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Properties

Label 20.0.45014308264...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{14}\cdot 269^{9}$
Root discriminant $38.25$
Ramified primes $5, 269$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T375

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2624639, -869298, 4404481, -1278310, 3056061, -738184, 1342246, -275664, 410478, -78690, 101911, -19773, 20397, -4419, 3340, -710, 423, -69, 32, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 32*x^18 - 69*x^17 + 423*x^16 - 710*x^15 + 3340*x^14 - 4419*x^13 + 20397*x^12 - 19773*x^11 + 101911*x^10 - 78690*x^9 + 410478*x^8 - 275664*x^7 + 1342246*x^6 - 738184*x^5 + 3056061*x^4 - 1278310*x^3 + 4404481*x^2 - 869298*x + 2624639)
 
gp: K = bnfinit(x^20 - 3*x^19 + 32*x^18 - 69*x^17 + 423*x^16 - 710*x^15 + 3340*x^14 - 4419*x^13 + 20397*x^12 - 19773*x^11 + 101911*x^10 - 78690*x^9 + 410478*x^8 - 275664*x^7 + 1342246*x^6 - 738184*x^5 + 3056061*x^4 - 1278310*x^3 + 4404481*x^2 - 869298*x + 2624639, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 32 x^{18} - 69 x^{17} + 423 x^{16} - 710 x^{15} + 3340 x^{14} - 4419 x^{13} + 20397 x^{12} - 19773 x^{11} + 101911 x^{10} - 78690 x^{9} + 410478 x^{8} - 275664 x^{7} + 1342246 x^{6} - 738184 x^{5} + 3056061 x^{4} - 1278310 x^{3} + 4404481 x^{2} - 869298 x + 2624639 \)

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:  \(45014308264858198791680908203125=5^{14}\cdot 269^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.25$
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:  $5, 269$
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}{23178771106883275563146144325377706799522619628640342652931551} a^{19} + \frac{9111916009151806518346155868579140786774770125629597886518884}{23178771106883275563146144325377706799522619628640342652931551} a^{18} - \frac{5167832901566740696830857696787330039356524984574024738432070}{23178771106883275563146144325377706799522619628640342652931551} a^{17} + \frac{7710529545295447282792734074903904984342125645385028150959425}{23178771106883275563146144325377706799522619628640342652931551} a^{16} + \frac{3081470168892588638366478655120386048923333421992720938058408}{23178771106883275563146144325377706799522619628640342652931551} a^{15} - \frac{9690363696099754963533018419610750305447370481930517867066383}{23178771106883275563146144325377706799522619628640342652931551} a^{14} - \frac{7423001380651316902378355297954619666544308273079811137851661}{23178771106883275563146144325377706799522619628640342652931551} a^{13} + \frac{3648344278273990111204971877149713160900253699182522691291966}{23178771106883275563146144325377706799522619628640342652931551} a^{12} - \frac{692925477149768134262120434670080299943399415320531164006067}{23178771106883275563146144325377706799522619628640342652931551} a^{11} - \frac{10333400923367084359183744426953101552438664690068929248620215}{23178771106883275563146144325377706799522619628640342652931551} a^{10} - \frac{5320564735760444555458713001111248013327032583207307201554928}{23178771106883275563146144325377706799522619628640342652931551} a^{9} - \frac{2329537414428836777918272370043069717991141494840867323359548}{23178771106883275563146144325377706799522619628640342652931551} a^{8} - \frac{10084067043325145647080513605519758042677498383544967222617443}{23178771106883275563146144325377706799522619628640342652931551} a^{7} - \frac{5109442305291520885946223331420366005029997429568670730006054}{23178771106883275563146144325377706799522619628640342652931551} a^{6} + \frac{7284411586081279768637957042438711791029729204773081592701522}{23178771106883275563146144325377706799522619628640342652931551} a^{5} - \frac{5425616353738550478511324993256668877042412872907802154400924}{23178771106883275563146144325377706799522619628640342652931551} a^{4} - \frac{6703137393049801345398573369316342095974339201614315923486228}{23178771106883275563146144325377706799522619628640342652931551} a^{3} + \frac{1810308693814705940707741603520589379710522267391716080630524}{23178771106883275563146144325377706799522619628640342652931551} a^{2} - \frac{126362316315425152262645466260523470650245114846572147655694}{23178771106883275563146144325377706799522619628640342652931551} a - \frac{4411236275579320621820174728715871673269313791698512543353235}{23178771106883275563146144325377706799522619628640342652931551}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -1 \) (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:  \( 13225817.0369 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T375:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 48 conjugacy class representatives for t20n375
Character table for t20n375 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.1.33625.1, 10.2.5653203125.1

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 $20$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$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}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
269Data not computed