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Properties

Label 20.4.90346941218...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{38}\cdot 5^{22}\cdot 13^{10}$
Root discriminant $79.03$
Ramified primes $2, 5, 13$
Class number Not computed
Class group Not computed
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![551342629319, 564365370, -1570569355, 2584199310, -2806975185, 2058448164, -1091601330, 356441820, -100249065, 7425090, -1485303, 770, 45, -540, 90, 228, -105, -30, 35, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 35*x^18 - 30*x^17 - 105*x^16 + 228*x^15 + 90*x^14 - 540*x^13 + 45*x^12 + 770*x^11 - 1485303*x^10 + 7425090*x^9 - 100249065*x^8 + 356441820*x^7 - 1091601330*x^6 + 2058448164*x^5 - 2806975185*x^4 + 2584199310*x^3 - 1570569355*x^2 + 564365370*x + 551342629319)
 
gp: K = bnfinit(x^20 - 10*x^19 + 35*x^18 - 30*x^17 - 105*x^16 + 228*x^15 + 90*x^14 - 540*x^13 + 45*x^12 + 770*x^11 - 1485303*x^10 + 7425090*x^9 - 100249065*x^8 + 356441820*x^7 - 1091601330*x^6 + 2058448164*x^5 - 2806975185*x^4 + 2584199310*x^3 - 1570569355*x^2 + 564365370*x + 551342629319, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 35 x^{18} - 30 x^{17} - 105 x^{16} + 228 x^{15} + 90 x^{14} - 540 x^{13} + 45 x^{12} + 770 x^{11} - 1485303 x^{10} + 7425090 x^{9} - 100249065 x^{8} + 356441820 x^{7} - 1091601330 x^{6} + 2058448164 x^{5} - 2806975185 x^{4} + 2584199310 x^{3} - 1570569355 x^{2} + 564365370 x + 551342629319 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(90346941218160640000000000000000000000=2^{38}\cdot 5^{22}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.03$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{13} a^{4} - \frac{2}{13} a^{3} - \frac{1}{13} a^{2} + \frac{2}{13} a + \frac{1}{13}$, $\frac{1}{13} a^{5} - \frac{5}{13} a^{3} + \frac{5}{13} a + \frac{2}{13}$, $\frac{1}{13} a^{6} + \frac{3}{13} a^{3} - \frac{1}{13} a + \frac{5}{13}$, $\frac{1}{13} a^{7} + \frac{6}{13} a^{3} + \frac{2}{13} a^{2} - \frac{1}{13} a - \frac{3}{13}$, $\frac{1}{169} a^{8} - \frac{4}{169} a^{7} + \frac{2}{169} a^{6} - \frac{5}{169} a^{5} - \frac{5}{169} a^{4} + \frac{57}{169} a^{3} + \frac{2}{169} a^{2} - \frac{61}{169} a - \frac{25}{169}$, $\frac{1}{169} a^{9} - \frac{1}{169} a^{7} + \frac{3}{169} a^{6} + \frac{1}{169} a^{5} - \frac{2}{169} a^{4} - \frac{82}{169} a^{3} + \frac{12}{169} a^{2} - \frac{61}{169} a + \frac{43}{169}$, $\frac{1}{845} a^{10} + \frac{5}{169} a^{7} - \frac{2}{169} a^{6} + \frac{19}{845} a^{5} + \frac{6}{169} a^{4} + \frac{32}{169} a^{3} + \frac{9}{169} a^{2} - \frac{1}{169} a - \frac{337}{845}$, $\frac{1}{845} a^{11} + \frac{5}{169} a^{7} - \frac{31}{845} a^{6} + \frac{5}{169} a^{5} + \frac{5}{169} a^{4} + \frac{49}{169} a^{3} + \frac{15}{169} a^{2} + \frac{83}{845} a + \frac{60}{169}$, $\frac{1}{10985} a^{12} - \frac{6}{10985} a^{11} - \frac{4}{10985} a^{10} + \frac{2}{2197} a^{9} - \frac{6}{2197} a^{8} - \frac{331}{10985} a^{7} + \frac{171}{10985} a^{6} - \frac{241}{10985} a^{5} - \frac{84}{2197} a^{4} - \frac{418}{2197} a^{3} + \frac{3818}{10985} a^{2} - \frac{4323}{10985} a - \frac{12}{10985}$, $\frac{1}{10985} a^{13} - \frac{1}{10985} a^{11} - \frac{1}{10985} a^{10} + \frac{6}{2197} a^{9} + \frac{9}{10985} a^{8} - \frac{12}{2197} a^{7} - \frac{359}{10985} a^{6} + \frac{136}{10985} a^{5} + \frac{14}{2197} a^{4} - \frac{5472}{10985} a^{3} + \frac{64}{2197} a^{2} - \frac{5488}{10985} a - \frac{4908}{10985}$, $\frac{1}{10985} a^{14} + \frac{6}{10985} a^{11} + \frac{19}{10985} a^{9} - \frac{5}{2197} a^{8} + \frac{83}{2197} a^{7} + \frac{294}{10985} a^{6} + \frac{36}{2197} a^{5} + \frac{88}{10985} a^{4} + \frac{868}{2197} a^{3} - \frac{1023}{2197} a^{2} + \frac{4068}{10985} a + \frac{684}{2197}$, $\frac{1}{10985} a^{15} - \frac{3}{10985} a^{11} + \frac{4}{10985} a^{10} - \frac{4}{2197} a^{9} + \frac{2}{2197} a^{8} + \frac{14}{2197} a^{7} - \frac{222}{10985} a^{6} + \frac{21}{845} a^{5} - \frac{19}{2197} a^{4} - \frac{283}{2197} a^{3} + \frac{119}{2197} a^{2} - \frac{269}{10985} a + \frac{576}{2197}$, $\frac{1}{714025} a^{16} - \frac{8}{714025} a^{15} + \frac{7}{714025} a^{14} + \frac{2}{54925} a^{13} + \frac{21}{714025} a^{12} + \frac{82}{714025} a^{11} + \frac{12}{142805} a^{10} - \frac{367}{714025} a^{9} - \frac{606}{714025} a^{8} - \frac{16676}{714025} a^{7} + \frac{571}{142805} a^{6} + \frac{12164}{714025} a^{5} - \frac{15644}{714025} a^{4} - \frac{859}{54925} a^{3} + \frac{42868}{714025} a^{2} + \frac{166304}{714025} a - \frac{313429}{714025}$, $\frac{1}{714025} a^{17} + \frac{8}{714025} a^{15} + \frac{17}{714025} a^{14} - \frac{31}{714025} a^{13} - \frac{2}{142805} a^{12} + \frac{261}{714025} a^{11} - \frac{17}{714025} a^{10} + \frac{423}{714025} a^{9} - \frac{1114}{714025} a^{8} + \frac{16282}{714025} a^{7} - \frac{24016}{714025} a^{6} - \frac{5822}{714025} a^{5} + \frac{27286}{714025} a^{4} + \frac{307652}{714025} a^{3} - \frac{5357}{714025} a^{2} - \frac{18674}{54925} a - \frac{131877}{714025}$, $\frac{1}{104827185471840524813028472525} a^{18} - \frac{9}{104827185471840524813028472525} a^{17} + \frac{5540662033956200579011}{20965437094368104962605694505} a^{16} - \frac{221626481358248023160236}{104827185471840524813028472525} a^{15} - \frac{601098586021066931410096}{20965437094368104962605694505} a^{14} - \frac{3711357018203519493211719}{104827185471840524813028472525} a^{13} - \frac{246327160277583001534674}{8063629651680040370232959425} a^{12} - \frac{2926955000451569167837909}{8063629651680040370232959425} a^{11} - \frac{27596554018392674159002804}{104827185471840524813028472525} a^{10} - \frac{34655609987677193919835994}{20965437094368104962605694505} a^{9} - \frac{25943725077410902516315539}{104827185471840524813028472525} a^{8} - \frac{1027762660422277072733157101}{104827185471840524813028472525} a^{7} + \frac{2309658606420241955020899362}{104827185471840524813028472525} a^{6} + \frac{2531225702993286075230121532}{104827185471840524813028472525} a^{5} + \frac{3421674583879233219463612}{20965437094368104962605694505} a^{4} - \frac{1945297525826230606741157329}{104827185471840524813028472525} a^{3} - \frac{13744998846662232739745032778}{104827185471840524813028472525} a^{2} + \frac{34684582874572384902672295429}{104827185471840524813028472525} a + \frac{875773906367679178763012053}{20965437094368104962605694505}$, $\frac{1}{79711109574256580488178902055635800975} a^{19} + \frac{76040492}{15942221914851316097635780411127160195} a^{18} - \frac{30705072872418005691377696831396}{79711109574256580488178902055635800975} a^{17} + \frac{6893340721147517883089420478188}{11387301367750940069739843150805114425} a^{16} + \frac{16632126469503782889997963008007}{455492054710037602789593726032204577} a^{15} + \frac{230030099099947077696322841731894}{15942221914851316097635780411127160195} a^{14} + \frac{2257419730287197836244681361952024}{79711109574256580488178902055635800975} a^{13} + \frac{1739051821859051014035956351939462}{79711109574256580488178902055635800975} a^{12} + \frac{5289586331411066004031032870398662}{79711109574256580488178902055635800975} a^{11} - \frac{9272121925886287598655570823564496}{79711109574256580488178902055635800975} a^{10} - \frac{222581299884806391608880607319931823}{79711109574256580488178902055635800975} a^{9} + \frac{217648083792373956284219808096069976}{79711109574256580488178902055635800975} a^{8} - \frac{1251364634801203233045852847796209459}{79711109574256580488178902055635800975} a^{7} - \frac{571722388611832445573099489539840729}{15942221914851316097635780411127160195} a^{6} + \frac{3035100940913649226665282734000295776}{79711109574256580488178902055635800975} a^{5} - \frac{1970250914573609630399534943261347717}{79711109574256580488178902055635800975} a^{4} + \frac{28061812608880507936695908476654345097}{79711109574256580488178902055635800975} a^{3} + \frac{26845226902294621656566067001491159763}{79711109574256580488178902055635800975} a^{2} + \frac{4229197397345655976438820018541729052}{11387301367750940069739843150805114425} a - \frac{7136482987188957214689828147793074343}{79711109574256580488178902055635800975}$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{26}) \), \(\Q(\sqrt{130}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{26})\), 5.1.50000.1, 10.2.1901020160000000000.1, 10.2.9505100800000000000.1, 10.2.12500000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$5$5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$