↑

Properties

Label 20.0.42871776200...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{15}\cdot 11^{8}$
Root discriminant $15.19$
Ramified primes $2, 5, 11$
Class number $1$
Class group Trivial
Galois group $C_5:F_5$ (as 20T27)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, 0, -40, -25, 111, -106, 8, 38, 16, -56, 30, 14, -11, -12, 10, 7, -14, 6, 2, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 2*x^18 + 6*x^17 - 14*x^16 + 7*x^15 + 10*x^14 - 12*x^13 - 11*x^12 + 14*x^11 + 30*x^10 - 56*x^9 + 16*x^8 + 38*x^7 + 8*x^6 - 106*x^5 + 111*x^4 - 25*x^3 - 40*x^2 + 25)
 
gp: K = bnfinit(x^20 - 3*x^19 + 2*x^18 + 6*x^17 - 14*x^16 + 7*x^15 + 10*x^14 - 12*x^13 - 11*x^12 + 14*x^11 + 30*x^10 - 56*x^9 + 16*x^8 + 38*x^7 + 8*x^6 - 106*x^5 + 111*x^4 - 25*x^3 - 40*x^2 + 25, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 2 x^{18} + 6 x^{17} - 14 x^{16} + 7 x^{15} + 10 x^{14} - 12 x^{13} - 11 x^{12} + 14 x^{11} + 30 x^{10} - 56 x^{9} + 16 x^{8} + 38 x^{7} + 8 x^{6} - 106 x^{5} + 111 x^{4} - 25 x^{3} - 40 x^{2} + 25 \)

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:  \(428717762000000000000000=2^{16}\cdot 5^{15}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.19$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{6} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{7} - \frac{2}{5} a^{2}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{8} - \frac{2}{5} a^{3}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{9} - \frac{2}{5} a^{4}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{17} + \frac{2}{25} a^{16} - \frac{1}{25} a^{14} - \frac{1}{25} a^{12} - \frac{2}{25} a^{11} + \frac{6}{25} a^{9} - \frac{1}{5} a^{8} - \frac{12}{25} a^{7} + \frac{11}{25} a^{6} - \frac{2}{5} a^{5} + \frac{12}{25} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{475} a^{18} - \frac{6}{475} a^{17} - \frac{31}{475} a^{16} + \frac{44}{475} a^{15} - \frac{47}{475} a^{14} - \frac{6}{475} a^{13} + \frac{31}{475} a^{12} + \frac{11}{475} a^{11} - \frac{29}{475} a^{10} + \frac{72}{475} a^{9} - \frac{227}{475} a^{8} + \frac{12}{475} a^{7} + \frac{62}{475} a^{6} - \frac{2}{25} a^{5} - \frac{9}{25} a^{4} - \frac{5}{19} a^{3} + \frac{5}{19} a^{2} - \frac{8}{19} a + \frac{3}{19}$, $\frac{1}{13775} a^{19} - \frac{7}{13775} a^{18} + \frac{146}{13775} a^{17} - \frac{723}{13775} a^{16} + \frac{384}{13775} a^{15} + \frac{164}{2755} a^{14} - \frac{1008}{13775} a^{13} + \frac{569}{13775} a^{12} + \frac{758}{13775} a^{11} - \frac{89}{13775} a^{10} + \frac{5287}{13775} a^{9} - \frac{6791}{13775} a^{8} - \frac{4662}{13775} a^{7} + \frac{2446}{13775} a^{6} + \frac{328}{725} a^{5} + \frac{3618}{13775} a^{4} - \frac{1109}{2755} a^{3} + \frac{196}{551} a^{2} + \frac{125}{551} a + \frac{225}{551}$

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:  \( -\frac{3354}{2755} a^{19} + \frac{1233}{551} a^{18} + \frac{89}{551} a^{17} - \frac{19816}{2755} a^{16} + \frac{24256}{2755} a^{15} + \frac{1031}{551} a^{14} - \frac{29017}{2755} a^{13} + \frac{359}{145} a^{12} + \frac{9404}{551} a^{11} + \frac{6592}{2755} a^{10} - \frac{95262}{2755} a^{9} + \frac{4181}{145} a^{8} + \frac{42468}{2755} a^{7} - \frac{83304}{2755} a^{6} - \frac{6696}{145} a^{5} + \frac{213689}{2755} a^{4} - \frac{120294}{2755} a^{3} - \frac{67493}{2755} a^{2} + \frac{11789}{551} a + \frac{14701}{551} \) (order $10$)
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:  \( 33038.0055622 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:F_5$ (as 20T27):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 100
The 10 conjugacy class representatives for $C_5:F_5$
Character table for $C_5:F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.2.292820000000.1 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
Degree 20 sibling: data not computed
Degree 25 sibling: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$11$11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$