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Properties

Label 20.8.41962377418...6576.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 29\cdot 53^{14}$
Root discriminant $38.12$
Ramified primes $2, 29, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T513

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![153, -720, 422, 8932, -39280, 77324, -81637, 42398, -667, -10874, 3184, 202, 1627, -930, -268, 46, 84, 10, -7, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 7*x^18 + 10*x^17 + 84*x^16 + 46*x^15 - 268*x^14 - 930*x^13 + 1627*x^12 + 202*x^11 + 3184*x^10 - 10874*x^9 - 667*x^8 + 42398*x^7 - 81637*x^6 + 77324*x^5 - 39280*x^4 + 8932*x^3 + 422*x^2 - 720*x + 153)
 
gp: K = bnfinit(x^20 - 4*x^19 - 7*x^18 + 10*x^17 + 84*x^16 + 46*x^15 - 268*x^14 - 930*x^13 + 1627*x^12 + 202*x^11 + 3184*x^10 - 10874*x^9 - 667*x^8 + 42398*x^7 - 81637*x^6 + 77324*x^5 - 39280*x^4 + 8932*x^3 + 422*x^2 - 720*x + 153, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 7 x^{18} + 10 x^{17} + 84 x^{16} + 46 x^{15} - 268 x^{14} - 930 x^{13} + 1627 x^{12} + 202 x^{11} + 3184 x^{10} - 10874 x^{9} - 667 x^{8} + 42398 x^{7} - 81637 x^{6} + 77324 x^{5} - 39280 x^{4} + 8932 x^{3} + 422 x^{2} - 720 x + 153 \)

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:  \(41962377418802115938311734296576=2^{20}\cdot 29\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.12$
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:  $2, 29, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{8} a^{16} - \frac{1}{4} a^{15} + \frac{3}{8} a^{12} - \frac{1}{4} a^{11} - \frac{3}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a - \frac{3}{8}$, $\frac{1}{24} a^{17} + \frac{1}{6} a^{14} + \frac{7}{24} a^{13} + \frac{3}{8} a^{11} + \frac{7}{24} a^{9} + \frac{1}{12} a^{8} - \frac{1}{3} a^{7} + \frac{1}{6} a^{6} + \frac{5}{12} a^{5} + \frac{5}{12} a^{4} + \frac{1}{24} a^{3} - \frac{1}{12} a^{2} + \frac{1}{24} a + \frac{1}{4}$, $\frac{1}{240} a^{18} + \frac{1}{80} a^{17} - \frac{3}{80} a^{16} + \frac{29}{120} a^{15} - \frac{53}{240} a^{14} - \frac{13}{80} a^{13} + \frac{13}{40} a^{12} - \frac{9}{80} a^{11} - \frac{43}{120} a^{10} - \frac{19}{240} a^{9} + \frac{79}{240} a^{8} - \frac{11}{60} a^{7} + \frac{59}{120} a^{6} + \frac{19}{60} a^{5} - \frac{59}{240} a^{4} + \frac{43}{240} a^{3} + \frac{59}{120} a^{2} - \frac{37}{80} a + \frac{7}{80}$, $\frac{1}{12460882594813197068636678545680} a^{19} - \frac{5795973958237029954758993237}{12460882594813197068636678545680} a^{18} + \frac{169290305275784789888690538101}{12460882594813197068636678545680} a^{17} - \frac{151283443156995734202968923463}{3115220648703299267159169636420} a^{16} - \frac{2617274880426474903012028119193}{12460882594813197068636678545680} a^{15} - \frac{354318078347465629082260262213}{4153627531604399022878892848560} a^{14} - \frac{387136579100831289137288987587}{778805162175824816789792409105} a^{13} - \frac{1581746897360053984095132849039}{4153627531604399022878892848560} a^{12} - \frac{293126133764199200042142046919}{3115220648703299267159169636420} a^{11} + \frac{1745181738119939428271561578877}{4153627531604399022878892848560} a^{10} + \frac{1222385132289141004066876404989}{12460882594813197068636678545680} a^{9} - \frac{284881579673075750079165234299}{2076813765802199511439446424280} a^{8} - \frac{2401387090692185267003566134901}{6230441297406598534318339272840} a^{7} + \frac{190342682587199784738615174829}{3115220648703299267159169636420} a^{6} - \frac{4511566017369117462447496628039}{12460882594813197068636678545680} a^{5} + \frac{2978581221927699244548267212803}{12460882594813197068636678545680} a^{4} + \frac{239995321858847141141075430028}{778805162175824816789792409105} a^{3} - \frac{112186020348603263433573864187}{4153627531604399022878892848560} a^{2} - \frac{1048491657881940659338269766309}{12460882594813197068636678545680} a + \frac{145240276940697731741080980761}{415362753160439902287889284856}$

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:  \( 387948686.111 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T513:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 128 conjugacy class representatives for t20n513 are not computed
Character table for t20n513 is not computed

Intermediate fields

\(\Q(\sqrt{53}) \), 5.5.2382032.1, 10.10.300726051798272.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
$53$53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$