↑

Properties

Label 20.0.66102413089...6037.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 7^{15}\cdot 11^{9}$
Root discriminant $21.93$
Ramified primes $3, 7, 11$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_5\times C_5:D_4$ (as 20T53)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5671, 1658, 32779, 7840, 13317, -13069, 5165, -10579, 5856, -1159, 325, 33, 491, -322, 270, -88, -46, 16, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 16*x^17 - 46*x^16 - 88*x^15 + 270*x^14 - 322*x^13 + 491*x^12 + 33*x^11 + 325*x^10 - 1159*x^9 + 5856*x^8 - 10579*x^7 + 5165*x^6 - 13069*x^5 + 13317*x^4 + 7840*x^3 + 32779*x^2 + 1658*x + 5671)
 
gp: K = bnfinit(x^20 - 2*x^19 + 16*x^17 - 46*x^16 - 88*x^15 + 270*x^14 - 322*x^13 + 491*x^12 + 33*x^11 + 325*x^10 - 1159*x^9 + 5856*x^8 - 10579*x^7 + 5165*x^6 - 13069*x^5 + 13317*x^4 + 7840*x^3 + 32779*x^2 + 1658*x + 5671, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 16 x^{17} - 46 x^{16} - 88 x^{15} + 270 x^{14} - 322 x^{13} + 491 x^{12} + 33 x^{11} + 325 x^{10} - 1159 x^{9} + 5856 x^{8} - 10579 x^{7} + 5165 x^{6} - 13069 x^{5} + 13317 x^{4} + 7840 x^{3} + 32779 x^{2} + 1658 x + 5671 \)

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:  \(661024130898095931054356037=3^{10}\cdot 7^{15}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.93$
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:  $3, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{37} a^{18} + \frac{4}{37} a^{17} + \frac{15}{37} a^{16} - \frac{4}{37} a^{15} + \frac{17}{37} a^{14} + \frac{13}{37} a^{13} + \frac{10}{37} a^{12} - \frac{9}{37} a^{11} + \frac{14}{37} a^{10} + \frac{13}{37} a^{9} + \frac{18}{37} a^{8} + \frac{16}{37} a^{7} + \frac{18}{37} a^{6} + \frac{4}{37} a^{5} - \frac{5}{37} a^{4} + \frac{5}{37} a^{2} - \frac{11}{37} a - \frac{3}{37}$, $\frac{1}{3287428458108901300610723292348370887079119377333} a^{19} - \frac{21550209849849950552869080609893467747103263165}{3287428458108901300610723292348370887079119377333} a^{18} - \frac{887600938423600339006480192807269811659620965244}{3287428458108901300610723292348370887079119377333} a^{17} - \frac{606547281543176752487542152458334977750857368655}{3287428458108901300610723292348370887079119377333} a^{16} + \frac{1295333345227817832589690094278034202175712553438}{3287428458108901300610723292348370887079119377333} a^{15} + \frac{1323136685212039511535938544039238473202599393199}{3287428458108901300610723292348370887079119377333} a^{14} - \frac{1590008036584122102198049053422358910883020822310}{3287428458108901300610723292348370887079119377333} a^{13} - \frac{1458979009249215820047516285196558809183696366275}{3287428458108901300610723292348370887079119377333} a^{12} + \frac{560634376262600006506244249005726858552964654134}{3287428458108901300610723292348370887079119377333} a^{11} - \frac{696577780813065090778890815769230318941137919646}{3287428458108901300610723292348370887079119377333} a^{10} - \frac{1491194089694642094422676932625234367352509256066}{3287428458108901300610723292348370887079119377333} a^{9} + \frac{168402262875442090702362616122730075497298266012}{3287428458108901300610723292348370887079119377333} a^{8} - \frac{711360849770981084791356890380322093046221682079}{3287428458108901300610723292348370887079119377333} a^{7} + \frac{1342036301615099339799720892026630067042993645396}{3287428458108901300610723292348370887079119377333} a^{6} - \frac{1446032964069637634198073766156622286320562893721}{3287428458108901300610723292348370887079119377333} a^{5} - \frac{1623264022032782026266341181062680804957220544359}{3287428458108901300610723292348370887079119377333} a^{4} + \frac{54226839057931986855263646395853772693528383589}{3287428458108901300610723292348370887079119377333} a^{3} + \frac{1306051674872350054995908337440565140684184075849}{3287428458108901300610723292348370887079119377333} a^{2} + \frac{336440123557833862151483825158681736268066261946}{3287428458108901300610723292348370887079119377333} a - \frac{185885054230347910218992742697794252315890587218}{3287428458108901300610723292348370887079119377333}$

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:  \( 69540.12284114186 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 4.0.33957.1, 10.0.246071287.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ R $20$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{10}$ $20$

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
3Data not computed
7Data not computed
$11$11.10.9.5$x^{10} - 8019$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$