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Properties

Label 20.0.20765915743...5584.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 761^{4}$
Root discriminant $13.06$
Ramified primes $2, 3, 761$
Class number $1$
Class group Trivial
Galois group $C_2\times D_5\wr C_2$ (as 20T100)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1, 0, -2, -6, 21, -22, 10, 0, -8, 10, -2, -6, 10, -14, 15, -12, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 8*x^18 - 12*x^17 + 15*x^16 - 14*x^15 + 10*x^14 - 6*x^13 - 2*x^12 + 10*x^11 - 8*x^10 + 10*x^8 - 22*x^7 + 21*x^6 - 6*x^5 - 2*x^4 + x^2 + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 8*x^18 - 12*x^17 + 15*x^16 - 14*x^15 + 10*x^14 - 6*x^13 - 2*x^12 + 10*x^11 - 8*x^10 + 10*x^8 - 22*x^7 + 21*x^6 - 6*x^5 - 2*x^4 + x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 8 x^{18} - 12 x^{17} + 15 x^{16} - 14 x^{15} + 10 x^{14} - 6 x^{13} - 2 x^{12} + 10 x^{11} - 8 x^{10} + 10 x^{8} - 22 x^{7} + 21 x^{6} - 6 x^{5} - 2 x^{4} + x^{2} + 1 \)

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:  \(20765915743590452035584=2^{20}\cdot 3^{10}\cdot 761^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.06$
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:  $2, 3, 761$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{21074387} a^{19} + \frac{3569178}{21074387} a^{18} + \frac{14436}{726703} a^{17} + \frac{442122}{21074387} a^{16} + \frac{5934433}{21074387} a^{15} + \frac{5896798}{21074387} a^{14} + \frac{7874990}{21074387} a^{13} + \frac{4351695}{21074387} a^{12} - \frac{350608}{21074387} a^{11} - \frac{7736973}{21074387} a^{10} - \frac{10355740}{21074387} a^{9} + \frac{3579140}{21074387} a^{8} + \frac{4118861}{21074387} a^{7} - \frac{969845}{21074387} a^{6} - \frac{954471}{21074387} a^{5} - \frac{6054178}{21074387} a^{4} - \frac{9027044}{21074387} a^{3} - \frac{7880798}{21074387} a^{2} + \frac{3036052}{21074387} a + \frac{4172321}{21074387}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{833707}{21074387} a^{19} - \frac{12612780}{21074387} a^{18} + \frac{1192572}{726703} a^{17} - \frac{54045537}{21074387} a^{16} + \frac{70943463}{21074387} a^{15} - \frac{73377948}{21074387} a^{14} + \frac{46208272}{21074387} a^{13} - \frac{22510620}{21074387} a^{12} - \frac{2596166}{21074387} a^{11} + \frac{58749662}{21074387} a^{10} - \frac{66657116}{21074387} a^{9} + \frac{10542263}{21074387} a^{8} + \frac{41555560}{21074387} a^{7} - \frac{110931321}{21074387} a^{6} + \frac{146145445}{21074387} a^{5} - \frac{52742572}{21074387} a^{4} - \frac{35430538}{21074387} a^{3} + \frac{43028030}{21074387} a^{2} - \frac{3594645}{21074387} a - \frac{2945499}{21074387} \) (order $12$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3535.25125607 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_5\wr C_2$ (as 20T100):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $C_2\times D_5\wr C_2$
Character table for $C_2\times D_5\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{12})\), 10.0.593019904.1

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 R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
2Data not computed
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
761Data not computed