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Properties

Label 20.0.32000000000...0000.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{25}$
Root discriminant $21.15$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group 20T138

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![80, 0, 0, 0, -40, 0, 80, 0, 325, 0, 292, 0, 100, 0, -30, 0, -20, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 20*x^16 - 30*x^14 + 100*x^12 + 292*x^10 + 325*x^8 + 80*x^6 - 40*x^4 + 80)
 
gp: K = bnfinit(x^20 - 20*x^16 - 30*x^14 + 100*x^12 + 292*x^10 + 325*x^8 + 80*x^6 - 40*x^4 + 80, 1)
 

Normalized defining polynomial

\( x^{20} - 20 x^{16} - 30 x^{14} + 100 x^{12} + 292 x^{10} + 325 x^{8} + 80 x^{6} - 40 x^{4} + 80 \)

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:  \(320000000000000000000000000=2^{30}\cdot 5^{25}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.15$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{6} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{7} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{64} a^{16} - \frac{1}{8} a^{14} + \frac{1}{32} a^{10} + \frac{1}{16} a^{8} - \frac{1}{2} a^{7} + \frac{7}{16} a^{6} - \frac{1}{2} a^{5} - \frac{11}{64} a^{4} - \frac{3}{8} a^{2} + \frac{3}{16}$, $\frac{1}{128} a^{17} + \frac{1}{16} a^{15} - \frac{1}{8} a^{13} + \frac{1}{64} a^{11} - \frac{3}{32} a^{9} - \frac{1}{4} a^{8} + \frac{11}{32} a^{7} - \frac{11}{128} a^{5} - \frac{3}{16} a^{3} + \frac{1}{4} a^{2} + \frac{3}{32} a - \frac{1}{2}$, $\frac{1}{44692736} a^{18} + \frac{13769}{22346368} a^{16} - \frac{132137}{1396648} a^{14} - \frac{2744079}{22346368} a^{12} - \frac{551015}{5586592} a^{10} + \frac{62325}{11173184} a^{8} - \frac{1}{2} a^{7} - \frac{17457619}{44692736} a^{6} - \frac{1}{2} a^{5} + \frac{338013}{22346368} a^{4} - \frac{228473}{11173184} a^{2} + \frac{2199855}{5586592}$, $\frac{1}{89385472} a^{19} + \frac{13769}{44692736} a^{17} + \frac{217025}{2793296} a^{15} + \frac{2842513}{44692736} a^{13} - \frac{551015}{11173184} a^{11} - \frac{2730971}{22346368} a^{9} - \frac{28630803}{89385472} a^{7} - \frac{1}{2} a^{6} - \frac{22008355}{44692736} a^{5} - \frac{228473}{22346368} a^{3} - \frac{1}{2} a^{2} + \frac{2199855}{11173184} a$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{575}{166144} a^{18} - \frac{625}{83072} a^{16} - \frac{345}{5192} a^{14} + \frac{3375}{83072} a^{12} + \frac{10771}{20768} a^{10} + \frac{12075}{41536} a^{8} - \frac{125805}{166144} a^{6} - \frac{160805}{83072} a^{4} - \frac{14375}{41536} a^{2} + \frac{5241}{20768} \) (order $4$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 518280.75159 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T138:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 28 conjugacy class representatives for t20n138
Character table for t20n138 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.1.250000.1, 10.0.250000000000.1

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 $20$ R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$2.4.6.8$x^{4} + 2 x^{3} + 2$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.8.16.4$x^{8} + 6 x^{6} + 6 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 20$$4$$2$$16$$D_4$$[2, 3]^{2}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.10.12.7$x^{10} + 10 x^{8} + 5 x^{7} + 15 x^{6} + 5 x^{4} + 5 x^{2} - 20 x + 7$$5$$2$$12$$D_{10}$$[3/2]_{2}^{2}$
5.10.13.8$x^{10} + 20 x^{4} + 10$$10$$1$$13$$D_{10}$$[3/2]_{2}^{2}$