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Properties

Label 20.0.94964489254...000.14
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $177.37$
Ramified primes $2, 3, 5, 7, 11$
Class number $44617760$ (GRH)
Class group $[2, 4, 5577220]$ (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![522296735401, 0, 1641786500795, 0, 1112289528945, 0, 278664715539, 0, 34106092153, 0, 2323887083, 0, 94428929, 0, 2345091, 0, 34937, 0, 287, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 287*x^18 + 34937*x^16 + 2345091*x^14 + 94428929*x^12 + 2323887083*x^10 + 34106092153*x^8 + 278664715539*x^6 + 1112289528945*x^4 + 1641786500795*x^2 + 522296735401)
 
gp: K = bnfinit(x^20 + 287*x^18 + 34937*x^16 + 2345091*x^14 + 94428929*x^12 + 2323887083*x^10 + 34106092153*x^8 + 278664715539*x^6 + 1112289528945*x^4 + 1641786500795*x^2 + 522296735401, 1)
 

Normalized defining polynomial

\( x^{20} + 287 x^{18} + 34937 x^{16} + 2345091 x^{14} + 94428929 x^{12} + 2323887083 x^{10} + 34106092153 x^{8} + 278664715539 x^{6} + 1112289528945 x^{4} + 1641786500795 x^{2} + 522296735401 \)

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:  \(949644892545940254829738055123773440000000000=2^{20}\cdot 3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $177.37$
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:  $2, 3, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4620=2^{2}\cdot 3\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{4620}(1,·)$, $\chi_{4620}(3079,·)$, $\chi_{4620}(1289,·)$, $\chi_{4620}(139,·)$, $\chi_{4620}(2129,·)$, $\chi_{4620}(2969,·)$, $\chi_{4620}(3611,·)$, $\chi_{4620}(29,·)$, $\chi_{4620}(2659,·)$, $\chi_{4620}(421,·)$, $\chi_{4620}(1681,·)$, $\chi_{4620}(1511,·)$, $\chi_{4620}(2239,·)$, $\chi_{4620}(841,·)$, $\chi_{4620}(1399,·)$, $\chi_{4620}(2549,·)$, $\chi_{4620}(3191,·)$, $\chi_{4620}(251,·)$, $\chi_{4620}(2941,·)$, $\chi_{4620}(4031,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{7} a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{49} a^{4}$, $\frac{1}{49} a^{5}$, $\frac{1}{343} a^{6}$, $\frac{1}{343} a^{7}$, $\frac{1}{2401} a^{8}$, $\frac{1}{2401} a^{9}$, $\frac{1}{16807} a^{10}$, $\frac{1}{722701} a^{11} - \frac{3}{14749} a^{9} + \frac{4}{14749} a^{7} + \frac{2}{301} a^{5} + \frac{20}{301} a^{3} + \frac{8}{43} a$, $\frac{1}{116354861} a^{12} - \frac{365}{16622123} a^{10} + \frac{348}{2374589} a^{8} - \frac{158}{339227} a^{6} + \frac{52}{6923} a^{4} - \frac{164}{6923} a^{2} - \frac{1}{23}$, $\frac{1}{116354861} a^{13} + \frac{3}{16622123} a^{11} - \frac{457}{2374589} a^{9} + \frac{325}{339227} a^{7} - \frac{418}{48461} a^{5} + \frac{39}{989} a^{3} - \frac{66}{989} a$, $\frac{1}{814484027} a^{14} - \frac{351}{16622123} a^{10} + \frac{270}{2374589} a^{8} + \frac{8}{48461} a^{6} + \frac{170}{48461} a^{4} + \frac{426}{6923} a^{2} + \frac{3}{23}$, $\frac{1}{814484027} a^{15} - \frac{6}{16622123} a^{11} - \frac{52}{2374589} a^{9} + \frac{447}{339227} a^{7} + \frac{55}{48461} a^{5} + \frac{403}{6923} a^{3} - \frac{78}{989} a$, $\frac{1}{5701388189} a^{16} - \frac{264}{16622123} a^{10} - \frac{432}{2374589} a^{8} + \frac{96}{339227} a^{6} - \frac{380}{48461} a^{4} - \frac{73}{6923} a^{2} - \frac{6}{23}$, $\frac{1}{5701388189} a^{17} - \frac{11}{16622123} a^{11} + \frac{27}{339227} a^{9} + \frac{17}{48461} a^{7} + \frac{195}{48461} a^{5} + \frac{6}{989} a^{3} - \frac{212}{989} a$, $\frac{1}{39909717323} a^{18} + \frac{130}{16622123} a^{10} - \frac{9}{2374589} a^{8} + \frac{435}{339227} a^{6} + \frac{90}{48461} a^{4} - \frac{38}{6923} a^{2} - \frac{11}{23}$, $\frac{1}{39909717323} a^{19} - \frac{8}{16622123} a^{11} - \frac{78}{2374589} a^{9} - \frac{117}{339227} a^{7} + \frac{136}{48461} a^{5} + \frac{169}{6923} a^{3} + \frac{401}{989} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{5577220}$, which has order $44617760$ (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:  \( 5362955.973727969 \) (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

\(\Q(\sqrt{-385}) \), \(\Q(\sqrt{165}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{-21}, \sqrt{165})\), \(\Q(\zeta_{11})^+\), 10.0.126816085896438400000.1, 10.10.1790566527853125.1, 10.0.896474439937004544.3

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 R R R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
$5$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$7$7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$