Properties

Label 20.0.25000000000...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{22}$
Root discriminant $11.75$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois Group $C_{10}\times D_5$ (as 20T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -12, 66, -208, 417, -554, 464, -154, -169, 290, -190, 24, 64, -54, 14, 6, -2, -6, 7, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 7*x^18 - 6*x^17 - 2*x^16 + 6*x^15 + 14*x^14 - 54*x^13 + 64*x^12 + 24*x^11 - 190*x^10 + 290*x^9 - 169*x^8 - 154*x^7 + 464*x^6 - 554*x^5 + 417*x^4 - 208*x^3 + 66*x^2 - 12*x + 1)
gp: K = bnfinit(x^20 - 4*x^19 + 7*x^18 - 6*x^17 - 2*x^16 + 6*x^15 + 14*x^14 - 54*x^13 + 64*x^12 + 24*x^11 - 190*x^10 + 290*x^9 - 169*x^8 - 154*x^7 + 464*x^6 - 554*x^5 + 417*x^4 - 208*x^3 + 66*x^2 - 12*x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 4 x^{19} \) \(\mathstrut +\mathstrut 7 x^{18} \) \(\mathstrut -\mathstrut 6 x^{17} \) \(\mathstrut -\mathstrut 2 x^{16} \) \(\mathstrut +\mathstrut 6 x^{15} \) \(\mathstrut +\mathstrut 14 x^{14} \) \(\mathstrut -\mathstrut 54 x^{13} \) \(\mathstrut +\mathstrut 64 x^{12} \) \(\mathstrut +\mathstrut 24 x^{11} \) \(\mathstrut -\mathstrut 190 x^{10} \) \(\mathstrut +\mathstrut 290 x^{9} \) \(\mathstrut -\mathstrut 169 x^{8} \) \(\mathstrut -\mathstrut 154 x^{7} \) \(\mathstrut +\mathstrut 464 x^{6} \) \(\mathstrut -\mathstrut 554 x^{5} \) \(\mathstrut +\mathstrut 417 x^{4} \) \(\mathstrut -\mathstrut 208 x^{3} \) \(\mathstrut +\mathstrut 66 x^{2} \) \(\mathstrut -\mathstrut 12 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(2500000000000000000000=2^{20}\cdot 5^{22}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.75$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 5$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} - \frac{1}{7} a^{15} - \frac{2}{7} a^{14} - \frac{3}{7} a^{13} + \frac{2}{7} a^{11} - \frac{2}{7} a^{9} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{3}{7} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} - \frac{1}{7} a^{3} - \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{17} - \frac{3}{7} a^{15} + \frac{2}{7} a^{14} - \frac{3}{7} a^{13} + \frac{2}{7} a^{12} + \frac{2}{7} a^{11} - \frac{2}{7} a^{10} - \frac{1}{7} a^{9} + \frac{2}{7} a^{8} - \frac{2}{7} a^{7} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} + \frac{3}{7} a^{4} - \frac{1}{7} a^{3} - \frac{2}{7} a^{2} + \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{18} - \frac{1}{7} a^{15} - \frac{2}{7} a^{14} + \frac{2}{7} a^{12} - \frac{3}{7} a^{11} - \frac{1}{7} a^{10} + \frac{3}{7} a^{9} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} + \frac{3}{7} a^{6} - \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{2}{7} a^{3} + \frac{1}{7} a^{2} - \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{22230187} a^{19} + \frac{316742}{22230187} a^{18} - \frac{1248133}{22230187} a^{17} + \frac{72912}{3175741} a^{16} - \frac{5356625}{22230187} a^{15} - \frac{2276879}{22230187} a^{14} + \frac{369525}{3175741} a^{13} + \frac{9538906}{22230187} a^{12} + \frac{10332540}{22230187} a^{11} + \frac{1228161}{3175741} a^{10} - \frac{222459}{22230187} a^{9} - \frac{6008298}{22230187} a^{8} + \frac{9247629}{22230187} a^{7} - \frac{1567752}{22230187} a^{6} - \frac{4433063}{22230187} a^{5} + \frac{8436821}{22230187} a^{4} + \frac{9943944}{22230187} a^{3} + \frac{1386475}{22230187} a^{2} - \frac{109510}{22230187} a - \frac{201610}{3175741}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( \frac{57119}{9107} a^{19} - \frac{26866}{1301} a^{18} + \frac{267410}{9107} a^{17} - \frac{156983}{9107} a^{16} - \frac{218307}{9107} a^{15} + \frac{181352}{9107} a^{14} + \frac{926865}{9107} a^{13} - \frac{2419212}{9107} a^{12} + \frac{1951750}{9107} a^{11} + \frac{2701613}{9107} a^{10} - \frac{1266459}{1301} a^{9} + \frac{10294686}{9107} a^{8} - \frac{2557279}{9107} a^{7} - \frac{10292841}{9107} a^{6} + \frac{19014662}{9107} a^{5} - \frac{18350921}{9107} a^{4} + \frac{11338829}{9107} a^{3} - \frac{4435672}{9107} a^{2} + \frac{1011806}{9107} a - \frac{107027}{9107} \) (order $4$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{18912006}{3175741} a^{19} - \frac{469207801}{22230187} a^{18} + \frac{696786026}{22230187} a^{17} - \frac{428962951}{22230187} a^{16} - \frac{520610511}{22230187} a^{15} + \frac{579232476}{22230187} a^{14} + \frac{2193417843}{22230187} a^{13} - \frac{882894234}{3175741} a^{12} + \frac{5381288298}{22230187} a^{11} + \frac{6240563925}{22230187} a^{10} - \frac{22659122460}{22230187} a^{9} + \frac{27131369652}{22230187} a^{8} - \frac{7737767500}{22230187} a^{7} - \frac{26068255952}{22230187} a^{6} + \frac{49358820704}{22230187} a^{5} - \frac{47734643902}{22230187} a^{4} + \frac{4147553537}{3175741} a^{3} - \frac{10830306582}{22230187} a^{2} + \frac{2158231877}{22230187} a - \frac{186884032}{22230187} \),  \( \frac{699802}{3175741} a^{19} - \frac{22620897}{22230187} a^{18} + \frac{52966113}{22230187} a^{17} - \frac{64451337}{22230187} a^{16} + \frac{17210346}{22230187} a^{15} + \frac{48013919}{22230187} a^{14} + \frac{14226421}{22230187} a^{13} - \frac{326622522}{22230187} a^{12} + \frac{596942336}{22230187} a^{11} - \frac{231490934}{22230187} a^{10} - \frac{154864866}{3175741} a^{9} + \frac{2486417714}{22230187} a^{8} - \frac{2263334792}{22230187} a^{7} - \frac{168318967}{22230187} a^{6} + \frac{3514112691}{22230187} a^{5} - \frac{5152925334}{22230187} a^{4} + \frac{4346126546}{22230187} a^{3} - \frac{2237892887}{22230187} a^{2} + \frac{672207672}{22230187} a - \frac{82141887}{22230187} \),  \( \frac{52251229}{22230187} a^{19} - \frac{20797040}{3175741} a^{18} + \frac{23654941}{3175741} a^{17} - \frac{46149449}{22230187} a^{16} - \frac{238074483}{22230187} a^{15} + \frac{45270500}{22230187} a^{14} + \frac{128707562}{3175741} a^{13} - \frac{1755538609}{22230187} a^{12} + \frac{803192232}{22230187} a^{11} + \frac{3049490289}{22230187} a^{10} - \frac{6612933162}{22230187} a^{9} + \frac{5702780623}{22230187} a^{8} + \frac{1190275056}{22230187} a^{7} - \frac{9241374918}{22230187} a^{6} + \frac{1768124500}{3175741} a^{5} - \frac{1358243418}{3175741} a^{4} + \frac{4498899555}{22230187} a^{3} - \frac{1184570064}{22230187} a^{2} + \frac{189536024}{22230187} a - \frac{32014916}{22230187} \),  \( \frac{8173423}{3175741} a^{19} - \frac{158673093}{22230187} a^{18} + \frac{178147573}{22230187} a^{17} - \frac{41944177}{22230187} a^{16} - \frac{271368593}{22230187} a^{15} + \frac{58237316}{22230187} a^{14} + \frac{141082189}{3175741} a^{13} - \frac{1930109845}{22230187} a^{12} + \frac{835795157}{22230187} a^{11} + \frac{3425910797}{22230187} a^{10} - \frac{7323623117}{22230187} a^{9} + \frac{6156286420}{22230187} a^{8} + \frac{1712114974}{22230187} a^{7} - \frac{10622219810}{22230187} a^{6} + \frac{13771978233}{22230187} a^{5} - \frac{9903826563}{22230187} a^{4} + \frac{3902205409}{22230187} a^{3} - \frac{275817383}{22230187} a^{2} - \frac{49290959}{3175741} a + \frac{78141851}{22230187} \),  \( \frac{9873327}{22230187} a^{19} - \frac{29487821}{22230187} a^{18} + \frac{40806067}{22230187} a^{17} - \frac{26879376}{22230187} a^{16} - \frac{31136725}{22230187} a^{15} + \frac{16848252}{22230187} a^{14} + \frac{147381681}{22230187} a^{13} - \frac{361219735}{22230187} a^{12} + \frac{296694275}{22230187} a^{11} + \frac{405535329}{22230187} a^{10} - \frac{1319217447}{22230187} a^{9} + \frac{1605215200}{22230187} a^{8} - \frac{521419289}{22230187} a^{7} - \frac{1408159952}{22230187} a^{6} + \frac{2897540052}{22230187} a^{5} - \frac{3086956584}{22230187} a^{4} + \frac{302056462}{3175741} a^{3} - \frac{140107556}{3175741} a^{2} + \frac{248327593}{22230187} a - \frac{646188}{3175741} \),  \( \frac{28897571}{22230187} a^{19} - \frac{93912633}{22230187} a^{18} + \frac{120266788}{22230187} a^{17} - \frac{51856058}{22230187} a^{16} - \frac{127394468}{22230187} a^{15} + \frac{11763251}{3175741} a^{14} + \frac{513062146}{22230187} a^{13} - \frac{1174552146}{22230187} a^{12} + \frac{764344487}{22230187} a^{11} + \frac{1613647156}{22230187} a^{10} - \frac{4381793009}{22230187} a^{9} + \frac{4429224405}{22230187} a^{8} - \frac{264181599}{22230187} a^{7} - \frac{5671056780}{22230187} a^{6} + \frac{8837479629}{22230187} a^{5} - \frac{7593446029}{22230187} a^{4} + \frac{4000439329}{22230187} a^{3} - \frac{1125834102}{22230187} a^{2} + \frac{65609254}{22230187} a + \frac{22439191}{22230187} \),  \( \frac{2268820}{22230187} a^{19} + \frac{27086701}{22230187} a^{18} - \frac{85662813}{22230187} a^{17} + \frac{106961244}{22230187} a^{16} - \frac{47741643}{22230187} a^{15} - \frac{146071993}{22230187} a^{14} + \frac{73513845}{22230187} a^{13} + \frac{481426565}{22230187} a^{12} - \frac{156348383}{3175741} a^{11} + \frac{713690426}{22230187} a^{10} + \frac{1585783509}{22230187} a^{9} - \frac{4013240642}{22230187} a^{8} + \frac{3940548769}{22230187} a^{7} + \frac{76547561}{22230187} a^{6} - \frac{5344137224}{22230187} a^{5} + \frac{7842683817}{22230187} a^{4} - \frac{6355450336}{22230187} a^{3} + \frac{3101351468}{22230187} a^{2} - \frac{791964833}{22230187} a + \frac{83600929}{22230187} \),  \( \frac{8133218}{3175741} a^{19} - \frac{199192626}{22230187} a^{18} + \frac{292073094}{22230187} a^{17} - \frac{176480909}{22230187} a^{16} - \frac{218931267}{22230187} a^{15} + \frac{228501197}{22230187} a^{14} + \frac{946872977}{22230187} a^{13} - \frac{2601304366}{22230187} a^{12} + \frac{2203257580}{22230187} a^{11} + \frac{382961071}{3175741} a^{10} - \frac{9497815111}{22230187} a^{9} + \frac{11275402905}{22230187} a^{8} - \frac{3160444828}{22230187} a^{7} - \frac{10788650057}{22230187} a^{6} + \frac{20521079744}{22230187} a^{5} - \frac{19997248903}{22230187} a^{4} + \frac{12492825309}{22230187} a^{3} - \frac{4946627349}{22230187} a^{2} + \frac{1140590520}{22230187} a - \frac{19179999}{3175741} \),  \( \frac{86489267}{22230187} a^{19} - \frac{279494459}{22230187} a^{18} + \frac{54063787}{3175741} a^{17} - \frac{195232594}{22230187} a^{16} - \frac{353777804}{22230187} a^{15} + \frac{242668081}{22230187} a^{14} + \frac{1459976232}{22230187} a^{13} - \frac{3547291975}{22230187} a^{12} + \frac{368115137}{3175741} a^{11} + \frac{4435405692}{22230187} a^{10} - \frac{1868254249}{3175741} a^{9} + \frac{14198364703}{22230187} a^{8} - \frac{2274031971}{22230187} a^{7} - \frac{15932916602}{22230187} a^{6} + \frac{27078442070}{22230187} a^{5} - \frac{24882965151}{22230187} a^{4} + \frac{14658774374}{22230187} a^{3} - \frac{5426156740}{22230187} a^{2} + \frac{1208261745}{22230187} a - \frac{150140205}{22230187} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 397.737246993 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_{10}\times D_5$ (as 20T24):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 100
The 40 conjugacy class representatives for $C_{10}\times D_5$
Character table for $C_{10}\times D_5$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), \(\Q(i, \sqrt{5})\), 10.0.400000000.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$5$5.10.17.5$x^{10} - 5 x^{8} + 55$$10$$1$$17$$C_{10}$$[2]_{2}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$