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Properties

Label 20.8.70699355089...5389.3
Degree $20$
Signature $[8, 6]$
Discriminant $11^{16}\cdot 109^{5}$
Root discriminant $22.00$
Ramified primes $11, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T303

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-23, -26, 331, -234, -576, 1015, -808, -83, 685, -361, 72, -182, 61, 76, -25, 30, -19, -4, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 3*x^18 - 4*x^17 - 19*x^16 + 30*x^15 - 25*x^14 + 76*x^13 + 61*x^12 - 182*x^11 + 72*x^10 - 361*x^9 + 685*x^8 - 83*x^7 - 808*x^6 + 1015*x^5 - 576*x^4 - 234*x^3 + 331*x^2 - 26*x - 23)
 
gp: K = bnfinit(x^20 + 3*x^18 - 4*x^17 - 19*x^16 + 30*x^15 - 25*x^14 + 76*x^13 + 61*x^12 - 182*x^11 + 72*x^10 - 361*x^9 + 685*x^8 - 83*x^7 - 808*x^6 + 1015*x^5 - 576*x^4 - 234*x^3 + 331*x^2 - 26*x - 23, 1)
 

Normalized defining polynomial

\( x^{20} + 3 x^{18} - 4 x^{17} - 19 x^{16} + 30 x^{15} - 25 x^{14} + 76 x^{13} + 61 x^{12} - 182 x^{11} + 72 x^{10} - 361 x^{9} + 685 x^{8} - 83 x^{7} - 808 x^{6} + 1015 x^{5} - 576 x^{4} - 234 x^{3} + 331 x^{2} - 26 x - 23 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(706993550892760357993595389=11^{16}\cdot 109^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.00$
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:  $11, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{67} a^{18} + \frac{10}{67} a^{17} + \frac{11}{67} a^{16} - \frac{10}{67} a^{15} + \frac{8}{67} a^{14} + \frac{25}{67} a^{13} + \frac{25}{67} a^{12} - \frac{31}{67} a^{11} - \frac{3}{67} a^{10} + \frac{27}{67} a^{9} + \frac{15}{67} a^{8} - \frac{15}{67} a^{7} + \frac{26}{67} a^{6} + \frac{16}{67} a^{5} - \frac{25}{67} a^{4} + \frac{30}{67} a^{3} + \frac{14}{67} a^{2} + \frac{27}{67} a - \frac{17}{67}$, $\frac{1}{98543148732622962785205597049} a^{19} - \frac{665125539393151075943148989}{98543148732622962785205597049} a^{18} - \frac{34317448808303638723130471428}{98543148732622962785205597049} a^{17} - \frac{3933345725355941113893757560}{98543148732622962785205597049} a^{16} + \frac{22046601307595237175718203585}{98543148732622962785205597049} a^{15} - \frac{28345009718506198476693800510}{98543148732622962785205597049} a^{14} + \frac{34294135222512917342292026465}{98543148732622962785205597049} a^{13} + \frac{10859341482263066533465042285}{98543148732622962785205597049} a^{12} + \frac{1218500959749966230511561160}{98543148732622962785205597049} a^{11} + \frac{28391426815652063196118710777}{98543148732622962785205597049} a^{10} - \frac{8685340392921720162301176230}{98543148732622962785205597049} a^{9} - \frac{44033851901828216851336468788}{98543148732622962785205597049} a^{8} - \frac{334692938756337608635859769}{4284484727505346208052417263} a^{7} - \frac{4512976583471119164039226196}{98543148732622962785205597049} a^{6} - \frac{34373138176298353856952762906}{98543148732622962785205597049} a^{5} - \frac{47427643850166583047905479116}{98543148732622962785205597049} a^{4} + \frac{29818905712470943655393424339}{98543148732622962785205597049} a^{3} + \frac{3421019726254824169351547751}{98543148732622962785205597049} a^{2} - \frac{31493794275108388338067381377}{98543148732622962785205597049} a + \frac{143892458917170349313504018}{4284484727505346208052417263}$

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:  $13$
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:  \( 591384.592085 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T303:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 50 conjugacy class representatives for t20n303 are not computed
Character table for t20n303 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.23365118029.1

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
Arithmetically equvalently 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 $20$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R $20$ $20$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $20$ $20$ $20$

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
11Data not computed
$109$$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$