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Properties

Label 20.0.81976336236...8176.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 163^{10}$
Root discriminant $44.23$
Ramified primes $2, 3, 163$
Class number $242$ (GRH)
Class group $[11, 22]$ (GRH)
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![622521, 0, -274590, 0, -21389, 0, 36816, 0, 2885, 0, -3306, 0, 357, 0, -12, 0, 83, 0, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 18*x^18 + 83*x^16 - 12*x^14 + 357*x^12 - 3306*x^10 + 2885*x^8 + 36816*x^6 - 21389*x^4 - 274590*x^2 + 622521)
 
gp: K = bnfinit(x^20 - 18*x^18 + 83*x^16 - 12*x^14 + 357*x^12 - 3306*x^10 + 2885*x^8 + 36816*x^6 - 21389*x^4 - 274590*x^2 + 622521, 1)
 

Normalized defining polynomial

\( x^{20} - 18 x^{18} + 83 x^{16} - 12 x^{14} + 357 x^{12} - 3306 x^{10} + 2885 x^{8} + 36816 x^{6} - 21389 x^{4} - 274590 x^{2} + 622521 \)

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:  \(819763362363033867895669628338176=2^{20}\cdot 3^{10}\cdot 163^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.23$
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:  $2, 3, 163$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{24} a^{10} - \frac{1}{8} a^{8} - \frac{5}{24} a^{6} - \frac{1}{2} a^{5} + \frac{3}{8} a^{4} - \frac{11}{24} a^{2} - \frac{1}{2} a + \frac{1}{8}$, $\frac{1}{24} a^{11} - \frac{1}{8} a^{9} - \frac{5}{24} a^{7} + \frac{3}{8} a^{5} - \frac{11}{24} a^{3} - \frac{1}{2} a^{2} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{24} a^{12} - \frac{1}{12} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} + \frac{3}{8}$, $\frac{1}{24} a^{13} - \frac{1}{12} a^{9} - \frac{1}{4} a^{7} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{48} a^{14} - \frac{1}{48} a^{10} + \frac{1}{16} a^{8} - \frac{1}{4} a^{7} + \frac{11}{48} a^{6} - \frac{1}{2} a^{5} + \frac{5}{16} a^{4} - \frac{1}{2} a^{3} + \frac{5}{24} a^{2} - \frac{1}{4} a + \frac{1}{16}$, $\frac{1}{48} a^{15} - \frac{1}{48} a^{11} + \frac{1}{16} a^{9} - \frac{1}{48} a^{7} - \frac{1}{4} a^{6} + \frac{5}{16} a^{5} - \frac{7}{24} a^{3} + \frac{5}{16} a - \frac{1}{4}$, $\frac{1}{144} a^{16} - \frac{1}{48} a^{12} - \frac{1}{48} a^{10} - \frac{5}{48} a^{8} - \frac{3}{16} a^{6} - \frac{7}{36} a^{4} - \frac{5}{48} a^{2}$, $\frac{1}{144} a^{17} - \frac{1}{48} a^{13} - \frac{1}{48} a^{11} - \frac{5}{48} a^{9} - \frac{3}{16} a^{7} - \frac{7}{36} a^{5} - \frac{5}{48} a^{3}$, $\frac{1}{1682674073161320096} a^{18} - \frac{1}{288} a^{17} - \frac{257464683640925}{280445678860220016} a^{16} - \frac{1}{96} a^{15} + \frac{115625833783843}{12747530857282728} a^{14} + \frac{1}{96} a^{13} + \frac{1236014167868861}{560891357720440032} a^{12} - \frac{5330426802192175}{280445678860220016} a^{10} - \frac{1}{24} a^{9} - \frac{197117932150367}{93481892953406672} a^{8} - \frac{1}{24} a^{7} + \frac{185046071241140909}{1682674073161320096} a^{6} - \frac{143}{288} a^{5} - \frac{65785745946761959}{140222839430110008} a^{4} - \frac{43}{96} a^{3} + \frac{20007140429518291}{93481892953406672} a^{2} + \frac{1}{32} a - \frac{72388289927901257}{186963785906813344}$, $\frac{1}{442543281241427185248} a^{19} + \frac{179576379495379877}{442543281241427185248} a^{17} - \frac{1}{288} a^{16} - \frac{27688293974697319}{13410402461861429856} a^{15} - \frac{1}{96} a^{14} + \frac{1148692504917960121}{73757213540237864208} a^{13} + \frac{1}{96} a^{12} - \frac{243271699063697961}{24585737846745954736} a^{11} + \frac{6578196862799543441}{73757213540237864208} a^{9} - \frac{1}{24} a^{8} + \frac{88455323492495390945}{442543281241427185248} a^{7} + \frac{5}{24} a^{6} + \frac{49369449236451123937}{442543281241427185248} a^{5} - \frac{143}{288} a^{4} - \frac{29087206087052887337}{147514427080475728416} a^{3} + \frac{5}{96} a^{2} - \frac{2178405192452107625}{12292868923372977368} a - \frac{7}{32}$

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

Class group and class number

$C_{11}\times C_{22}$, which has order $242$ (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:  \( -1 \) (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:  \( 2636356.57032 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-489}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-163}) \), \(\Q(\sqrt{3}, \sqrt{-163})\), 5.1.3825936.1 x5, 10.0.28631509956043776.1, 10.2.175653435313152.1 x5, 10.0.2385959163003648.1 x5

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

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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$163$163.4.2.1$x^{4} + 3423 x^{2} + 3214849$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
163.4.2.1$x^{4} + 3423 x^{2} + 3214849$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
163.4.2.1$x^{4} + 3423 x^{2} + 3214849$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
163.4.2.1$x^{4} + 3423 x^{2} + 3214849$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
163.4.2.1$x^{4} + 3423 x^{2} + 3214849$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$