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Properties

Label 20.0.71965450154...1664.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 1609^{4}$
Root discriminant $12.38$
Ramified primes $2, 1609$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T279

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 8, -18, 24, 0, -30, 52, -32, 4, 24, -12, 1, 2, 16, -14, 6, 0, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 2*x^18 + 6*x^16 - 14*x^15 + 16*x^14 + 2*x^13 + x^12 - 12*x^11 + 24*x^10 + 4*x^9 - 32*x^8 + 52*x^7 - 30*x^6 + 24*x^4 - 18*x^3 + 8*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 2*x^18 + 6*x^16 - 14*x^15 + 16*x^14 + 2*x^13 + x^12 - 12*x^11 + 24*x^10 + 4*x^9 - 32*x^8 + 52*x^7 - 30*x^6 + 24*x^4 - 18*x^3 + 8*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 2 x^{18} + 6 x^{16} - 14 x^{15} + 16 x^{14} + 2 x^{13} + x^{12} - 12 x^{11} + 24 x^{10} + 4 x^{9} - 32 x^{8} + 52 x^{7} - 30 x^{6} + 24 x^{4} - 18 x^{3} + 8 x^{2} - 4 x + 1 \)

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:  \(7196545015488566001664=2^{30}\cdot 1609^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.38$
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:  $2, 1609$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{11} a^{18} + \frac{3}{11} a^{17} - \frac{1}{11} a^{16} - \frac{4}{11} a^{15} + \frac{4}{11} a^{14} + \frac{1}{11} a^{13} + \frac{4}{11} a^{12} + \frac{4}{11} a^{11} + \frac{4}{11} a^{10} + \frac{2}{11} a^{9} - \frac{5}{11} a^{8} - \frac{2}{11} a^{7} + \frac{4}{11} a^{6} - \frac{2}{11} a^{5} - \frac{2}{11} a^{4} + \frac{4}{11} a^{3} + \frac{3}{11} a^{2} + \frac{2}{11} a - \frac{3}{11}$, $\frac{1}{577504644211} a^{19} + \frac{14961917859}{577504644211} a^{18} + \frac{279456753010}{577504644211} a^{17} + \frac{256210863195}{577504644211} a^{16} + \frac{121812977046}{577504644211} a^{15} + \frac{251964673821}{577504644211} a^{14} + \frac{136725760599}{577504644211} a^{13} + \frac{70308554190}{577504644211} a^{12} - \frac{133809138413}{577504644211} a^{11} + \frac{35441940450}{577504644211} a^{10} - \frac{109900683336}{577504644211} a^{9} - \frac{281004045640}{577504644211} a^{8} + \frac{244582903249}{577504644211} a^{7} + \frac{811322744}{52500422201} a^{6} + \frac{7903001004}{577504644211} a^{5} - \frac{242262460503}{577504644211} a^{4} - \frac{81586978138}{577504644211} a^{3} - \frac{1783883917}{577504644211} a^{2} + \frac{20118307119}{577504644211} a + \frac{204227800517}{577504644211}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{67712926826}{52500422201} a^{19} - \frac{97035022998}{52500422201} a^{18} + \frac{74306720579}{52500422201} a^{17} + \frac{54600965394}{52500422201} a^{16} + \frac{422633175625}{52500422201} a^{15} - \frac{705703745770}{52500422201} a^{14} + \frac{644264265723}{52500422201} a^{13} + \frac{590052077619}{52500422201} a^{12} + \frac{281864390484}{52500422201} a^{11} - \frac{639670847470}{52500422201} a^{10} + \frac{1235490245950}{52500422201} a^{9} + \frac{1056457875083}{52500422201} a^{8} - \frac{1785618575940}{52500422201} a^{7} + \frac{2544081661998}{52500422201} a^{6} - \frac{435779796013}{52500422201} a^{5} - \frac{564412588962}{52500422201} a^{4} + \frac{1447316320638}{52500422201} a^{3} - \frac{418144531731}{52500422201} a^{2} + \frac{185256315495}{52500422201} a - \frac{111093563455}{52500422201} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 713.051816016 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T279:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 36 conjugacy class representatives for t20n279
Character table for t20n279 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.1.1609.1, 10.0.2651014144.4, 10.0.2651014144.3, 10.2.2651014144.2

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
1609Data not computed