↑

Properties

Label 20.0.16434012871...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{23}\cdot 13^{10}$
Root discriminant $22.95$
Ramified primes $5, 13$
Class number $2$
Class group $[2]$
Galois group $F_5$ (as 20T5)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![295, -1925, 5425, -9200, 11445, -11300, 8095, -3530, 705, -305, 930, -1115, 755, -360, 145, -70, 45, -30, 15, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 15*x^18 - 30*x^17 + 45*x^16 - 70*x^15 + 145*x^14 - 360*x^13 + 755*x^12 - 1115*x^11 + 930*x^10 - 305*x^9 + 705*x^8 - 3530*x^7 + 8095*x^6 - 11300*x^5 + 11445*x^4 - 9200*x^3 + 5425*x^2 - 1925*x + 295)
 
gp: K = bnfinit(x^20 - 5*x^19 + 15*x^18 - 30*x^17 + 45*x^16 - 70*x^15 + 145*x^14 - 360*x^13 + 755*x^12 - 1115*x^11 + 930*x^10 - 305*x^9 + 705*x^8 - 3530*x^7 + 8095*x^6 - 11300*x^5 + 11445*x^4 - 9200*x^3 + 5425*x^2 - 1925*x + 295, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 15 x^{18} - 30 x^{17} + 45 x^{16} - 70 x^{15} + 145 x^{14} - 360 x^{13} + 755 x^{12} - 1115 x^{11} + 930 x^{10} - 305 x^{9} + 705 x^{8} - 3530 x^{7} + 8095 x^{6} - 11300 x^{5} + 11445 x^{4} - 9200 x^{3} + 5425 x^{2} - 1925 x + 295 \)

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:  \(1643401287186145782470703125=5^{23}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.95$
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:  $5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{1416} a^{18} + \frac{167}{1416} a^{17} + \frac{5}{59} a^{16} - \frac{5}{24} a^{15} - \frac{199}{1416} a^{14} + \frac{33}{472} a^{13} + \frac{35}{236} a^{12} - \frac{71}{472} a^{11} - \frac{259}{1416} a^{10} - \frac{16}{59} a^{9} + \frac{287}{1416} a^{8} + \frac{109}{472} a^{7} - \frac{68}{177} a^{6} - \frac{57}{472} a^{5} + \frac{149}{472} a^{4} + \frac{601}{1416} a^{3} + \frac{35}{177} a^{2} + \frac{51}{472} a - \frac{5}{24}$, $\frac{1}{44474610595407335881100112} a^{19} - \frac{2085390350221834295305}{11118652648851833970275028} a^{18} + \frac{3243417984633424839955453}{14824870198469111960366704} a^{17} - \frac{9750901016327713724257339}{44474610595407335881100112} a^{16} + \frac{4984403005096003047839881}{22237305297703667940550056} a^{15} - \frac{179269217388436577401534}{926554387404319497522919} a^{14} + \frac{3435202701338938918149599}{14824870198469111960366704} a^{13} - \frac{442434493484520255421069}{14824870198469111960366704} a^{12} + \frac{3435362002992020518883885}{11118652648851833970275028} a^{11} - \frac{5836998560236245174693361}{14824870198469111960366704} a^{10} - \frac{277150962685214865859775}{753806959244192133577968} a^{9} - \frac{2439622883540433041253001}{7412435099234555980183352} a^{8} + \frac{18994889245032583913965775}{44474610595407335881100112} a^{7} + \frac{4387312078733784625190115}{14824870198469111960366704} a^{6} - \frac{201110991796522748077977}{1853108774808638995045838} a^{5} + \frac{640069415202119538976834}{2779663162212958492568757} a^{4} + \frac{17652293417352867825625657}{44474610595407335881100112} a^{3} + \frac{4844242606543463990303719}{14824870198469111960366704} a^{2} - \frac{4070866656660858973723481}{22237305297703667940550056} a + \frac{116676006702505097930453}{251268986414730711192656}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 117824.99403 \)
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

\(\Q(\sqrt{5}) \), 4.0.21125.1, 5.1.528125.1 x5, 10.2.1394580078125.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.528125.1
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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{20}$

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
5Data not computed
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$