714...1088.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{24}\cdot 3^{10}\cdot 7^{4}\cdot 37^{5}$
Root discriminant $14.48$
Ramified primes $2, 3, 7, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1036

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 16, 32, 12, -70, -124, -22, 118, 127, 46, -54, -103, -49, 24, 41, 12, 1, -9, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - x^18 - 9*x^17 + x^16 + 12*x^15 + 41*x^14 + 24*x^13 - 49*x^12 - 103*x^11 - 54*x^10 + 46*x^9 + 127*x^8 + 118*x^7 - 22*x^6 - 124*x^5 - 70*x^4 + 12*x^3 + 32*x^2 + 16*x + 4)
 
gp: K = bnfinit(x^20 - x^19 - x^18 - 9*x^17 + x^16 + 12*x^15 + 41*x^14 + 24*x^13 - 49*x^12 - 103*x^11 - 54*x^10 + 46*x^9 + 127*x^8 + 118*x^7 - 22*x^6 - 124*x^5 - 70*x^4 + 12*x^3 + 32*x^2 + 16*x + 4, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - x^{18} - 9 x^{17} + x^{16} + 12 x^{15} + 41 x^{14} + 24 x^{13} - 49 x^{12} - 103 x^{11} - 54 x^{10} + 46 x^{9} + 127 x^{8} + 118 x^{7} - 22 x^{6} - 124 x^{5} - 70 x^{4} + 12 x^{3} + 32 x^{2} + 16 x + 4 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(164942747145088782041088=2^{24}\cdot 3^{10}\cdot 7^{4}\cdot 37^{5}\)
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:  $2, 3, 7, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{674} a^{18} - \frac{11}{674} a^{17} - \frac{83}{337} a^{16} - \frac{21}{337} a^{15} - \frac{49}{337} a^{14} + \frac{73}{674} a^{13} + \frac{156}{337} a^{12} + \frac{41}{337} a^{11} - \frac{30}{337} a^{10} + \frac{189}{674} a^{9} + \frac{65}{674} a^{8} - \frac{7}{674} a^{7} + \frac{183}{674} a^{6} - \frac{62}{337} a^{5} + \frac{95}{674} a^{4} - \frac{123}{337} a^{2} - \frac{112}{337} a - \frac{87}{337}$, $\frac{1}{371767492658} a^{19} - \frac{79181380}{185883746329} a^{18} + \frac{11402511605}{371767492658} a^{17} - \frac{8195885315}{371767492658} a^{16} + \frac{9992986335}{371767492658} a^{15} + \frac{29165331807}{185883746329} a^{14} + \frac{12193677565}{185883746329} a^{13} - \frac{23475217579}{371767492658} a^{12} + \frac{26203645394}{185883746329} a^{11} + \frac{20363171375}{185883746329} a^{10} + \frac{72196177832}{185883746329} a^{9} - \frac{2077598561}{371767492658} a^{8} - \frac{31053558715}{371767492658} a^{7} - \frac{93189839735}{371767492658} a^{6} - \frac{58002168213}{371767492658} a^{5} + \frac{24443091410}{185883746329} a^{4} - \frac{10619091869}{185883746329} a^{3} - \frac{4074582259}{185883746329} a^{2} - \frac{36303647162}{185883746329} a + \frac{44925167669}{185883746329}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{302111}{1116818} a^{19} - \frac{150572}{558409} a^{18} - \frac{74272}{558409} a^{17} - \frac{2805305}{1116818} a^{16} + \frac{179313}{1116818} a^{15} + \frac{1087476}{558409} a^{14} + \frac{5910996}{558409} a^{13} + \frac{12026}{1657} a^{12} - \frac{8632973}{1116818} a^{11} - \frac{24415003}{1116818} a^{10} - \frac{17412213}{1116818} a^{9} + \frac{1866605}{1116818} a^{8} + \frac{12766778}{558409} a^{7} + \frac{29708577}{1116818} a^{6} + \frac{935338}{558409} a^{5} - \frac{20026961}{1116818} a^{4} - \frac{6109484}{558409} a^{3} - \frac{421624}{558409} a^{2} + \frac{1764582}{558409} a + \frac{964792}{558409} \) (order $6$)
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:  \( 7879.29030942 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1036:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 396 conjugacy class representatives for t20n1036 are not computed
Character table for t20n1036 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 10.0.16691899392.1

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 $20$ R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.16.10$x^{8} + 2 x^{6} + 4 x^{5} + 6 x^{4} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
2.12.8.2$x^{12} - 8 x^{3} + 16$$3$$4$$8$$C_3\times (C_3 : C_4)$$[\ ]_{3}^{12}$
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.8.0.1$x^{8} - x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
$37$37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.3.2.3$x^{3} - 148$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.3$x^{3} - 148$$3$$1$$2$$C_3$$[\ ]_{3}$
37.8.0.1$x^{8} - x + 18$$1$$8$$0$$C_8$$[\ ]^{8}$