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Properties

Label 20.8.19833781175...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{4}\cdot 5^{13}\cdot 6329^{5}$
Root discriminant $29.17$
Ramified primes $2, 5, 6329$
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![-1, -2, 28, 36, -125, 30, 377, -59, -487, 423, 820, -310, -623, 170, 358, 10, -59, -11, -7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^18 - 11*x^17 - 59*x^16 + 10*x^15 + 358*x^14 + 170*x^13 - 623*x^12 - 310*x^11 + 820*x^10 + 423*x^9 - 487*x^8 - 59*x^7 + 377*x^6 + 30*x^5 - 125*x^4 + 36*x^3 + 28*x^2 - 2*x - 1)
 
gp: K = bnfinit(x^20 - 7*x^18 - 11*x^17 - 59*x^16 + 10*x^15 + 358*x^14 + 170*x^13 - 623*x^12 - 310*x^11 + 820*x^10 + 423*x^9 - 487*x^8 - 59*x^7 + 377*x^6 + 30*x^5 - 125*x^4 + 36*x^3 + 28*x^2 - 2*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{18} - 11 x^{17} - 59 x^{16} + 10 x^{15} + 358 x^{14} + 170 x^{13} - 623 x^{12} - 310 x^{11} + 820 x^{10} + 423 x^{9} - 487 x^{8} - 59 x^{7} + 377 x^{6} + 30 x^{5} - 125 x^{4} + 36 x^{3} + 28 x^{2} - 2 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(198337811759061770488281250000=2^{4}\cdot 5^{13}\cdot 6329^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.17$
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:  $2, 5, 6329$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{70898492468862962795469709} a^{19} + \frac{6206032133341937617856616}{70898492468862962795469709} a^{18} + \frac{5694108179413174327318761}{70898492468862962795469709} a^{17} + \frac{27331829870564891150071177}{70898492468862962795469709} a^{16} + \frac{23926703697734710317820886}{70898492468862962795469709} a^{15} - \frac{26535621274375841076709422}{70898492468862962795469709} a^{14} - \frac{2898513560014468454510372}{70898492468862962795469709} a^{13} - \frac{4819127317738998536762959}{70898492468862962795469709} a^{12} - \frac{32460721892051110327202916}{70898492468862962795469709} a^{11} + \frac{13275341206362695658110482}{70898492468862962795469709} a^{10} - \frac{17832715890882605428937928}{70898492468862962795469709} a^{9} - \frac{25329876334124515507236357}{70898492468862962795469709} a^{8} - \frac{130729880842245760174420}{70898492468862962795469709} a^{7} + \frac{23492244169070917195523151}{70898492468862962795469709} a^{6} - \frac{9992030024107418698109417}{70898492468862962795469709} a^{5} + \frac{8723194324356323540059155}{70898492468862962795469709} a^{4} + \frac{32552183990307223975534969}{70898492468862962795469709} a^{3} - \frac{8027235070776030981627418}{70898492468862962795469709} a^{2} + \frac{30058522925253129013267706}{70898492468862962795469709} a - \frac{20847444245111453833168854}{70898492468862962795469709}$

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:  $13$
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:  \( 17541014.4038 \) (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

\(\Q(\sqrt{5}) \), 10.6.625878765625.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.8.0.1}{8} }$ $16{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.8.0.1}{8} }$ $16{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
2Data not computed
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
6329Data not computed