Properties

Label 20.4.493...664.2
Degree $20$
Signature $[4, 8]$
Discriminant $4.934\times 10^{25}$
Root discriminant $19.26$
Ramified primes $2, 11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^4:C_5$ (as 20T23)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 14*x^17 - 36*x^16 - 24*x^15 + 76*x^14 - 174*x^13 - 197*x^12 + 210*x^11 + 34*x^10 - 176*x^9 + 19*x^8 - 58*x^7 - 100*x^6 + 16*x^5 + 13*x^4 - 2*x^3 + 10*x^2 + 4*x - 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 14*x^17 - 36*x^16 - 24*x^15 + 76*x^14 - 174*x^13 - 197*x^12 + 210*x^11 + 34*x^10 - 176*x^9 + 19*x^8 - 58*x^7 - 100*x^6 + 16*x^5 + 13*x^4 - 2*x^3 + 10*x^2 + 4*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 4, 10, -2, 13, 16, -100, -58, 19, -176, 34, 210, -197, -174, 76, -24, -36, 14, 0, -2, 1]);
 

\( x^{20} - 2 x^{19} + 14 x^{17} - 36 x^{16} - 24 x^{15} + 76 x^{14} - 174 x^{13} - 197 x^{12} + 210 x^{11} + 34 x^{10} - 176 x^{9} + 19 x^{8} - 58 x^{7} - 100 x^{6} + 16 x^{5} + 13 x^{4} - 2 x^{3} + 10 x^{2} + 4 x - 1 \)

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[4, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(49338146756019243307761664\)\(\medspace = 2^{30}\cdot 11^{16}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $19.26$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $4$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} + \frac{5}{11} a^{12} - \frac{2}{11} a^{11} + \frac{1}{11} a^{10} - \frac{1}{11} a^{8} + \frac{1}{11} a^{7} + \frac{3}{11} a^{5} - \frac{2}{11} a^{4} + \frac{2}{11} a^{3} - \frac{2}{11} a^{2} - \frac{2}{11} a + \frac{2}{11}$, $\frac{1}{11} a^{14} - \frac{5}{11} a^{12} - \frac{5}{11} a^{10} - \frac{1}{11} a^{9} - \frac{5}{11} a^{8} - \frac{5}{11} a^{7} + \frac{3}{11} a^{6} + \frac{5}{11} a^{5} + \frac{1}{11} a^{4} - \frac{1}{11} a^{3} - \frac{3}{11} a^{2} + \frac{1}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{15} + \frac{3}{11} a^{12} - \frac{4}{11} a^{11} + \frac{4}{11} a^{10} - \frac{5}{11} a^{9} + \frac{1}{11} a^{8} - \frac{3}{11} a^{7} + \frac{5}{11} a^{6} + \frac{5}{11} a^{5} - \frac{4}{11} a^{3} + \frac{2}{11} a^{2} + \frac{2}{11} a - \frac{1}{11}$, $\frac{1}{11} a^{16} + \frac{3}{11} a^{12} - \frac{1}{11} a^{11} + \frac{3}{11} a^{10} + \frac{1}{11} a^{9} + \frac{2}{11} a^{7} + \frac{5}{11} a^{6} + \frac{2}{11} a^{5} + \frac{2}{11} a^{4} - \frac{4}{11} a^{3} - \frac{3}{11} a^{2} + \frac{5}{11} a + \frac{5}{11}$, $\frac{1}{11} a^{17} - \frac{5}{11} a^{12} - \frac{2}{11} a^{11} - \frac{2}{11} a^{10} + \frac{5}{11} a^{8} + \frac{2}{11} a^{7} + \frac{2}{11} a^{6} + \frac{4}{11} a^{5} + \frac{2}{11} a^{4} + \frac{2}{11} a^{3} + \frac{5}{11}$, $\frac{1}{121} a^{18} + \frac{2}{121} a^{17} + \frac{4}{121} a^{16} + \frac{3}{121} a^{14} - \frac{1}{121} a^{13} - \frac{6}{121} a^{12} - \frac{29}{121} a^{11} - \frac{36}{121} a^{10} + \frac{39}{121} a^{9} - \frac{7}{121} a^{8} - \frac{52}{121} a^{7} + \frac{26}{121} a^{6} + \frac{56}{121} a^{5} - \frac{13}{121} a^{4} - \frac{18}{121} a^{3} + \frac{26}{121} a^{2} + \frac{20}{121} a - \frac{47}{121}$, $\frac{1}{33972148677841} a^{19} + \frac{117763882196}{33972148677841} a^{18} + \frac{413636325220}{33972148677841} a^{17} + \frac{1222272052372}{33972148677841} a^{16} - \frac{1145141287155}{33972148677841} a^{15} + \frac{401611167424}{33972148677841} a^{14} + \frac{1512310380232}{33972148677841} a^{13} - \frac{2289135694117}{33972148677841} a^{12} + \frac{9116397299993}{33972148677841} a^{11} + \frac{550831379436}{3088377152531} a^{10} + \frac{745377593097}{33972148677841} a^{9} + \frac{633333929291}{3088377152531} a^{8} - \frac{3156794737262}{33972148677841} a^{7} + \frac{13164755328393}{33972148677841} a^{6} + \frac{56376942606}{3088377152531} a^{5} - \frac{958196935655}{3088377152531} a^{4} + \frac{2860033257896}{33972148677841} a^{3} + \frac{12325588528600}{33972148677841} a^{2} + \frac{16534732396230}{33972148677841} a - \frac{16485588049519}{33972148677841}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 90919.8145635 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{4}\cdot(2\pi)^{8}\cdot 90919.8145635 \cdot 1}{2\sqrt{49338146756019243307761664}}\approx 0.251533630318$ (assuming GRH)

Galois group

$C_2^4:C_5$ (as 20T23):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 80
The 8 conjugacy class representatives for $C_2^4:C_5$
Character table for $C_2^4:C_5$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.219503494144.1, 10.2.219503494144.1, 10.6.219503494144.2

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 16 sibling: data not computed
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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$