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Properties

Label 20.0.10984189329...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{20}\cdot 5^{8}\cdot 73^{8}$
Root discriminant $31.77$
Ramified primes $3, 5, 73$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group 20T230

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 16, -38, 202, -401, 927, -933, 1347, -772, 1331, -772, 1347, -933, 927, -401, 202, -38, 16, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 16*x^18 - 38*x^17 + 202*x^16 - 401*x^15 + 927*x^14 - 933*x^13 + 1347*x^12 - 772*x^11 + 1331*x^10 - 772*x^9 + 1347*x^8 - 933*x^7 + 927*x^6 - 401*x^5 + 202*x^4 - 38*x^3 + 16*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 16*x^18 - 38*x^17 + 202*x^16 - 401*x^15 + 927*x^14 - 933*x^13 + 1347*x^12 - 772*x^11 + 1331*x^10 - 772*x^9 + 1347*x^8 - 933*x^7 + 927*x^6 - 401*x^5 + 202*x^4 - 38*x^3 + 16*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 16 x^{18} - 38 x^{17} + 202 x^{16} - 401 x^{15} + 927 x^{14} - 933 x^{13} + 1347 x^{12} - 772 x^{11} + 1331 x^{10} - 772 x^{9} + 1347 x^{8} - 933 x^{7} + 927 x^{6} - 401 x^{5} + 202 x^{4} - 38 x^{3} + 16 x^{2} - 2 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1098418932986448505871281640625=3^{20}\cdot 5^{8}\cdot 73^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.77$
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:  $3, 5, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{13} - \frac{1}{9} a^{12} - \frac{1}{9} a^{10} + \frac{1}{9} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{9} a^{4} - \frac{1}{3} a^{3} - \frac{1}{9} a^{2} - \frac{4}{9} a + \frac{4}{9}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{13} - \frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{8} + \frac{4}{9} a^{7} - \frac{1}{3} a^{6} + \frac{2}{9} a^{5} - \frac{4}{9} a^{4} - \frac{4}{9} a^{3} + \frac{4}{9} a^{2} - \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{2}{9} a^{7} - \frac{4}{9} a^{6} + \frac{2}{9} a^{5} - \frac{1}{3} a^{4} - \frac{2}{9} a^{3} + \frac{4}{9} a^{2} - \frac{4}{9} a + \frac{2}{9}$, $\frac{1}{27} a^{17} + \frac{1}{27} a^{16} + \frac{1}{27} a^{15} - \frac{1}{27} a^{14} - \frac{1}{27} a^{13} - \frac{1}{27} a^{12} - \frac{1}{27} a^{11} + \frac{2}{27} a^{10} - \frac{4}{27} a^{9} + \frac{1}{27} a^{8} - \frac{2}{27} a^{7} + \frac{4}{27} a^{6} + \frac{7}{27} a^{5} - \frac{8}{27} a^{4} + \frac{4}{27} a^{3} - \frac{7}{27} a^{2} + \frac{8}{27} a - \frac{4}{27}$, $\frac{1}{46935639} a^{18} - \frac{841783}{46935639} a^{17} - \frac{836587}{46935639} a^{16} - \frac{60907}{15645213} a^{15} + \frac{2075530}{46935639} a^{14} - \frac{618689}{46935639} a^{13} + \frac{7673590}{46935639} a^{12} + \frac{6119272}{46935639} a^{11} + \frac{4886725}{46935639} a^{10} + \frac{903227}{15645213} a^{9} + \frac{1410011}{46935639} a^{8} - \frac{13002655}{46935639} a^{7} + \frac{14627018}{46935639} a^{6} + \frac{16764881}{46935639} a^{5} + \frac{9028958}{46935639} a^{4} + \frac{559150}{5215071} a^{3} + \frac{18285340}{46935639} a^{2} - \frac{2580140}{46935639} a - \frac{8691784}{46935639}$, $\frac{1}{72515562255} a^{19} - \frac{106}{72515562255} a^{18} + \frac{60188992}{4834370817} a^{17} + \frac{239083852}{72515562255} a^{16} - \frac{1513391101}{72515562255} a^{15} - \frac{1260347669}{24171854085} a^{14} + \frac{135223388}{4834370817} a^{13} - \frac{1811211241}{24171854085} a^{12} - \frac{3378501917}{24171854085} a^{11} - \frac{7106189188}{72515562255} a^{10} - \frac{5261869952}{72515562255} a^{9} + \frac{1036109807}{24171854085} a^{8} - \frac{9394026539}{24171854085} a^{7} + \frac{2178438817}{4834370817} a^{6} - \frac{5823727546}{24171854085} a^{5} - \frac{15975181394}{72515562255} a^{4} - \frac{35061378247}{72515562255} a^{3} + \frac{834661133}{4834370817} a^{2} + \frac{26560082431}{72515562255} a - \frac{17267996761}{72515562255}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( -\frac{4494054401}{24171854085} a^{19} + \frac{7503901166}{24171854085} a^{18} - \frac{4572761209}{1611456939} a^{17} + \frac{146242436563}{24171854085} a^{16} - \frac{845779165804}{24171854085} a^{15} + \frac{495974566229}{8057284695} a^{14} - \frac{233271727619}{1611456939} a^{13} + \frac{98664508699}{895253855} a^{12} - \frac{1443680247208}{8057284695} a^{11} + \frac{1107197083493}{24171854085} a^{10} - \frac{4385030133218}{24171854085} a^{9} + \frac{407543786608}{8057284695} a^{8} - \frac{167378804259}{895253855} a^{7} + \frac{126300060380}{1611456939} a^{6} - \frac{786468833759}{8057284695} a^{5} + \frac{73539968254}{24171854085} a^{4} - \frac{42139256053}{24171854085} a^{3} - \frac{17191951748}{1611456939} a^{2} - \frac{2366410541}{24171854085} a - \frac{832754704}{24171854085} \) (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:  \( 9746537.71101 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T230:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1920
The 24 conjugacy class representatives for t20n230
Character table for t20n230 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.10791225.1, 10.0.349351611001875.1, 10.8.349351611001875.1, 10.2.1048054833005625.1

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

Sibling fields

Degree 10 sibling: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$3$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.12.14.6$x^{12} + 3 x^{11} + 3 x^{10} - 6 x^{9} + 3 x^{8} + 9 x^{7} + 9 x^{4} + 9 x^{3} + 9$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$73$$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.6.4.2$x^{6} - 73 x^{3} + 58619$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
73.6.4.2$x^{6} - 73 x^{3} + 58619$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$