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Properties

Label 20.2.58482521049...3792.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{15}\cdot 17^{10}\cdot 97^{4}$
Root discriminant $17.31$
Ramified primes $2, 17, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T174

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9, 69, -286, 739, -1259, 1425, -935, 134, 312, -367, 245, -5, -104, 26, 33, -12, -10, 2, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 6*x^18 + 2*x^17 - 10*x^16 - 12*x^15 + 33*x^14 + 26*x^13 - 104*x^12 - 5*x^11 + 245*x^10 - 367*x^9 + 312*x^8 + 134*x^7 - 935*x^6 + 1425*x^5 - 1259*x^4 + 739*x^3 - 286*x^2 + 69*x - 9)
 
gp: K = bnfinit(x^20 - 4*x^19 + 6*x^18 + 2*x^17 - 10*x^16 - 12*x^15 + 33*x^14 + 26*x^13 - 104*x^12 - 5*x^11 + 245*x^10 - 367*x^9 + 312*x^8 + 134*x^7 - 935*x^6 + 1425*x^5 - 1259*x^4 + 739*x^3 - 286*x^2 + 69*x - 9, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 6 x^{18} + 2 x^{17} - 10 x^{16} - 12 x^{15} + 33 x^{14} + 26 x^{13} - 104 x^{12} - 5 x^{11} + 245 x^{10} - 367 x^{9} + 312 x^{8} + 134 x^{7} - 935 x^{6} + 1425 x^{5} - 1259 x^{4} + 739 x^{3} - 286 x^{2} + 69 x - 9 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-5848252104937817609633792=-\,2^{15}\cdot 17^{10}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.31$
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:  $2, 17, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{16} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{16} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{1835084914643903929857} a^{19} + \frac{127146701684233754840}{1835084914643903929857} a^{18} - \frac{57227600481810536653}{611694971547967976619} a^{17} + \frac{819459879776080333562}{1835084914643903929857} a^{16} + \frac{64160256034418933696}{1835084914643903929857} a^{15} - \frac{27363409974653637349}{203898323849322658873} a^{14} + \frac{83642410478161235726}{203898323849322658873} a^{13} - \frac{166921226294922662119}{1835084914643903929857} a^{12} - \frac{899855135781968791037}{1835084914643903929857} a^{11} + \frac{746419320116462135410}{1835084914643903929857} a^{10} - \frac{464855216094975188056}{1835084914643903929857} a^{9} - \frac{320603509344995973391}{1835084914643903929857} a^{8} + \frac{187708284236742227584}{611694971547967976619} a^{7} + \frac{243775466840548177898}{1835084914643903929857} a^{6} - \frac{26373473746272754724}{1835084914643903929857} a^{5} + \frac{166846719566164457519}{611694971547967976619} a^{4} + \frac{389904588425968406785}{1835084914643903929857} a^{3} - \frac{482674249141435551170}{1835084914643903929857} a^{2} + \frac{534994432935291732119}{1835084914643903929857} a + \frac{58313352412903774078}{611694971547967976619}$

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:  $10$
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:  \( 22427.2922237 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T174:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 960
The 35 conjugacy class representatives for t20n174
Character table for t20n174 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.2.2312.1, 5.1.1649.1, 10.2.13359434513.1

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 24 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ $20$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ $20$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
2.10.15.13$x^{10} - 2 x^{8} - 4 x^{6} - 48 x^{2} - 96$$2$$5$$15$$C_{10}$$[3]^{5}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.12.6.1$x^{12} + 117912 x^{6} - 1419857 x^{2} + 3475809936$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$