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Properties

Label 20.0.16306500045...8624.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{10}\cdot 71^{6}\cdot 179^{6}$
Root discriminant $51.36$
Ramified primes $2, 3, 71, 179$
Class number $528$ (GRH)
Class group $[2, 264]$ (GRH)
Galois group 20T199

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -48, 240, -448, 1776, -3128, 7176, -6720, 8920, -5712, 6696, -3320, 3032, -964, 836, -214, 152, -24, 16, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 16*x^18 - 24*x^17 + 152*x^16 - 214*x^15 + 836*x^14 - 964*x^13 + 3032*x^12 - 3320*x^11 + 6696*x^10 - 5712*x^9 + 8920*x^8 - 6720*x^7 + 7176*x^6 - 3128*x^5 + 1776*x^4 - 448*x^3 + 240*x^2 - 48*x + 16)
 
gp: K = bnfinit(x^20 - 2*x^19 + 16*x^18 - 24*x^17 + 152*x^16 - 214*x^15 + 836*x^14 - 964*x^13 + 3032*x^12 - 3320*x^11 + 6696*x^10 - 5712*x^9 + 8920*x^8 - 6720*x^7 + 7176*x^6 - 3128*x^5 + 1776*x^4 - 448*x^3 + 240*x^2 - 48*x + 16, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 16 x^{18} - 24 x^{17} + 152 x^{16} - 214 x^{15} + 836 x^{14} - 964 x^{13} + 3032 x^{12} - 3320 x^{11} + 6696 x^{10} - 5712 x^{9} + 8920 x^{8} - 6720 x^{7} + 7176 x^{6} - 3128 x^{5} + 1776 x^{4} - 448 x^{3} + 240 x^{2} - 48 x + 16 \)

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:  \(16306500045643672482006601944858624=2^{16}\cdot 3^{10}\cdot 71^{6}\cdot 179^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.36$
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:  $2, 3, 71, 179$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{4} a^{12}$, $\frac{1}{4} a^{13}$, $\frac{1}{4} a^{14}$, $\frac{1}{8} a^{15}$, $\frac{1}{8} a^{16}$, $\frac{1}{8} a^{17}$, $\frac{1}{8} a^{18}$, $\frac{1}{977485293114559186544} a^{19} + \frac{444513091035120545}{488742646557279593272} a^{18} - \frac{12965076484335224269}{488742646557279593272} a^{17} + \frac{15023419419417436129}{488742646557279593272} a^{16} + \frac{4296051786061501733}{488742646557279593272} a^{15} + \frac{2176568583138696939}{61092830819659949159} a^{14} - \frac{3694261642591790493}{122185661639319898318} a^{13} - \frac{19379156574198114425}{244371323278639796636} a^{12} - \frac{22898734415389331283}{244371323278639796636} a^{11} - \frac{6019716318038976051}{122185661639319898318} a^{10} + \frac{11561361298889397951}{122185661639319898318} a^{9} + \frac{20317982719569830483}{122185661639319898318} a^{8} - \frac{6641082331080727160}{61092830819659949159} a^{7} - \frac{18607166495056702525}{122185661639319898318} a^{6} + \frac{24475791553205961191}{122185661639319898318} a^{5} + \frac{36106186192468062871}{122185661639319898318} a^{4} + \frac{7137435760644790135}{61092830819659949159} a^{3} + \frac{9167865847799348557}{61092830819659949159} a^{2} - \frac{9751543420303425119}{61092830819659949159} a + \frac{18537436950085663666}{61092830819659949159}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{264}$, which has order $528$ (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:  \( \frac{19379458600911261787}{977485293114559186544} a^{19} - \frac{18287275933837406421}{488742646557279593272} a^{18} + \frac{152178323311954249501}{488742646557279593272} a^{17} - \frac{213362447195349149351}{488742646557279593272} a^{16} + \frac{717623439010512918601}{244371323278639796636} a^{15} - \frac{942945860024724412393}{244371323278639796636} a^{14} + \frac{3879037477022392929085}{244371323278639796636} a^{13} - \frac{2058140265895828810529}{122185661639319898318} a^{12} + \frac{6926934250257708455505}{122185661639319898318} a^{11} - \frac{6986342270477713099243}{122185661639319898318} a^{10} + \frac{7374801562458088486841}{61092830819659949159} a^{9} - \frac{11224200156616557910307}{122185661639319898318} a^{8} + \frac{18745926098495440685385}{122185661639319898318} a^{7} - \frac{6224045495233539939203}{61092830819659949159} a^{6} + \frac{6922178072459874911635}{61092830819659949159} a^{5} - \frac{4005287841154121338871}{122185661639319898318} a^{4} + \frac{1027497761772304659912}{61092830819659949159} a^{3} + \frac{127497081178805214035}{61092830819659949159} a^{2} + \frac{126289756178889682990}{61092830819659949159} a + \frac{36744016867698072232}{61092830819659949159} \) (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:  \( 8563512.73108 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T199:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1440
The 22 conjugacy class representatives for t20n199
Character table for t20n199 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 10.10.525501674708224.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 20 siblings: data not computed
Degree 24 siblings: data not computed
Degree 30 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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$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}$
$71$71.4.2.2$x^{4} - 71 x^{2} + 55451$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
71.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
71.4.2.2$x^{4} - 71 x^{2} + 55451$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$179$179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.6.0.1$x^{6} - x + 50$$1$$6$$0$$C_6$$[\ ]^{6}$
179.12.6.1$x^{12} + 573533900 x^{6} - 183765996899 x^{2} + 82235283612302500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$