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Properties

Label 20.10.1609107269...0000.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{20}\cdot 5^{10}\cdot 11^{4}\cdot 181^{4}$
Root discriminant $20.43$
Ramified primes $2, 5, 11, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T299

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, -35, -10, 222, -262, -276, 732, -183, -708, 623, 240, -469, 84, 147, -88, -10, 22, -6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 6*x^18 + 22*x^17 - 10*x^16 - 88*x^15 + 147*x^14 + 84*x^13 - 469*x^12 + 240*x^11 + 623*x^10 - 708*x^9 - 183*x^8 + 732*x^7 - 276*x^6 - 262*x^5 + 222*x^4 - 10*x^3 - 35*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - 6*x^18 + 22*x^17 - 10*x^16 - 88*x^15 + 147*x^14 + 84*x^13 - 469*x^12 + 240*x^11 + 623*x^10 - 708*x^9 - 183*x^8 + 732*x^7 - 276*x^6 - 262*x^5 + 222*x^4 - 10*x^3 - 35*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 6 x^{18} + 22 x^{17} - 10 x^{16} - 88 x^{15} + 147 x^{14} + 84 x^{13} - 469 x^{12} + 240 x^{11} + 623 x^{10} - 708 x^{9} - 183 x^{8} + 732 x^{7} - 276 x^{6} - 262 x^{5} + 222 x^{4} - 10 x^{3} - 35 x^{2} + 6 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-160910726907504640000000000=-\,2^{20}\cdot 5^{10}\cdot 11^{4}\cdot 181^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.43$
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:  $2, 5, 11, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{1442181690648887} a^{19} - \frac{380796761777037}{1442181690648887} a^{18} - \frac{212215351829809}{1442181690648887} a^{17} + \frac{538289119304410}{1442181690648887} a^{16} - \frac{526025768235029}{1442181690648887} a^{15} - \frac{327025590686620}{1442181690648887} a^{14} + \frac{52268994533162}{1442181690648887} a^{13} + \frac{13209669567215}{38977883531051} a^{12} - \frac{34948138731579}{1442181690648887} a^{11} - \frac{137319404053644}{1442181690648887} a^{10} - \frac{128887861465888}{1442181690648887} a^{9} - \frac{479372993052885}{1442181690648887} a^{8} - \frac{578482557658765}{1442181690648887} a^{7} - \frac{584203050770148}{1442181690648887} a^{6} - \frac{279972291965294}{1442181690648887} a^{5} + \frac{307752641804850}{1442181690648887} a^{4} + \frac{325765261751816}{1442181690648887} a^{3} + \frac{220668242095263}{1442181690648887} a^{2} - \frac{483651043195083}{1442181690648887} a - \frac{8350763380914}{1442181690648887}$

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:  $14$
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:  \( 458094.158162 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T299:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5000
The 230 conjugacy class representatives for t20n299 are not computed
Character table for t20n299 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.400.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R $20$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
5Data not computed
$11$11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$181$181.5.0.1$x^{5} - x + 10$$1$$5$$0$$C_5$$[\ ]^{5}$
181.5.4.4$x^{5} + 1448$$5$$1$$4$$C_5$$[\ ]_{5}$
181.5.0.1$x^{5} - x + 10$$1$$5$$0$$C_5$$[\ ]^{5}$
181.5.0.1$x^{5} - x + 10$$1$$5$$0$$C_5$$[\ ]^{5}$