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Properties

Label 20.4.34109519776...8125.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{15}\cdot 97^{2}\cdot 1039^{2}\cdot 1049^{2}$
Root discriminant $21.22$
Ramified primes $5, 97, 1039, 1049$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1039

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-359, 682, 1015, -2729, -623, 4751, -1344, -4537, 3089, 2135, -2694, 23, 892, -313, 55, -59, -4, 30, -7, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 7*x^18 + 30*x^17 - 4*x^16 - 59*x^15 + 55*x^14 - 313*x^13 + 892*x^12 + 23*x^11 - 2694*x^10 + 2135*x^9 + 3089*x^8 - 4537*x^7 - 1344*x^6 + 4751*x^5 - 623*x^4 - 2729*x^3 + 1015*x^2 + 682*x - 359)
 
gp: K = bnfinit(x^20 - 3*x^19 - 7*x^18 + 30*x^17 - 4*x^16 - 59*x^15 + 55*x^14 - 313*x^13 + 892*x^12 + 23*x^11 - 2694*x^10 + 2135*x^9 + 3089*x^8 - 4537*x^7 - 1344*x^6 + 4751*x^5 - 623*x^4 - 2729*x^3 + 1015*x^2 + 682*x - 359, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 7 x^{18} + 30 x^{17} - 4 x^{16} - 59 x^{15} + 55 x^{14} - 313 x^{13} + 892 x^{12} + 23 x^{11} - 2694 x^{10} + 2135 x^{9} + 3089 x^{8} - 4537 x^{7} - 1344 x^{6} + 4751 x^{5} - 623 x^{4} - 2729 x^{3} + 1015 x^{2} + 682 x - 359 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(341095197764547393798828125=5^{15}\cdot 97^{2}\cdot 1039^{2}\cdot 1049^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.22$
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:  $5, 97, 1039, 1049$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{154870694414699035007668399} a^{19} - \frac{38430971692797328060744376}{154870694414699035007668399} a^{18} + \frac{16309994848077140708856409}{154870694414699035007668399} a^{17} - \frac{23738889828884716187176014}{154870694414699035007668399} a^{16} + \frac{72971099103257704044643127}{154870694414699035007668399} a^{15} + \frac{67803600956042358803364710}{154870694414699035007668399} a^{14} + \frac{12873520023981137087709100}{154870694414699035007668399} a^{13} + \frac{37979477880092176312758314}{154870694414699035007668399} a^{12} - \frac{63992151019456633123572251}{154870694414699035007668399} a^{11} + \frac{29549215385917916893935560}{154870694414699035007668399} a^{10} - \frac{33337352908595661045901798}{154870694414699035007668399} a^{9} + \frac{70988150295195284887139910}{154870694414699035007668399} a^{8} - \frac{60679995927656763864369070}{154870694414699035007668399} a^{7} - \frac{25173861355304987514592138}{154870694414699035007668399} a^{6} - \frac{17512492154620018184559809}{154870694414699035007668399} a^{5} + \frac{3719902580481651277579041}{154870694414699035007668399} a^{4} + \frac{21403838590767969600118896}{154870694414699035007668399} a^{3} - \frac{20437838187304809240244243}{154870694414699035007668399} a^{2} - \frac{67959685540992024096908153}{154870694414699035007668399} a - \frac{46848382148936401206254926}{154870694414699035007668399}$

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:  $11$
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:  \( 162768.021028 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1039:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 378 conjugacy class representatives for t20n1039 are not computed
Character table for t20n1039 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.4.3405971875.1

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 $20$ $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$97$97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
1039Data not computed
1049Data not computed