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Properties

Label 20.0.17911192835...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{10}\cdot 211^{10}$
Root discriminant $64.96$
Ramified primes $2, 5, 211$
Class number $11564$ (GRH)
Class group $[2, 5782]$ (GRH)
Galois group 20T538

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![44521, 0, 351315, 0, 1073921, 0, 1646074, 0, 1364983, 0, 619709, 0, 153351, 0, 21179, 0, 1623, 0, 64, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 64*x^18 + 1623*x^16 + 21179*x^14 + 153351*x^12 + 619709*x^10 + 1364983*x^8 + 1646074*x^6 + 1073921*x^4 + 351315*x^2 + 44521)
 
gp: K = bnfinit(x^20 + 64*x^18 + 1623*x^16 + 21179*x^14 + 153351*x^12 + 619709*x^10 + 1364983*x^8 + 1646074*x^6 + 1073921*x^4 + 351315*x^2 + 44521, 1)
 

Normalized defining polynomial

\( x^{20} + 64 x^{18} + 1623 x^{16} + 21179 x^{14} + 153351 x^{12} + 619709 x^{10} + 1364983 x^{8} + 1646074 x^{6} + 1073921 x^{4} + 351315 x^{2} + 44521 \)

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:  \(1791119283562577700931594240000000000=2^{20}\cdot 5^{10}\cdot 211^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.96$
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:  $2, 5, 211$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{8733467642349963970549} a^{18} - \frac{2653805784251450140166}{8733467642349963970549} a^{16} + \frac{30302825044263873550}{8733467642349963970549} a^{14} + \frac{3812339087793086433819}{8733467642349963970549} a^{12} + \frac{433848657127123395955}{8733467642349963970549} a^{10} + \frac{751330483523143024595}{8733467642349963970549} a^{8} + \frac{2697151535596013577929}{8733467642349963970549} a^{6} - \frac{725271096739775349509}{8733467642349963970549} a^{4} - \frac{2069621483497289023504}{8733467642349963970549} a^{2} - \frac{322393546322147017}{41390841906871867159}$, $\frac{1}{8733467642349963970549} a^{19} - \frac{2653805784251450140166}{8733467642349963970549} a^{17} + \frac{30302825044263873550}{8733467642349963970549} a^{15} + \frac{3812339087793086433819}{8733467642349963970549} a^{13} + \frac{433848657127123395955}{8733467642349963970549} a^{11} + \frac{751330483523143024595}{8733467642349963970549} a^{9} + \frac{2697151535596013577929}{8733467642349963970549} a^{7} - \frac{725271096739775349509}{8733467642349963970549} a^{5} - \frac{2069621483497289023504}{8733467642349963970549} a^{3} - \frac{322393546322147017}{41390841906871867159} a$

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

Class group and class number

$C_{2}\times C_{5782}$, which has order $11564$ (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:  \( 454775.127492 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T538:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 25600
The 88 conjugacy class representatives for t20n538 are not computed
Character table for t20n538 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.6194123253125.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
211Data not computed