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Properties

Label 20.8.67108864000...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{48}\cdot 5^{22}$
Root discriminant $31.00$
Ramified primes $2, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T201

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, 0, -5450, 0, 9525, 0, -3400, 0, 650, 0, -1340, 0, 410, 0, -200, 0, 45, 0, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^18 + 45*x^16 - 200*x^14 + 410*x^12 - 1340*x^10 + 650*x^8 - 3400*x^6 + 9525*x^4 - 5450*x^2 + 25)
 
gp: K = bnfinit(x^20 - 10*x^18 + 45*x^16 - 200*x^14 + 410*x^12 - 1340*x^10 + 650*x^8 - 3400*x^6 + 9525*x^4 - 5450*x^2 + 25, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{18} + 45 x^{16} - 200 x^{14} + 410 x^{12} - 1340 x^{10} + 650 x^{8} - 3400 x^{6} + 9525 x^{4} - 5450 x^{2} + 25 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(671088640000000000000000000000=2^{48}\cdot 5^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.00$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{8} a^{8} - \frac{1}{2} a^{2} + \frac{3}{8}$, $\frac{1}{8} a^{9} - \frac{1}{2} a^{3} + \frac{3}{8} a$, $\frac{1}{320} a^{10} - \frac{1}{64} a^{8} + \frac{1}{32} a^{6} + \frac{3}{32} a^{4} + \frac{13}{64} a^{2} - \frac{1}{2} a - \frac{17}{64}$, $\frac{1}{320} a^{11} - \frac{1}{64} a^{9} + \frac{1}{32} a^{7} + \frac{3}{32} a^{5} + \frac{13}{64} a^{3} - \frac{1}{2} a^{2} - \frac{17}{64} a$, $\frac{1}{320} a^{12} - \frac{3}{64} a^{8} - \frac{5}{64} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{5}{64}$, $\frac{1}{320} a^{13} - \frac{3}{64} a^{9} - \frac{5}{64} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{5}{64} a - \frac{1}{2}$, $\frac{1}{640} a^{14} - \frac{1}{640} a^{12} - \frac{1}{640} a^{10} + \frac{5}{128} a^{8} - \frac{9}{128} a^{6} - \frac{7}{128} a^{4} - \frac{1}{2} a^{3} + \frac{49}{128} a^{2} - \frac{1}{2} a - \frac{57}{128}$, $\frac{1}{1280} a^{15} - \frac{1}{1280} a^{14} + \frac{1}{1280} a^{13} - \frac{1}{1280} a^{12} - \frac{1}{1280} a^{11} + \frac{1}{1280} a^{10} - \frac{1}{256} a^{9} + \frac{1}{256} a^{8} + \frac{23}{256} a^{7} + \frac{9}{256} a^{6} + \frac{15}{256} a^{5} + \frac{17}{256} a^{4} + \frac{17}{256} a^{3} - \frac{113}{256} a^{2} + \frac{93}{256} a - \frac{61}{256}$, $\frac{1}{6400} a^{16} - \frac{1}{640} a^{12} + \frac{19}{320} a^{8} - \frac{1}{8} a^{7} + \frac{3}{40} a^{6} - \frac{1}{8} a^{5} + \frac{29}{128} a^{4} + \frac{3}{8} a^{3} + \frac{3}{8} a^{2} + \frac{3}{8} a + \frac{91}{256}$, $\frac{1}{6400} a^{17} - \frac{1}{640} a^{13} + \frac{19}{320} a^{9} + \frac{3}{40} a^{7} - \frac{1}{8} a^{6} - \frac{3}{128} a^{5} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{3}{8} a^{2} - \frac{101}{256} a - \frac{3}{8}$, $\frac{1}{2291200} a^{18} + \frac{41}{2291200} a^{16} - \frac{37}{229120} a^{14} + \frac{241}{229120} a^{12} - \frac{99}{114560} a^{10} - \frac{1}{16} a^{9} - \frac{563}{28640} a^{8} - \frac{1}{8} a^{7} + \frac{25973}{229120} a^{6} - \frac{1}{8} a^{5} + \frac{8099}{45824} a^{4} + \frac{1}{8} a^{3} - \frac{42029}{91648} a^{2} - \frac{5}{16} a + \frac{1439}{91648}$, $\frac{1}{2291200} a^{19} + \frac{41}{2291200} a^{17} - \frac{37}{229120} a^{15} + \frac{241}{229120} a^{13} - \frac{99}{114560} a^{11} - \frac{563}{28640} a^{9} - \frac{1}{16} a^{8} + \frac{25973}{229120} a^{7} - \frac{3357}{45824} a^{5} - \frac{1}{4} a^{4} + \frac{3795}{91648} a^{3} + \frac{1}{4} a^{2} + \frac{24351}{91648} a + \frac{1}{16}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $13$
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:  \( 73814480.5605 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T201:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1440
The 13 conjugacy class representatives for t20n201
Character table for t20n201

Intermediate fields

\(\Q(\sqrt{5}) \), 10.4.163840000000000.1

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

Sibling fields

Degree 10 sibling: data not computed
Degree 12 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 sibling: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.22.38$x^{8} + 2 x^{4} + 16 x + 52$$4$$2$$22$$C_8:C_2$$[2, 3, 4]^{2}$
2.8.22.38$x^{8} + 2 x^{4} + 16 x + 52$$4$$2$$22$$C_8:C_2$$[2, 3, 4]^{2}$
$5$5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$