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Properties

Label 20.8.13390863387...0625.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{10}\cdot 1039^{2}\cdot 1049^{5}$
Root discriminant $25.49$
Ramified primes $5, 1039, 1049$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![89, -158, -920, -543, -1088, -757, -578, -213, 1251, 690, 67, -1030, -364, 69, 146, 96, -7, -1, -7, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 7*x^18 - x^17 - 7*x^16 + 96*x^15 + 146*x^14 + 69*x^13 - 364*x^12 - 1030*x^11 + 67*x^10 + 690*x^9 + 1251*x^8 - 213*x^7 - 578*x^6 - 757*x^5 - 1088*x^4 - 543*x^3 - 920*x^2 - 158*x + 89)
 
gp: K = bnfinit(x^20 - 2*x^19 - 7*x^18 - x^17 - 7*x^16 + 96*x^15 + 146*x^14 + 69*x^13 - 364*x^12 - 1030*x^11 + 67*x^10 + 690*x^9 + 1251*x^8 - 213*x^7 - 578*x^6 - 757*x^5 - 1088*x^4 - 543*x^3 - 920*x^2 - 158*x + 89, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 7 x^{18} - x^{17} - 7 x^{16} + 96 x^{15} + 146 x^{14} + 69 x^{13} - 364 x^{12} - 1030 x^{11} + 67 x^{10} + 690 x^{9} + 1251 x^{8} - 213 x^{7} - 578 x^{6} - 757 x^{5} - 1088 x^{4} - 543 x^{3} - 920 x^{2} - 158 x + 89 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13390863387968309382119140625=5^{10}\cdot 1039^{2}\cdot 1049^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.49$
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:  $5, 1039, 1049$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{227269492889674298303560788656792903267} a^{19} - \frac{27080074727411013628596385085206422124}{227269492889674298303560788656792903267} a^{18} + \frac{53788064522889315523849349736020684919}{227269492889674298303560788656792903267} a^{17} + \frac{112486734414383582252582302356229243154}{227269492889674298303560788656792903267} a^{16} + \frac{56652437267058936763283151442287461184}{227269492889674298303560788656792903267} a^{15} + \frac{2980547013168175168094765382504608368}{7836879065161182700122785815751479423} a^{14} + \frac{95766913719760670483795077932204337580}{227269492889674298303560788656792903267} a^{13} + \frac{111443127500191468808534708631878780367}{227269492889674298303560788656792903267} a^{12} + \frac{76667638792927530884793616540195222661}{227269492889674298303560788656792903267} a^{11} + \frac{154300453688525044507058950546946681}{7331273964183041880760025440541706557} a^{10} + \frac{3609699005975478964762550249491142702}{7836879065161182700122785815751479423} a^{9} - \frac{19809870669022643193893126773568426300}{227269492889674298303560788656792903267} a^{8} + \frac{2389681228659609771806344641522282520}{7836879065161182700122785815751479423} a^{7} + \frac{48542078189125366117358758416913053776}{227269492889674298303560788656792903267} a^{6} - \frac{15497409848998689015247361019736837914}{227269492889674298303560788656792903267} a^{5} - \frac{111922029757201131074584495132807929821}{227269492889674298303560788656792903267} a^{4} + \frac{61284289956320385620760857828479045406}{227269492889674298303560788656792903267} a^{3} - \frac{2458167624403580115166048064362600240}{7331273964183041880760025440541706557} a^{2} - \frac{21403107161814526914753896580618907371}{227269492889674298303560788656792903267} a - \frac{107015602351311984898903901226445595922}{227269492889674298303560788656792903267}$

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:  $13$
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:  \( 2561196.6963 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 115200
The 119 conjugacy class representatives for t20n781 are not computed
Character table for t20n781 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.26225.1, 10.4.3405971875.1

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 24 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
1039Data not computed
1049Data not computed