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Properties

Label 20.12.3935781606...2368.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{24}\cdot 17\cdot 53^{14}$
Root discriminant $42.63$
Ramified primes $2, 17, 53$
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![-4, 38, -27, -426, 789, 1524, -3856, -1598, 6680, -108, -5071, 1054, 1877, -578, -416, 72, 60, 16, -7, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 7*x^18 + 16*x^17 + 60*x^16 + 72*x^15 - 416*x^14 - 578*x^13 + 1877*x^12 + 1054*x^11 - 5071*x^10 - 108*x^9 + 6680*x^8 - 1598*x^7 - 3856*x^6 + 1524*x^5 + 789*x^4 - 426*x^3 - 27*x^2 + 38*x - 4)
 
gp: K = bnfinit(x^20 - 4*x^19 - 7*x^18 + 16*x^17 + 60*x^16 + 72*x^15 - 416*x^14 - 578*x^13 + 1877*x^12 + 1054*x^11 - 5071*x^10 - 108*x^9 + 6680*x^8 - 1598*x^7 - 3856*x^6 + 1524*x^5 + 789*x^4 - 426*x^3 - 27*x^2 + 38*x - 4, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 7 x^{18} + 16 x^{17} + 60 x^{16} + 72 x^{15} - 416 x^{14} - 578 x^{13} + 1877 x^{12} + 1054 x^{11} - 5071 x^{10} - 108 x^{9} + 6680 x^{8} - 1598 x^{7} - 3856 x^{6} + 1524 x^{5} + 789 x^{4} - 426 x^{3} - 27 x^{2} + 38 x - 4 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(393578160617730190869682473402368=2^{24}\cdot 17\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.63$
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:  $2, 17, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{5}$, $\frac{1}{10} a^{18} + \frac{1}{10} a^{17} + \frac{1}{10} a^{16} - \frac{1}{10} a^{15} - \frac{1}{5} a^{14} - \frac{1}{10} a^{13} - \frac{1}{5} a^{12} - \frac{1}{10} a^{11} + \frac{1}{10} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{2} a^{7} - \frac{2}{5} a^{6} - \frac{3}{10} a^{5} - \frac{3}{10} a^{4} - \frac{1}{2} a^{2} - \frac{1}{10} a - \frac{1}{5}$, $\frac{1}{5551367955785970498420} a^{19} - \frac{167136318036483742073}{5551367955785970498420} a^{18} + \frac{60623407799934480253}{1387841988946492624605} a^{17} + \frac{35582002272776590477}{185045598526199016614} a^{16} - \frac{140879951751622917333}{925227992630995083070} a^{15} - \frac{3811334010264487774}{462613996315497541535} a^{14} + \frac{189749047533221192051}{2775683977892985249210} a^{13} - \frac{66421615606980105729}{462613996315497541535} a^{12} + \frac{18450826404234802759}{1110273591157194099684} a^{11} - \frac{1370558117208134467633}{5551367955785970498420} a^{10} - \frac{3354983612257423961}{19685701970872235810} a^{9} + \frac{10790158184088165431}{462613996315497541535} a^{8} + \frac{338401895726481235933}{2775683977892985249210} a^{7} - \frac{545494279334432925748}{1387841988946492624605} a^{6} - \frac{281719039965801679511}{925227992630995083070} a^{5} + \frac{255094606660209317047}{925227992630995083070} a^{4} + \frac{156852669375698020751}{370091197052398033228} a^{3} + \frac{890028963209382925313}{1850455985261990166140} a^{2} + \frac{71860041995271704901}{462613996315497541535} a - \frac{603170156058097983793}{1387841988946492624605}$

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:  $15$
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:  \( 3270260517.95 \) (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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ 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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.8.12.19$x^{8} + 12 x^{4} + 80$$4$$2$$12$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
2.8.12.19$x^{8} + 12 x^{4} + 80$$4$$2$$12$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$53$53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$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}$