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Properties

Label 20.18.2056246033...4256.1
Degree $20$
Signature $[18, 1]$
Discriminant $-\,2^{48}\cdot 31^{7}\cdot 227^{4}$
Root discriminant $51.96$
Ramified primes $2, 31, 227$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1037

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-62, -992, 2080, 16520, -12407, -76720, 20782, 139896, 13068, -120040, -41946, 49224, 25770, -9536, -6638, 808, 806, -24, -46, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 46*x^18 - 24*x^17 + 806*x^16 + 808*x^15 - 6638*x^14 - 9536*x^13 + 25770*x^12 + 49224*x^11 - 41946*x^10 - 120040*x^9 + 13068*x^8 + 139896*x^7 + 20782*x^6 - 76720*x^5 - 12407*x^4 + 16520*x^3 + 2080*x^2 - 992*x - 62)
 
gp: K = bnfinit(x^20 - 46*x^18 - 24*x^17 + 806*x^16 + 808*x^15 - 6638*x^14 - 9536*x^13 + 25770*x^12 + 49224*x^11 - 41946*x^10 - 120040*x^9 + 13068*x^8 + 139896*x^7 + 20782*x^6 - 76720*x^5 - 12407*x^4 + 16520*x^3 + 2080*x^2 - 992*x - 62, 1)
 

Normalized defining polynomial

\( x^{20} - 46 x^{18} - 24 x^{17} + 806 x^{16} + 808 x^{15} - 6638 x^{14} - 9536 x^{13} + 25770 x^{12} + 49224 x^{11} - 41946 x^{10} - 120040 x^{9} + 13068 x^{8} + 139896 x^{7} + 20782 x^{6} - 76720 x^{5} - 12407 x^{4} + 16520 x^{3} + 2080 x^{2} - 992 x - 62 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 1]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-20562460332300807986799747065184256=-\,2^{48}\cdot 31^{7}\cdot 227^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.96$
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:  $2, 31, 227$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{12} + \frac{1}{4} a^{8} + \frac{1}{4} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{13} + \frac{1}{4} a^{9} + \frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{16} + \frac{1}{8} a^{14} - \frac{1}{8} a^{12} + \frac{1}{8} a^{10} + \frac{3}{8} a^{8} + \frac{3}{8} a^{6} - \frac{3}{8} a^{4} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{100245429127554572143391930207725232} a^{19} - \frac{2330923604729996811399265631578161}{100245429127554572143391930207725232} a^{18} + \frac{5429787034902049742340249378547815}{100245429127554572143391930207725232} a^{17} + \frac{8372256946762443900468939336281829}{100245429127554572143391930207725232} a^{16} + \frac{8389813781820486642822058524371997}{100245429127554572143391930207725232} a^{15} - \frac{24789767411689938502513109282526413}{100245429127554572143391930207725232} a^{14} - \frac{4130808066583842089540088229153325}{100245429127554572143391930207725232} a^{13} - \frac{12460349315290101082126837323296711}{100245429127554572143391930207725232} a^{12} - \frac{17247100655720684926981597407689307}{100245429127554572143391930207725232} a^{11} - \frac{3864682621917072609996691298389421}{100245429127554572143391930207725232} a^{10} + \frac{16180026993629479573285084511828623}{100245429127554572143391930207725232} a^{9} - \frac{45015911916180269056056259773088867}{100245429127554572143391930207725232} a^{8} + \frac{34018692324157499515922352284072371}{100245429127554572143391930207725232} a^{7} + \frac{40349083849881303909512021938967037}{100245429127554572143391930207725232} a^{6} - \frac{2374574603388870972132104793345307}{100245429127554572143391930207725232} a^{5} - \frac{27910494463058553088279717179508769}{100245429127554572143391930207725232} a^{4} + \frac{22618521558897866629716168358681655}{50122714563777286071695965103862616} a^{3} - \frac{17723314011579280357413683671679883}{50122714563777286071695965103862616} a^{2} + \frac{565360385596555788753090806348895}{50122714563777286071695965103862616} a - \frac{17422418670221640970885963104178931}{50122714563777286071695965103862616}$

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:  $18$
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:  \( 296328893050 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1037:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 384 conjugacy class representatives for t20n1037 are not computed
Character table for t20n1037 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.207699287474176.1

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 $16{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ $16{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
2.12.26.27$x^{12} - 2 x^{10} + 4 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{3} + 2$$12$$1$$26$12T48$[4/3, 4/3, 2, 3]_{3}^{2}$
31Data not computed
227Data not computed