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Properties

Label 20.0.25389989167...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{38}\cdot 3^{18}\cdot 5^{22}$
Root discriminant $58.92$
Ramified primes $2, 3, 5$
Class number $10$ (GRH)
Class group $[10]$ (GRH)
Galois group $C_2^2\times F_5$ (as 20T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10404, 0, 22320, 0, 22500, 0, 10080, 0, 1080, 0, -276, 0, 1020, 0, 450, 0, 120, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 120*x^16 + 450*x^14 + 1020*x^12 - 276*x^10 + 1080*x^8 + 10080*x^6 + 22500*x^4 + 22320*x^2 + 10404)
 
gp: K = bnfinit(x^20 + 120*x^16 + 450*x^14 + 1020*x^12 - 276*x^10 + 1080*x^8 + 10080*x^6 + 22500*x^4 + 22320*x^2 + 10404, 1)
 

Normalized defining polynomial

\( x^{20} + 120 x^{16} + 450 x^{14} + 1020 x^{12} - 276 x^{10} + 1080 x^{8} + 10080 x^{6} + 22500 x^{4} + 22320 x^{2} + 10404 \)

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:  \(253899891671040000000000000000000000=2^{38}\cdot 3^{18}\cdot 5^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.92$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{6} a^{10}$, $\frac{1}{6} a^{11}$, $\frac{1}{6} a^{12}$, $\frac{1}{6} a^{13}$, $\frac{1}{6} a^{14}$, $\frac{1}{6} a^{15}$, $\frac{1}{3252} a^{16} - \frac{119}{1626} a^{14} + \frac{46}{813} a^{12} + \frac{17}{813} a^{10} - \frac{88}{271} a^{8} - \frac{217}{542} a^{6} + \frac{125}{271} a^{4} + \frac{54}{271} a^{2} - \frac{79}{271}$, $\frac{1}{9756} a^{17} - \frac{65}{813} a^{15} + \frac{121}{1626} a^{13} - \frac{79}{1626} a^{11} - \frac{88}{813} a^{9} + \frac{325}{1626} a^{7} + \frac{132}{271} a^{5} + \frac{18}{271} a^{3} + \frac{64}{271} a$, $\frac{1}{10292600911332348} a^{18} + \frac{143252368411}{1143622323481372} a^{16} - \frac{111688799344979}{1715433485222058} a^{14} + \frac{52638145102847}{1715433485222058} a^{12} - \frac{106952339363971}{1715433485222058} a^{10} + \frac{159908895285763}{1715433485222058} a^{8} + \frac{151969065924247}{571811161740686} a^{6} - \frac{59523015951101}{285905580870343} a^{4} - \frac{5875677248279}{40843654410049} a^{2} + \frac{54017912564326}{285905580870343}$, $\frac{1}{174974215492649916} a^{19} + \frac{149397728947}{3430866970444116} a^{17} - \frac{692737598234697}{9720789749591662} a^{15} + \frac{838143221923117}{14581184624387493} a^{13} + \frac{10521548394709}{571811161740686} a^{11} + \frac{7622994057930805}{29162369248774986} a^{9} + \frac{3124965106118441}{9720789749591662} a^{7} - \frac{1225300384444566}{4860394874795831} a^{5} + \frac{255463425139267}{694342124970833} a^{3} - \frac{1828105911320146}{4860394874795831} a$

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

Class group and class number

$C_{10}$, which has order $10$ (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:  \( 2439011616.4501934 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times F_5$ (as 20T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_2^2\times F_5$
Character table for $C_2^2\times F_5$

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-2}, \sqrt{-15})\), 5.1.4050000.3, 10.0.33592320000000000.77, 10.0.246037500000000.9, 10.2.503884800000000000.65

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$2$2.10.19.33$x^{10} - 6 x^{4} + 4 x^{2} - 14$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
2.10.19.33$x^{10} - 6 x^{4} + 4 x^{2} - 14$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
$3$3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
5Data not computed