↑

Properties

Label 20.16.6633051720...4976.2
Degree $20$
Signature $[16, 2]$
Discriminant $2^{48}\cdot 31^{6}\cdot 227^{4}$
Root discriminant $43.76$
Ramified primes $2, 31, 227$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1025

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 152, -2128, 8860, -10940, -14064, 46526, -25088, -35780, 51256, -12984, -16620, 15723, -5468, -50, 968, -425, 48, 20, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 20*x^18 + 48*x^17 - 425*x^16 + 968*x^15 - 50*x^14 - 5468*x^13 + 15723*x^12 - 16620*x^11 - 12984*x^10 + 51256*x^9 - 35780*x^8 - 25088*x^7 + 46526*x^6 - 14064*x^5 - 10940*x^4 + 8860*x^3 - 2128*x^2 + 152*x + 2)
 
gp: K = bnfinit(x^20 - 8*x^19 + 20*x^18 + 48*x^17 - 425*x^16 + 968*x^15 - 50*x^14 - 5468*x^13 + 15723*x^12 - 16620*x^11 - 12984*x^10 + 51256*x^9 - 35780*x^8 - 25088*x^7 + 46526*x^6 - 14064*x^5 - 10940*x^4 + 8860*x^3 - 2128*x^2 + 152*x + 2, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 20 x^{18} + 48 x^{17} - 425 x^{16} + 968 x^{15} - 50 x^{14} - 5468 x^{13} + 15723 x^{12} - 16620 x^{11} - 12984 x^{10} + 51256 x^{9} - 35780 x^{8} - 25088 x^{7} + 46526 x^{6} - 14064 x^{5} - 10940 x^{4} + 8860 x^{3} - 2128 x^{2} + 152 x + 2 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(663305172009703483445153131134976=2^{48}\cdot 31^{6}\cdot 227^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.76$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{17} + \frac{1}{4} a^{15} - \frac{1}{4} a^{13} + \frac{1}{4} a^{12} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{356782754350744673045181212} a^{19} - \frac{30316572349459649584550207}{356782754350744673045181212} a^{18} + \frac{10475834839545357948437803}{178391377175372336522590606} a^{17} + \frac{189401936985312218136479}{1971175438401904270967852} a^{16} - \frac{63818480321807004180068591}{178391377175372336522590606} a^{15} - \frac{14643866621294789074865543}{356782754350744673045181212} a^{14} + \frac{40905495081413318985676555}{356782754350744673045181212} a^{13} + \frac{8886609836869338481046287}{89195688587686168261295303} a^{12} - \frac{40954184158655528250992555}{89195688587686168261295303} a^{11} + \frac{16309708610297379020971874}{89195688587686168261295303} a^{10} - \frac{4995198747273620846067048}{89195688587686168261295303} a^{9} - \frac{3646327969156102743367782}{89195688587686168261295303} a^{8} + \frac{11659587814695924083697962}{89195688587686168261295303} a^{7} + \frac{31014203772068565045256290}{89195688587686168261295303} a^{6} + \frac{48628581763012097057241129}{178391377175372336522590606} a^{5} + \frac{18573044827992805926018433}{178391377175372336522590606} a^{4} - \frac{387317750578502995720494}{89195688587686168261295303} a^{3} - \frac{87993771289894179591949645}{178391377175372336522590606} a^{2} - \frac{11771772106728152349699553}{178391377175372336522590606} a + \frac{22914406946990884347479474}{89195688587686168261295303}$

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:  $17$
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:  \( 9400609439.19 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 216 conjugacy class representatives for t20n1025 are not computed
Character table for t20n1025 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 sibling: 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 $16{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ $16{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ $16{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ R $16{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$2.8.22.7$x^{8} + 2 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.12.26.64$x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{6} - 2 x^{4} + 4 x^{3} + 2$$12$$1$$26$$S_3 \times C_2^2$$[2, 3]_{3}^{2}$
31Data not computed
227Data not computed