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Properties

Label 20.0.14022744709...8125.2
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 11^{16}$
Root discriminant $22.77$
Ramified primes $5, 11$
Class number $5$
Class group $[5]$
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, 14, 54, 214, -88, -424, -114, 346, -894, -466, 1348, 1538, -782, 58, 88, -32, 18, -2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 2*x^18 + 18*x^17 - 32*x^16 + 88*x^15 + 58*x^14 - 782*x^13 + 1538*x^12 + 1348*x^11 - 466*x^10 - 894*x^9 + 346*x^8 - 114*x^7 - 424*x^6 - 88*x^5 + 214*x^4 + 54*x^3 + 14*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - 2*x^18 + 18*x^17 - 32*x^16 + 88*x^15 + 58*x^14 - 782*x^13 + 1538*x^12 + 1348*x^11 - 466*x^10 - 894*x^9 + 346*x^8 - 114*x^7 - 424*x^6 - 88*x^5 + 214*x^4 + 54*x^3 + 14*x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 2 x^{18} + 18 x^{17} - 32 x^{16} + 88 x^{15} + 58 x^{14} - 782 x^{13} + 1538 x^{12} + 1348 x^{11} - 466 x^{10} - 894 x^{9} + 346 x^{8} - 114 x^{7} - 424 x^{6} - 88 x^{5} + 214 x^{4} + 54 x^{3} + 14 x^{2} + 4 x + 1 \)

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:  \(1402274470934209014892578125=5^{15}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.77$
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:  $5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{8} - \frac{1}{10} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{4} - \frac{1}{2} a^{3} - \frac{3}{10} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{10} a^{9} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{10} a^{5} + \frac{1}{10} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a + \frac{1}{10}$, $\frac{1}{20} a^{10} - \frac{1}{5} a^{5} + \frac{3}{20}$, $\frac{1}{100} a^{11} + \frac{1}{25} a^{9} + \frac{1}{25} a^{8} - \frac{4}{25} a^{7} + \frac{1}{10} a^{6} - \frac{3}{50} a^{5} - \frac{8}{25} a^{4} + \frac{12}{25} a^{3} + \frac{12}{25} a^{2} + \frac{1}{100} a - \frac{1}{50}$, $\frac{1}{100} a^{12} - \frac{1}{100} a^{10} + \frac{1}{25} a^{9} + \frac{1}{25} a^{8} - \frac{1}{10} a^{7} + \frac{1}{25} a^{6} - \frac{3}{25} a^{5} - \frac{8}{25} a^{4} + \frac{12}{25} a^{3} + \frac{41}{100} a^{2} + \frac{2}{25} a + \frac{1}{20}$, $\frac{1}{100} a^{13} - \frac{1}{100} a^{10} - \frac{1}{50} a^{9} + \frac{1}{25} a^{8} + \frac{2}{25} a^{7} - \frac{1}{50} a^{6} + \frac{11}{50} a^{5} - \frac{17}{50} a^{4} + \frac{19}{100} a^{3} - \frac{6}{25} a^{2} + \frac{23}{50} a - \frac{17}{100}$, $\frac{1}{500} a^{14} + \frac{1}{500} a^{13} + \frac{1}{500} a^{12} + \frac{1}{500} a^{11} + \frac{3}{250} a^{10} - \frac{3}{250} a^{9} + \frac{7}{250} a^{8} - \frac{29}{125} a^{7} - \frac{23}{250} a^{6} + \frac{27}{250} a^{5} - \frac{191}{500} a^{4} + \frac{249}{500} a^{3} + \frac{239}{500} a^{2} - \frac{71}{500} a - \frac{4}{125}$, $\frac{1}{1000} a^{15} + \frac{13}{1000} a^{10} - \frac{13}{200} a^{5} + \frac{251}{1000}$, $\frac{1}{1000} a^{16} + \frac{3}{1000} a^{11} - \frac{1}{25} a^{9} - \frac{1}{25} a^{8} + \frac{4}{25} a^{7} - \frac{33}{200} a^{6} + \frac{3}{50} a^{5} + \frac{8}{25} a^{4} - \frac{12}{25} a^{3} - \frac{12}{25} a^{2} + \frac{241}{1000} a + \frac{1}{50}$, $\frac{1}{95000} a^{17} + \frac{11}{47500} a^{16} + \frac{3}{11875} a^{15} + \frac{3}{23750} a^{14} + \frac{221}{47500} a^{13} - \frac{53}{19000} a^{12} + \frac{169}{47500} a^{11} - \frac{32}{11875} a^{10} - \frac{333}{23750} a^{9} + \frac{536}{11875} a^{8} - \frac{22457}{95000} a^{7} - \frac{7301}{47500} a^{6} - \frac{1903}{23750} a^{5} + \frac{3386}{11875} a^{4} - \frac{18821}{47500} a^{3} + \frac{17129}{95000} a^{2} + \frac{3377}{9500} a + \frac{7123}{23750}$, $\frac{1}{95000} a^{18} + \frac{3}{19000} a^{16} - \frac{41}{95000} a^{15} - \frac{3}{23750} a^{14} + \frac{271}{95000} a^{13} + \frac{139}{47500} a^{12} + \frac{193}{95000} a^{11} - \frac{33}{19000} a^{10} - \frac{9}{625} a^{9} + \frac{113}{5000} a^{8} - \frac{726}{11875} a^{7} - \frac{18203}{95000} a^{6} - \frac{15683}{95000} a^{5} + \frac{6003}{23750} a^{4} - \frac{2153}{5000} a^{3} + \frac{12861}{47500} a^{2} + \frac{45267}{95000} a + \frac{31241}{95000}$, $\frac{1}{95000} a^{19} + \frac{9}{95000} a^{16} + \frac{1}{11875} a^{15} + \frac{91}{95000} a^{14} + \frac{149}{47500} a^{13} + \frac{46}{11875} a^{12} - \frac{59}{19000} a^{11} - \frac{47}{23750} a^{10} + \frac{3127}{95000} a^{9} - \frac{216}{11875} a^{8} - \frac{2037}{23750} a^{7} + \frac{17147}{95000} a^{6} + \frac{174}{11875} a^{5} - \frac{10227}{95000} a^{4} - \frac{8349}{47500} a^{3} - \frac{8267}{23750} a^{2} + \frac{9571}{95000} a + \frac{157}{950}$

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

Class group and class number

$C_{5}$, which has order $5$

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:  \( -\frac{33}{2375} a^{19} + \frac{77}{2375} a^{18} + \frac{22}{2375} a^{17} - \frac{4521}{19000} a^{16} + \frac{1287}{2375} a^{15} - \frac{3663}{2375} a^{14} - \frac{143}{2375} a^{13} + \frac{24332}{2375} a^{12} - \frac{95957}{3800} a^{11} - \frac{10043}{2375} a^{10} - \frac{7931}{2375} a^{9} + \frac{3179}{2375} a^{8} - \frac{3586}{2375} a^{7} + \frac{342397}{19000} a^{6} + \frac{11}{125} a^{5} + \frac{2651}{2375} a^{4} + \frac{671}{2375} a^{3} + \frac{176}{2375} a^{2} - \frac{74459}{19000} a + \frac{11}{2375} \) (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:  \( 643711.144226 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.1830125.1 x5, 10.2.16746787578125.1 x5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 5 sibling: 5.1.1830125.1
Degree 10 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ 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
$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$$[\ ]_{4}$
$11$11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$