↑

Properties

Label 20.20.3439976290...1072.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{16}\cdot 17^{10}\cdot 53^{13}$
Root discriminant $94.80$
Ramified primes $2, 17, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2:F_5$ (as 20T19)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-152, -11276, -34480, 334262, -347964, -745555, 1205290, 390998, -1261035, 126754, 567584, -154754, -123202, 46828, 12626, -6424, -424, 413, -16, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 - 16*x^18 + 413*x^17 - 424*x^16 - 6424*x^15 + 12626*x^14 + 46828*x^13 - 123202*x^12 - 154754*x^11 + 567584*x^10 + 126754*x^9 - 1261035*x^8 + 390998*x^7 + 1205290*x^6 - 745555*x^5 - 347964*x^4 + 334262*x^3 - 34480*x^2 - 11276*x - 152)
 
gp: K = bnfinit(x^20 - 10*x^19 - 16*x^18 + 413*x^17 - 424*x^16 - 6424*x^15 + 12626*x^14 + 46828*x^13 - 123202*x^12 - 154754*x^11 + 567584*x^10 + 126754*x^9 - 1261035*x^8 + 390998*x^7 + 1205290*x^6 - 745555*x^5 - 347964*x^4 + 334262*x^3 - 34480*x^2 - 11276*x - 152, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} - 16 x^{18} + 413 x^{17} - 424 x^{16} - 6424 x^{15} + 12626 x^{14} + 46828 x^{13} - 123202 x^{12} - 154754 x^{11} + 567584 x^{10} + 126754 x^{9} - 1261035 x^{8} + 390998 x^{7} + 1205290 x^{6} - 745555 x^{5} - 347964 x^{4} + 334262 x^{3} - 34480 x^{2} - 11276 x - 152 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3439976290039196425657587129823518851072=2^{16}\cdot 17^{10}\cdot 53^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $94.80$
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:  $2, 17, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7}$, $\frac{1}{34} a^{14} - \frac{3}{17} a^{13} - \frac{3}{34} a^{12} - \frac{2}{17} a^{11} - \frac{7}{34} a^{10} - \frac{1}{17} a^{9} + \frac{5}{34} a^{8} + \frac{7}{17} a^{7} + \frac{13}{34} a^{6} + \frac{6}{17} a^{5} + \frac{9}{34} a^{4} - \frac{7}{17} a^{3} + \frac{2}{17} a^{2} - \frac{4}{17} a + \frac{2}{17}$, $\frac{1}{34} a^{15} - \frac{5}{34} a^{13} - \frac{5}{34} a^{12} + \frac{3}{34} a^{11} + \frac{7}{34} a^{10} - \frac{7}{34} a^{9} - \frac{7}{34} a^{8} - \frac{5}{34} a^{7} + \frac{5}{34} a^{6} + \frac{13}{34} a^{5} - \frac{11}{34} a^{4} - \frac{6}{17} a^{3} - \frac{1}{34} a^{2} - \frac{5}{17} a - \frac{5}{17}$, $\frac{1}{10812} a^{16} - \frac{14}{2703} a^{15} - \frac{31}{5406} a^{14} - \frac{431}{5406} a^{13} - \frac{185}{1802} a^{12} - \frac{1301}{5406} a^{11} + \frac{17}{159} a^{10} + \frac{275}{5406} a^{9} + \frac{9}{53} a^{8} - \frac{451}{1802} a^{7} + \frac{1111}{5406} a^{6} - \frac{1111}{5406} a^{5} - \frac{385}{10812} a^{4} - \frac{2351}{5406} a^{3} - \frac{1417}{5406} a^{2} + \frac{1229}{2703} a - \frac{767}{2703}$, $\frac{1}{183804} a^{17} - \frac{2}{45951} a^{16} + \frac{505}{45951} a^{15} + \frac{233}{45951} a^{14} + \frac{1998}{15317} a^{13} - \frac{14585}{91902} a^{12} + \frac{11429}{91902} a^{11} + \frac{989}{91902} a^{10} - \frac{2449}{30634} a^{9} + \frac{3170}{15317} a^{8} + \frac{440}{45951} a^{7} - \frac{7567}{91902} a^{6} + \frac{55457}{183804} a^{5} - \frac{27173}{91902} a^{4} + \frac{7529}{91902} a^{3} + \frac{20804}{45951} a^{2} + \frac{7027}{45951} a + \frac{6967}{15317}$, $\frac{1}{457590167220} a^{18} - \frac{8669}{5383413732} a^{17} - \frac{491311}{152530055740} a^{16} + \frac{3316167929}{228795083610} a^{15} + \frac{2291996431}{228795083610} a^{14} + \frac{586682391}{7626502787} a^{13} + \frac{56974812431}{228795083610} a^{12} - \frac{32004336671}{228795083610} a^{11} + \frac{9698362676}{114397541805} a^{10} - \frac{2458217375}{22879508361} a^{9} - \frac{15819213436}{114397541805} a^{8} + \frac{2643675653}{6729267165} a^{7} + \frac{1094677171}{152530055740} a^{6} - \frac{6064417189}{91518033444} a^{5} + \frac{217384275749}{457590167220} a^{4} + \frac{579221173}{15253005574} a^{3} + \frac{26981509313}{76265027870} a^{2} - \frac{21799242812}{114397541805} a - \frac{34418729554}{114397541805}$, $\frac{1}{105230637985081740} a^{19} - \frac{27841}{35076879328360580} a^{18} - \frac{90748594069}{52615318992540870} a^{17} - \frac{651626679497}{26307659496270435} a^{16} - \frac{723512644594351}{52615318992540870} a^{15} + \frac{3417540442007}{1031672921422370} a^{14} + \frac{3918193929534257}{17538439664180290} a^{13} + \frac{6187868730149873}{26307659496270435} a^{12} - \frac{64510698667288}{5261531899254087} a^{11} + \frac{176637400557779}{1031672921422370} a^{10} + \frac{33621007182199}{26307659496270435} a^{9} + \frac{402321141575924}{26307659496270435} a^{8} - \frac{12700964767773473}{35076879328360580} a^{7} - \frac{32638908782946289}{105230637985081740} a^{6} - \frac{7908924159800171}{17538439664180290} a^{5} - \frac{21329192433552421}{52615318992540870} a^{4} + \frac{5513814564725279}{52615318992540870} a^{3} - \frac{1932397152129087}{17538439664180290} a^{2} + \frac{9313528678639952}{26307659496270435} a - \frac{2094561465722368}{26307659496270435}$

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:  $19$
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:  \( 301584210044000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:F_5$ (as 20T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 14 conjugacy class representatives for $C_2^2:F_5$
Character table for $C_2^2:F_5$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.15317.1, 5.5.2382032.1, 10.10.8056377164681869568.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$53$53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$