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Properties

Label 20.0.72240803899...4989.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 37^{2}\cdot 109^{5}\cdot 241^{2}$
Root discriminant $13.90$
Ramified primes $3, 37, 109, 241$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -2, -19, 19, 42, 19, -166, 396, -691, 995, -1208, 1183, -944, 622, -346, 170, -73, 26, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 26*x^18 - 73*x^17 + 170*x^16 - 346*x^15 + 622*x^14 - 944*x^13 + 1183*x^12 - 1208*x^11 + 995*x^10 - 691*x^9 + 396*x^8 - 166*x^7 + 19*x^6 + 42*x^5 + 19*x^4 - 19*x^3 - 2*x^2 + x + 1)
 
gp: K = bnfinit(x^20 - 7*x^19 + 26*x^18 - 73*x^17 + 170*x^16 - 346*x^15 + 622*x^14 - 944*x^13 + 1183*x^12 - 1208*x^11 + 995*x^10 - 691*x^9 + 396*x^8 - 166*x^7 + 19*x^6 + 42*x^5 + 19*x^4 - 19*x^3 - 2*x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 26 x^{18} - 73 x^{17} + 170 x^{16} - 346 x^{15} + 622 x^{14} - 944 x^{13} + 1183 x^{12} - 1208 x^{11} + 995 x^{10} - 691 x^{9} + 396 x^{8} - 166 x^{7} + 19 x^{6} + 42 x^{5} + 19 x^{4} - 19 x^{3} - 2 x^{2} + x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(72240803899347741904989=3^{10}\cdot 37^{2}\cdot 109^{5}\cdot 241^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.90$
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:  $3, 37, 109, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{4065228542923783} a^{19} - \frac{12858034435664}{4065228542923783} a^{18} + \frac{9633253582333}{32009673566329} a^{17} + \frac{511689590997413}{4065228542923783} a^{16} - \frac{1500612018030663}{4065228542923783} a^{15} + \frac{498440793338599}{4065228542923783} a^{14} + \frac{101462569421497}{4065228542923783} a^{13} + \frac{1335279280328439}{4065228542923783} a^{12} + \frac{1974572181032248}{4065228542923783} a^{11} - \frac{1314950986798677}{4065228542923783} a^{10} - \frac{422824981473073}{4065228542923783} a^{9} + \frac{981391752329764}{4065228542923783} a^{8} - \frac{1560473994704942}{4065228542923783} a^{7} - \frac{1588378361270241}{4065228542923783} a^{6} - \frac{2019826360862286}{4065228542923783} a^{5} + \frac{1259678216254071}{4065228542923783} a^{4} + \frac{274206403665953}{4065228542923783} a^{3} - \frac{481061387624563}{4065228542923783} a^{2} + \frac{291858318291886}{4065228542923783} a + \frac{682576089397595}{4065228542923783}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{184003090}{469784017} a^{19} + \frac{1288527076}{469784017} a^{18} - \frac{4774001786}{469784017} a^{17} + \frac{13375406569}{469784017} a^{16} - \frac{31087434676}{469784017} a^{15} + \frac{63100915521}{469784017} a^{14} - \frac{113118991060}{469784017} a^{13} + \frac{170851925051}{469784017} a^{12} - \frac{212402268626}{469784017} a^{11} + \frac{213934530268}{469784017} a^{10} - \frac{171258238827}{469784017} a^{9} + \frac{112943784001}{469784017} a^{8} - \frac{57903364293}{469784017} a^{7} + \frac{17411293212}{469784017} a^{6} + \frac{5769996447}{469784017} a^{5} - \frac{13703433500}{469784017} a^{4} - \frac{1193534096}{469784017} a^{3} + \frac{1962896392}{469784017} a^{2} - \frac{21938048}{469784017} a + \frac{90469611}{469784017} \) (order $6$)
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:  \( 3087.8504129 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 115200
The 119 conjugacy class representatives for t20n781 are not computed
Character table for t20n781 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.981.1, 10.0.236184579.1

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 $20$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ $20$

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
3Data not computed
$37$37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.5.0.1$x^{5} - x + 13$$1$$5$$0$$C_5$$[\ ]^{5}$
37.5.0.1$x^{5} - x + 13$$1$$5$$0$$C_5$$[\ ]^{5}$
37.6.0.1$x^{6} - x + 20$$1$$6$$0$$C_6$$[\ ]^{6}$
$109$109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
109.4.0.1$x^{4} - x + 30$$1$$4$$0$$C_4$$[\ ]^{4}$
109.4.0.1$x^{4} - x + 30$$1$$4$$0$$C_4$$[\ ]^{4}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
241Data not computed