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Properties

Label 20.0.82419567486...9456.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{24}\cdot 53^{12}$
Root discriminant $24.88$
Ramified primes $2, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T196

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 8, 4, 19, -44, 32, -44, 29, -24, 48, -24, 29, -44, 32, -44, 19, 4, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 8*x^18 + 4*x^17 + 19*x^16 - 44*x^15 + 32*x^14 - 44*x^13 + 29*x^12 - 24*x^11 + 48*x^10 - 24*x^9 + 29*x^8 - 44*x^7 + 32*x^6 - 44*x^5 + 19*x^4 + 4*x^3 + 8*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 8*x^18 + 4*x^17 + 19*x^16 - 44*x^15 + 32*x^14 - 44*x^13 + 29*x^12 - 24*x^11 + 48*x^10 - 24*x^9 + 29*x^8 - 44*x^7 + 32*x^6 - 44*x^5 + 19*x^4 + 4*x^3 + 8*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 8 x^{18} + 4 x^{17} + 19 x^{16} - 44 x^{15} + 32 x^{14} - 44 x^{13} + 29 x^{12} - 24 x^{11} + 48 x^{10} - 24 x^{9} + 29 x^{8} - 44 x^{7} + 32 x^{6} - 44 x^{5} + 19 x^{4} + 4 x^{3} + 8 x^{2} - 4 x + 1 \)

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:  \(8241956748638414149261459456=2^{24}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.88$
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:  $2, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{4} a^{10} + \frac{1}{4} a^{6} + \frac{3}{8} a^{4} + \frac{3}{8} a^{2} + \frac{1}{8}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{7} + \frac{3}{8} a^{5} + \frac{3}{8} a^{3} + \frac{1}{8} a$, $\frac{1}{780776} a^{18} - \frac{20645}{390388} a^{17} - \frac{6963}{195194} a^{16} + \frac{6389}{97597} a^{15} - \frac{4678}{97597} a^{14} + \frac{37799}{390388} a^{13} - \frac{47109}{780776} a^{12} - \frac{11621}{195194} a^{11} + \frac{19077}{390388} a^{10} + \frac{8999}{195194} a^{9} + \frac{19077}{390388} a^{8} + \frac{42988}{97597} a^{7} + \frac{148085}{780776} a^{6} + \frac{37799}{390388} a^{5} - \frac{116309}{390388} a^{4} - \frac{84819}{195194} a^{3} - \frac{111523}{390388} a^{2} + \frac{174549}{390388} a + \frac{195195}{780776}$, $\frac{1}{11711640} a^{19} - \frac{1}{2342328} a^{18} - \frac{320941}{5855820} a^{17} - \frac{139613}{3903880} a^{16} + \frac{329969}{5855820} a^{15} - \frac{167239}{3903880} a^{14} - \frac{721321}{11711640} a^{13} + \frac{114863}{2927910} a^{12} + \frac{340927}{1951940} a^{11} - \frac{105384}{487985} a^{10} + \frac{117779}{1951940} a^{9} + \frac{117767}{1951940} a^{8} + \frac{3326657}{11711640} a^{7} + \frac{2045489}{11711640} a^{6} - \frac{449691}{975970} a^{5} - \frac{3649307}{11711640} a^{4} + \frac{101094}{487985} a^{3} + \frac{211049}{688920} a^{2} + \frac{501815}{2342328} a + \frac{2606963}{5855820}$

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:  \( \frac{2354809}{5855820} a^{19} - \frac{3437489}{2342328} a^{18} + \frac{31517459}{11711640} a^{17} + \frac{5163543}{1951940} a^{16} + \frac{97394669}{11711640} a^{15} - \frac{7098359}{487985} a^{14} + \frac{92885527}{11711640} a^{13} - \frac{155692199}{11711640} a^{12} + \frac{10258881}{1951940} a^{11} - \frac{12724633}{1951940} a^{10} + \frac{7507648}{487985} a^{9} - \frac{6927119}{1951940} a^{8} + \frac{25651099}{2927910} a^{7} - \frac{146866313}{11711640} a^{6} + \frac{30737183}{3903880} a^{5} - \frac{38849449}{2927910} a^{4} + \frac{4753611}{3903880} a^{3} + \frac{24601537}{5855820} a^{2} + \frac{6769069}{2342328} a - \frac{7956667}{11711640} \) (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:  \( 4265922.29823 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T196:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1280
The 44 conjugacy class representatives for t20n196
Character table for t20n196 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.5.2382032.1, 10.0.22696305796096.1, 10.6.45392611592192.1, 10.4.45392611592192.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$53$53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$