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Properties

Label 20.0.77932852281...6864.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{2}\cdot 41^{2}\cdot 53^{12}$
Root discriminant $35.04$
Ramified primes $2, 3, 41, 53$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 20T196

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5625, 0, 6549, 0, 976, 0, -2749, 0, 2807, 0, -84, 0, 571, 0, -93, 0, -36, 0, 1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + x^18 - 36*x^16 - 93*x^14 + 571*x^12 - 84*x^10 + 2807*x^8 - 2749*x^6 + 976*x^4 + 6549*x^2 + 5625)
 
gp: K = bnfinit(x^20 + x^18 - 36*x^16 - 93*x^14 + 571*x^12 - 84*x^10 + 2807*x^8 - 2749*x^6 + 976*x^4 + 6549*x^2 + 5625, 1)
 

Normalized defining polynomial

\( x^{20} + x^{18} - 36 x^{16} - 93 x^{14} + 571 x^{12} - 84 x^{10} + 2807 x^{8} - 2749 x^{6} + 976 x^{4} + 6549 x^{2} + 5625 \)

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:  \(7793285228134410479011038756864=2^{20}\cdot 3^{2}\cdot 41^{2}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.04$
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:  $2, 3, 41, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{3} - \frac{3}{8} a^{2} - \frac{3}{8} a + \frac{1}{8}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{8} a^{14} - \frac{1}{4} a^{7} - \frac{1}{2} a^{4} + \frac{3}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{15} - \frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{16} a^{16} - \frac{1}{16} a^{14} - \frac{1}{16} a^{12} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{7}{16} a^{4} - \frac{1}{4} a^{3} + \frac{7}{16} a^{2} - \frac{1}{4} a - \frac{1}{16}$, $\frac{1}{32} a^{17} - \frac{1}{32} a^{16} - \frac{1}{32} a^{15} + \frac{1}{32} a^{14} + \frac{1}{32} a^{13} - \frac{1}{32} a^{12} - \frac{1}{16} a^{11} + \frac{1}{16} a^{10} + \frac{1}{16} a^{9} - \frac{1}{16} a^{8} + \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{7}{32} a^{5} + \frac{7}{32} a^{4} - \frac{9}{32} a^{3} + \frac{9}{32} a^{2} + \frac{13}{32} a - \frac{13}{32}$, $\frac{1}{14599080061442112} a^{18} - \frac{103566684763063}{7299540030721056} a^{16} - \frac{92611633681915}{2433180010240352} a^{14} + \frac{81474722787931}{4866360020480704} a^{12} - \frac{383836027744697}{3649770015360528} a^{10} - \frac{19188467938977}{608295002560088} a^{8} + \frac{2028638393191583}{14599080061442112} a^{6} - \frac{2096669423797319}{7299540030721056} a^{4} - \frac{2521413768221671}{7299540030721056} a^{2} + \frac{994187368717629}{4866360020480704}$, $\frac{1}{2189862009216316800} a^{19} - \frac{1}{29198160122884224} a^{18} + \frac{16320398384359313}{1094931004608158400} a^{17} + \frac{103566684763063}{14599080061442112} a^{16} - \frac{19558051715604731}{364977001536052800} a^{15} + \frac{92611633681915}{4866360020480704} a^{14} + \frac{15897144789350219}{729954003072105600} a^{13} - \frac{81474722787931}{9732720040961408} a^{12} - \frac{20457571112227601}{547465502304079200} a^{11} + \frac{383836027744697}{7299540030721056} a^{10} + \frac{13363301588382959}{91244250384013200} a^{9} - \frac{284959033341067}{1216590005120176} a^{8} - \frac{151261702251950593}{2189862009216316800} a^{7} + \frac{5270901637529473}{29198160122884224} a^{6} + \frac{21626835676046113}{1094931004608158400} a^{5} - \frac{5202870606923737}{14599080061442112} a^{4} - \frac{447793355642206087}{1094931004608158400} a^{3} + \frac{6171183783582199}{14599080061442112} a^{2} + \frac{279593298541237933}{729954003072105600} a - \frac{3427367378957981}{9732720040961408}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( -\frac{55463191}{506040051600} a^{19} - \frac{13778941}{506040051600} a^{17} + \frac{706437367}{168680017200} a^{15} + \frac{624960473}{84340008600} a^{13} - \frac{19071171743}{253020025800} a^{11} + \frac{3683905999}{84340008600} a^{9} - \frac{119992706687}{506040051600} a^{7} + \frac{221815083859}{506040051600} a^{5} + \frac{118745673809}{506040051600} a^{3} - \frac{19877950091}{21085002150} a \) (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:  \( 21875805.5441 \) (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, 10.0.697911403229952.1, 10.10.2791645612919808.1

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ 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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$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$$[2]^{4}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
$53$$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$