↑

Properties

Label 20.4.10054202156...4353.2
Degree $20$
Signature $[4, 8]$
Discriminant $17^{15}\cdot 37^{8}$
Root discriminant $35.49$
Ramified primes $17, 37$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 20T135

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 18, 5, -160, -302, 381, 1599, 250, -4973, -9788, -9877, -5657, -924, 881, 276, -216, -85, 12, 0, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 12*x^17 - 85*x^16 - 216*x^15 + 276*x^14 + 881*x^13 - 924*x^12 - 5657*x^11 - 9877*x^10 - 9788*x^9 - 4973*x^8 + 250*x^7 + 1599*x^6 + 381*x^5 - 302*x^4 - 160*x^3 + 5*x^2 + 18*x + 1)
 
gp: K = bnfinit(x^20 - x^19 + 12*x^17 - 85*x^16 - 216*x^15 + 276*x^14 + 881*x^13 - 924*x^12 - 5657*x^11 - 9877*x^10 - 9788*x^9 - 4973*x^8 + 250*x^7 + 1599*x^6 + 381*x^5 - 302*x^4 - 160*x^3 + 5*x^2 + 18*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 12 x^{17} - 85 x^{16} - 216 x^{15} + 276 x^{14} + 881 x^{13} - 924 x^{12} - 5657 x^{11} - 9877 x^{10} - 9788 x^{9} - 4973 x^{8} + 250 x^{7} + 1599 x^{6} + 381 x^{5} - 302 x^{4} - 160 x^{3} + 5 x^{2} + 18 x + 1 \)

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:  \(10054202156858080231167941574353=17^{15}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.49$
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:  $17, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{17} a^{18} + \frac{2}{17} a^{17} + \frac{8}{17} a^{16} + \frac{6}{17} a^{15} + \frac{4}{17} a^{12} - \frac{8}{17} a^{11} - \frac{5}{17} a^{10} + \frac{7}{17} a^{9} - \frac{6}{17} a^{8} - \frac{4}{17} a^{6} - \frac{7}{17} a^{4} + \frac{3}{17} a^{3} - \frac{1}{17} a^{2} - \frac{4}{17} a + \frac{8}{17}$, $\frac{1}{52559920535318754987352039} a^{19} - \frac{706395850149179434029691}{52559920535318754987352039} a^{18} + \frac{5910790679276107006512319}{52559920535318754987352039} a^{17} + \frac{6224609623995801830889445}{52559920535318754987352039} a^{16} - \frac{24162015260906642579529012}{52559920535318754987352039} a^{15} + \frac{1105453686450808204026716}{3091760031489338528667767} a^{14} - \frac{22182298213521488260215943}{52559920535318754987352039} a^{13} + \frac{7262560440024591157659717}{52559920535318754987352039} a^{12} - \frac{6717873619503244878818708}{52559920535318754987352039} a^{11} - \frac{9728406048599899426991159}{52559920535318754987352039} a^{10} - \frac{10271419290767282970674306}{52559920535318754987352039} a^{9} - \frac{10925039365568787617864678}{52559920535318754987352039} a^{8} - \frac{10295301665899023654987420}{52559920535318754987352039} a^{7} - \frac{10843971807051662009417337}{52559920535318754987352039} a^{6} + \frac{25296713651345799438028122}{52559920535318754987352039} a^{5} + \frac{7903513884982488737693473}{52559920535318754987352039} a^{4} + \frac{1856829774791204443152462}{52559920535318754987352039} a^{3} + \frac{9982638981328090478520066}{52559920535318754987352039} a^{2} + \frac{6204836753971878280901415}{52559920535318754987352039} a - \frac{2495873246824939644658578}{52559920535318754987352039}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 9405206.70203 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T135:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 22 conjugacy class representatives for t20n135
Character table for t20n135 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 5.5.6725897.1, 10.10.769040737728353.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$17$17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
$37$37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$