↑

Properties

Label 20.20.2829356741...3125.1
Degree $20$
Signature $[20, 0]$
Discriminant $5^{27}\cdot 11^{14}$
Root discriminant $47.05$
Ramified primes $5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5\times C_5:C_4$ (as 20T25)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-103589, 592885, -484030, -1829365, 2195650, 2212556, -2831085, -1588305, 1778530, 754985, -617264, -228115, 124360, 40885, -15180, -4174, 1140, 225, -50, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 50*x^18 + 225*x^17 + 1140*x^16 - 4174*x^15 - 15180*x^14 + 40885*x^13 + 124360*x^12 - 228115*x^11 - 617264*x^10 + 754985*x^9 + 1778530*x^8 - 1588305*x^7 - 2831085*x^6 + 2212556*x^5 + 2195650*x^4 - 1829365*x^3 - 484030*x^2 + 592885*x - 103589)
 
gp: K = bnfinit(x^20 - 5*x^19 - 50*x^18 + 225*x^17 + 1140*x^16 - 4174*x^15 - 15180*x^14 + 40885*x^13 + 124360*x^12 - 228115*x^11 - 617264*x^10 + 754985*x^9 + 1778530*x^8 - 1588305*x^7 - 2831085*x^6 + 2212556*x^5 + 2195650*x^4 - 1829365*x^3 - 484030*x^2 + 592885*x - 103589, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 50 x^{18} + 225 x^{17} + 1140 x^{16} - 4174 x^{15} - 15180 x^{14} + 40885 x^{13} + 124360 x^{12} - 228115 x^{11} - 617264 x^{10} + 754985 x^{9} + 1778530 x^{8} - 1588305 x^{7} - 2831085 x^{6} + 2212556 x^{5} + 2195650 x^{4} - 1829365 x^{3} - 484030 x^{2} + 592885 x - 103589 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2829356741780348122119903564453125=5^{27}\cdot 11^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.05$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{19} a^{18} - \frac{6}{19} a^{17} + \frac{5}{19} a^{16} + \frac{2}{19} a^{15} - \frac{4}{19} a^{14} - \frac{6}{19} a^{13} + \frac{1}{19} a^{12} + \frac{6}{19} a^{11} - \frac{9}{19} a^{10} - \frac{2}{19} a^{9} + \frac{6}{19} a^{8} - \frac{8}{19} a^{7} - \frac{5}{19} a^{6} - \frac{7}{19} a^{5} + \frac{5}{19} a^{3} + \frac{5}{19} a^{2} + \frac{5}{19} a + \frac{7}{19}$, $\frac{1}{845588391879354345143648500390339} a^{19} - \frac{224428104120826702415157158164}{845588391879354345143648500390339} a^{18} + \frac{26677349492397367649226392741578}{845588391879354345143648500390339} a^{17} + \frac{209817158299727689158378013057279}{845588391879354345143648500390339} a^{16} - \frac{211953889077131739306734502032009}{845588391879354345143648500390339} a^{15} + \frac{16861860375667652056866956361502}{44504652204176544481244657915281} a^{14} + \frac{320630279953944106631412691075206}{845588391879354345143648500390339} a^{13} + \frac{259629325240426508000986888216983}{845588391879354345143648500390339} a^{12} + \frac{203193390044495537700324174380800}{845588391879354345143648500390339} a^{11} - \frac{380348981271496377810835664340677}{845588391879354345143648500390339} a^{10} - \frac{405040829912575035256310553521404}{845588391879354345143648500390339} a^{9} - \frac{374486541198980773360673146181082}{845588391879354345143648500390339} a^{8} + \frac{58833164480051625594065551115491}{845588391879354345143648500390339} a^{7} + \frac{375458871966136087945094469600088}{845588391879354345143648500390339} a^{6} - \frac{272395950362761114875895043967474}{845588391879354345143648500390339} a^{5} - \frac{236734820634415580606140150963708}{845588391879354345143648500390339} a^{4} + \frac{352835359985240987496003646067087}{845588391879354345143648500390339} a^{3} - \frac{258376014019852200468441900255169}{845588391879354345143648500390339} a^{2} + \frac{203891714388596229646735831547541}{845588391879354345143648500390339} a + \frac{3395701196134609997234509640378}{11909695660272596410473922540709}$

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:  $19$
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:  \( 29332960138.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:C_4$ (as 20T25):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 100
The 40 conjugacy class representatives for $C_5\times C_5:C_4$
Character table for $C_5\times C_5:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.15125.1, 10.10.17872314453125.1

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

Sibling fields

Degree 20 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 $20$ $20$ R $20$ R $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.10.9.9$x^{10} + 297$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.5.2$x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$