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Properties

Label 20.0.52033547103...1552.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{34}\cdot 13^{13}$
Root discriminant $17.21$
Ramified primes $2, 13$
Class number $1$
Class group Trivial
Galois group $C_2^2:F_5$ (as 20T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, 0, -18, -3, 36, 36, -168, 246, -204, 176, -316, 510, -588, 548, -440, 289, -142, 48, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 48*x^18 - 142*x^17 + 289*x^16 - 440*x^15 + 548*x^14 - 588*x^13 + 510*x^12 - 316*x^11 + 176*x^10 - 204*x^9 + 246*x^8 - 168*x^7 + 36*x^6 + 36*x^5 - 3*x^4 - 18*x^3 + 2*x + 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 48*x^18 - 142*x^17 + 289*x^16 - 440*x^15 + 548*x^14 - 588*x^13 + 510*x^12 - 316*x^11 + 176*x^10 - 204*x^9 + 246*x^8 - 168*x^7 + 36*x^6 + 36*x^5 - 3*x^4 - 18*x^3 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 48 x^{18} - 142 x^{17} + 289 x^{16} - 440 x^{15} + 548 x^{14} - 588 x^{13} + 510 x^{12} - 316 x^{11} + 176 x^{10} - 204 x^{9} + 246 x^{8} - 168 x^{7} + 36 x^{6} + 36 x^{5} - 3 x^{4} - 18 x^{3} + 2 x + 1 \)

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:  \(5203354710344962567831552=2^{34}\cdot 13^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.21$
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:  $2, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{1564} a^{18} - \frac{183}{1564} a^{17} - \frac{27}{391} a^{16} + \frac{80}{391} a^{15} - \frac{7}{391} a^{14} - \frac{5}{34} a^{13} - \frac{3}{23} a^{12} - \frac{105}{782} a^{11} - \frac{127}{782} a^{10} - \frac{106}{391} a^{9} + \frac{156}{391} a^{8} + \frac{301}{782} a^{7} - \frac{301}{782} a^{6} - \frac{174}{391} a^{5} - \frac{49}{782} a^{4} + \frac{174}{391} a^{3} - \frac{729}{1564} a^{2} + \frac{3}{68} a - \frac{205}{782}$, $\frac{1}{1133641940} a^{19} - \frac{138831}{1133641940} a^{18} + \frac{122517599}{1133641940} a^{17} + \frac{59757987}{566820970} a^{16} + \frac{12923653}{56682097} a^{15} - \frac{8215806}{56682097} a^{14} - \frac{28449331}{566820970} a^{13} - \frac{77228453}{566820970} a^{12} + \frac{64268184}{283410485} a^{11} - \frac{55624261}{566820970} a^{10} - \frac{147451311}{566820970} a^{9} + \frac{137103182}{283410485} a^{8} + \frac{63380727}{283410485} a^{7} - \frac{101288453}{566820970} a^{6} + \frac{26437911}{566820970} a^{5} - \frac{60026669}{283410485} a^{4} + \frac{478106043}{1133641940} a^{3} - \frac{93972761}{1133641940} a^{2} - \frac{339890569}{1133641940} a + \frac{63514174}{283410485}$

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:  \( -\frac{4647789}{56682097} a^{19} + \frac{30387808}{56682097} a^{18} - \frac{74920169}{56682097} a^{17} + \frac{10002271}{56682097} a^{16} + \frac{749973455}{113364194} a^{15} - \frac{986790324}{56682097} a^{14} + \frac{1338113864}{56682097} a^{13} - \frac{1308176110}{56682097} a^{12} + \frac{2290163397}{113364194} a^{11} - \frac{14309945}{2464439} a^{10} - \frac{1264556436}{56682097} a^{9} + \frac{1633977761}{56682097} a^{8} - \frac{286537867}{113364194} a^{7} - \frac{603608886}{56682097} a^{6} - \frac{115882470}{56682097} a^{5} + \frac{960165414}{56682097} a^{4} - \frac{1953594057}{113364194} a^{3} - \frac{122160917}{56682097} a^{2} + \frac{158507188}{56682097} a + \frac{86506641}{56682097} \) (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:  \( 49980.9250389 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:F_5$ (as 20T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 14 conjugacy class representatives for $C_2^2:F_5$
Character table for $C_2^2:F_5$

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.832.1, 5.1.35152.1, 10.0.79082438656.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$