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Properties

Label 20.0.75060310901...0000.6
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 5^{10}\cdot 31^{18}$
Root discriminant $196.68$
Ramified primes $2, 5, 31$
Class number $12617728$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 2, 2, 24644]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![541461977281, 0, 222248107898, 0, 44215198547, 0, 5619072258, 0, 499279507, 0, 32388528, 0, 1557872, 0, 56078, 0, 1617, 0, 38, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 38*x^18 + 1617*x^16 + 56078*x^14 + 1557872*x^12 + 32388528*x^10 + 499279507*x^8 + 5619072258*x^6 + 44215198547*x^4 + 222248107898*x^2 + 541461977281)
 
gp: K = bnfinit(x^20 + 38*x^18 + 1617*x^16 + 56078*x^14 + 1557872*x^12 + 32388528*x^10 + 499279507*x^8 + 5619072258*x^6 + 44215198547*x^4 + 222248107898*x^2 + 541461977281, 1)
 

Normalized defining polynomial

\( x^{20} + 38 x^{18} + 1617 x^{16} + 56078 x^{14} + 1557872 x^{12} + 32388528 x^{10} + 499279507 x^{8} + 5619072258 x^{6} + 44215198547 x^{4} + 222248107898 x^{2} + 541461977281 \)

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:  \(7506031090115774979391989500178595840000000000=2^{40}\cdot 5^{10}\cdot 31^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $196.68$
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:  $2, 5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1240=2^{3}\cdot 5\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{1240}(1,·)$, $\chi_{1240}(709,·)$, $\chi_{1240}(711,·)$, $\chi_{1240}(841,·)$, $\chi_{1240}(779,·)$, $\chi_{1240}(271,·)$, $\chi_{1240}(721,·)$, $\chi_{1240}(659,·)$, $\chi_{1240}(1179,·)$, $\chi_{1240}(151,·)$, $\chi_{1240}(281,·)$, $\chi_{1240}(219,·)$, $\chi_{1240}(29,·)$, $\chi_{1240}(991,·)$, $\chi_{1240}(481,·)$, $\chi_{1240}(419,·)$, $\chi_{1240}(1069,·)$, $\chi_{1240}(309,·)$, $\chi_{1240}(829,·)$, $\chi_{1240}(511,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{4} + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{5} + \frac{1}{5} a$, $\frac{1}{10} a^{10} - \frac{1}{10} a^{8} - \frac{1}{5} a^{6} - \frac{3}{10} a^{4} + \frac{1}{10} a^{2} - \frac{1}{10}$, $\frac{1}{10} a^{11} - \frac{1}{10} a^{9} - \frac{1}{5} a^{7} - \frac{3}{10} a^{5} + \frac{1}{10} a^{3} - \frac{1}{10} a$, $\frac{1}{50} a^{12} - \frac{3}{50} a^{8} - \frac{3}{10} a^{6} - \frac{11}{25} a^{4} - \frac{1}{5} a^{2} - \frac{1}{50}$, $\frac{1}{50} a^{13} - \frac{3}{50} a^{9} - \frac{3}{10} a^{7} - \frac{11}{25} a^{5} - \frac{1}{5} a^{3} - \frac{1}{50} a$, $\frac{1}{50} a^{14} + \frac{1}{25} a^{10} + \frac{9}{25} a^{6} - \frac{3}{10} a^{4} + \frac{2}{25} a^{2} + \frac{3}{10}$, $\frac{1}{100} a^{15} - \frac{1}{100} a^{14} - \frac{1}{100} a^{13} - \frac{3}{100} a^{11} - \frac{1}{50} a^{10} - \frac{1}{50} a^{9} + \frac{43}{100} a^{7} - \frac{9}{50} a^{6} + \frac{21}{50} a^{5} - \frac{7}{20} a^{4} - \frac{41}{100} a^{3} - \frac{1}{25} a^{2} + \frac{11}{100} a + \frac{7}{20}$, $\frac{1}{30500} a^{16} - \frac{19}{3050} a^{14} - \frac{1}{100} a^{13} + \frac{111}{30500} a^{12} - \frac{1}{20} a^{11} + \frac{39}{3050} a^{10} - \frac{1}{50} a^{9} - \frac{2589}{30500} a^{8} + \frac{1}{4} a^{7} - \frac{281}{3050} a^{6} + \frac{7}{100} a^{5} + \frac{333}{15250} a^{4} - \frac{9}{20} a^{3} - \frac{3}{6100} a^{2} + \frac{23}{50} a - \frac{4589}{30500}$, $\frac{1}{30500} a^{17} + \frac{23}{6100} a^{15} - \frac{97}{15250} a^{13} - \frac{1}{100} a^{12} - \frac{21}{1220} a^{11} + \frac{2901}{30500} a^{9} - \frac{7}{100} a^{8} + \frac{2061}{6100} a^{7} - \frac{7}{20} a^{6} + \frac{319}{7625} a^{5} + \frac{21}{50} a^{4} - \frac{626}{1525} a^{3} + \frac{1}{10} a^{2} + \frac{2433}{15250} a + \frac{41}{100}$, $\frac{1}{675226376262202766021258147946114500} a^{18} - \frac{472852063745233616870639285286}{168806594065550691505314536986528625} a^{16} + \frac{2175627596337765367279941848025563}{337613188131101383010629073973057250} a^{14} + \frac{23181463547629675840844994298713}{5534642428378711196895558589722250} a^{12} - \frac{1}{20} a^{11} - \frac{15247303771170333494656897379271589}{675226376262202766021258147946114500} a^{10} + \frac{1}{20} a^{9} - \frac{16668657687334478283674669316693981}{168806594065550691505314536986528625} a^{8} - \frac{2}{5} a^{7} - \frac{112754878877566754013059212202432227}{337613188131101383010629073973057250} a^{6} - \frac{7}{20} a^{5} + \frac{44506531015207571714098096131105779}{168806594065550691505314536986528625} a^{4} + \frac{9}{20} a^{3} - \frac{194129902276268642507383702763297109}{675226376262202766021258147946114500} a^{2} + \frac{1}{20} a + \frac{226180227427790821395735038316970941}{675226376262202766021258147946114500}$, $\frac{1}{496859251935155545551848616842816839794500} a^{19} - \frac{1945340150704517907264105997907076463}{496859251935155545551848616842816839794500} a^{17} - \frac{1159519287953189618481355961783025518399}{496859251935155545551848616842816839794500} a^{15} + \frac{733167782664651747509492522375657979878}{124214812983788886387962154210704209948625} a^{13} + \frac{73150995705577016834289202613421878603}{248429625967577772775924308421408419897250} a^{11} - \frac{1}{20} a^{10} + \frac{29917061059504246192318376676941430404937}{496859251935155545551848616842816839794500} a^{9} + \frac{1}{20} a^{8} + \frac{311317350228226093974136063824883848771}{8145233638281238451669649456439620324500} a^{7} - \frac{2}{5} a^{6} + \frac{57494115852673123954919922092581464337923}{124214812983788886387962154210704209948625} a^{5} - \frac{7}{20} a^{4} - \frac{227041386374409935797868237973101439636989}{496859251935155545551848616842816839794500} a^{3} + \frac{9}{20} a^{2} - \frac{58443010883026459384722894838014869413853}{496859251935155545551848616842816839794500} a + \frac{1}{20}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{24644}$, which has order $12617728$ (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:  \( 190570522.65858147 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-310}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{31}) \), \(\Q(\sqrt{-10}, \sqrt{31})\), 5.5.923521.1, 10.0.2707417309252710400000.1, 10.0.87336042233958400000.1, 10.10.27074173092527104.1

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

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
31Data not computed