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Properties

Label 20.20.5233381311...6896.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{30}\cdot 3^{6}\cdot 401^{8}$
Root discriminant $43.25$
Ramified primes $2, 3, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:D_5$ (as 20T85)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -22, 128, 804, -1887, -9850, 2876, 34834, 10210, -48164, -27774, 26248, 20419, -5434, -5942, 388, 791, -4, -48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 48*x^18 - 4*x^17 + 791*x^16 + 388*x^15 - 5942*x^14 - 5434*x^13 + 20419*x^12 + 26248*x^11 - 27774*x^10 - 48164*x^9 + 10210*x^8 + 34834*x^7 + 2876*x^6 - 9850*x^5 - 1887*x^4 + 804*x^3 + 128*x^2 - 22*x - 1)
 
gp: K = bnfinit(x^20 - 48*x^18 - 4*x^17 + 791*x^16 + 388*x^15 - 5942*x^14 - 5434*x^13 + 20419*x^12 + 26248*x^11 - 27774*x^10 - 48164*x^9 + 10210*x^8 + 34834*x^7 + 2876*x^6 - 9850*x^5 - 1887*x^4 + 804*x^3 + 128*x^2 - 22*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 48 x^{18} - 4 x^{17} + 791 x^{16} + 388 x^{15} - 5942 x^{14} - 5434 x^{13} + 20419 x^{12} + 26248 x^{11} - 27774 x^{10} - 48164 x^{9} + 10210 x^{8} + 34834 x^{7} + 2876 x^{6} - 9850 x^{5} - 1887 x^{4} + 804 x^{3} + 128 x^{2} - 22 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(523338131151796421636247986896896=2^{30}\cdot 3^{6}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.25$
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:  $2, 3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{17} - \frac{2}{9} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{4}{9} a^{12} + \frac{1}{3} a^{11} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{2}{9} a^{7} + \frac{4}{9} a^{6} + \frac{4}{9} a^{5} - \frac{1}{9} a^{4} + \frac{1}{3} a^{3} + \frac{4}{9} a^{2} + \frac{4}{9} a - \frac{4}{9}$, $\frac{1}{551333020244539827426686703} a^{19} + \frac{26344029388236521630114960}{551333020244539827426686703} a^{18} - \frac{86508696379124701826235647}{551333020244539827426686703} a^{17} + \frac{290192187437931726951853}{551333020244539827426686703} a^{16} - \frac{129075770985039143363222348}{551333020244539827426686703} a^{15} + \frac{20927675128737713628982720}{61259224471615536380742967} a^{14} - \frac{34714331209949020400404220}{551333020244539827426686703} a^{13} + \frac{20227501189139958539600140}{551333020244539827426686703} a^{12} + \frac{54998108189126773240685213}{183777673414846609142228901} a^{11} - \frac{99343241946169289713449263}{551333020244539827426686703} a^{10} + \frac{250520944724356530904453118}{551333020244539827426686703} a^{9} + \frac{176460872929014625388029157}{551333020244539827426686703} a^{8} + \frac{175388142926522384356895495}{551333020244539827426686703} a^{7} - \frac{170261290557653707123755724}{551333020244539827426686703} a^{6} + \frac{89829072562956360809864201}{183777673414846609142228901} a^{5} - \frac{226363802430148373604390109}{551333020244539827426686703} a^{4} + \frac{223069719643692161354976154}{551333020244539827426686703} a^{3} - \frac{82963501519020328119056638}{551333020244539827426686703} a^{2} + \frac{56248146653571813488822026}{183777673414846609142228901} a + \frac{24698445770300898856020452}{551333020244539827426686703}$

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:  $19$
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:  \( 16440361041.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T85):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

5.5.160801.1, 10.10.79432586038272.1, 10.10.714893274344448.2, 10.10.238297758114816.1

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 R R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed