↑

Properties

Label 20.0.31551644091...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{35}\cdot 251^{2}\cdot 4051^{2}$
Root discriminant $133.34$
Ramified primes $2, 5, 251, 4051$
Class number $38367360$ (GRH)
Class group $[2, 2, 2, 12, 399660]$ (GRH)
Galois group $C_4\times C_2^4:C_5$ (as 20T75)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5169421368005, 0, 5948819670525, 0, 2534690146825, 0, 513229518325, 0, 55944354250, 0, 3518361755, 0, 133406025, 0, 3089950, 0, 42620, 0, 320, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 320*x^18 + 42620*x^16 + 3089950*x^14 + 133406025*x^12 + 3518361755*x^10 + 55944354250*x^8 + 513229518325*x^6 + 2534690146825*x^4 + 5948819670525*x^2 + 5169421368005)
 
gp: K = bnfinit(x^20 + 320*x^18 + 42620*x^16 + 3089950*x^14 + 133406025*x^12 + 3518361755*x^10 + 55944354250*x^8 + 513229518325*x^6 + 2534690146825*x^4 + 5948819670525*x^2 + 5169421368005, 1)
 

Normalized defining polynomial

\( x^{20} + 320 x^{18} + 42620 x^{16} + 3089950 x^{14} + 133406025 x^{12} + 3518361755 x^{10} + 55944354250 x^{8} + 513229518325 x^{6} + 2534690146825 x^{4} + 5948819670525 x^{2} + 5169421368005 \)

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:  \(3155164409182739257812500000000000000000000=2^{20}\cdot 5^{35}\cdot 251^{2}\cdot 4051^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $133.34$
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:  $2, 5, 251, 4051$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{101} a^{16} - \frac{31}{101} a^{14} - \frac{8}{101} a^{12} - \frac{8}{101} a^{10} - \frac{13}{101} a^{8} + \frac{42}{101} a^{6} - \frac{14}{101} a^{4} - \frac{16}{101} a^{2} + \frac{47}{101}$, $\frac{1}{101} a^{17} - \frac{31}{101} a^{15} - \frac{8}{101} a^{13} - \frac{8}{101} a^{11} - \frac{13}{101} a^{9} + \frac{42}{101} a^{7} - \frac{14}{101} a^{5} - \frac{16}{101} a^{3} + \frac{47}{101} a$, $\frac{1}{4646471356354126959243969266768604318503043198392749} a^{18} + \frac{21738000948891620982627141085606980430872308589869}{4646471356354126959243969266768604318503043198392749} a^{16} - \frac{1673009388479757324000700431910413211758102732105333}{4646471356354126959243969266768604318503043198392749} a^{14} - \frac{2318456618591067998719941090330163358215000335030507}{4646471356354126959243969266768604318503043198392749} a^{12} - \frac{244556699919204390289749701205178605116570401502547}{4646471356354126959243969266768604318503043198392749} a^{10} + \frac{126460369284492567016828786789760402498048625840401}{4646471356354126959243969266768604318503043198392749} a^{8} + \frac{2067644785572527378453565052060978079792281257171593}{4646471356354126959243969266768604318503043198392749} a^{6} + \frac{2172896608249019945294755786521922707548752510328325}{4646471356354126959243969266768604318503043198392749} a^{4} + \frac{621997707069666239309328280661745337649397281968126}{4646471356354126959243969266768604318503043198392749} a^{2} + \frac{78282060386881851185706893421431167044449954}{4569695895611950577589881664916344809360969549}$, $\frac{1}{4646471356354126959243969266768604318503043198392749} a^{19} + \frac{21738000948891620982627141085606980430872308589869}{4646471356354126959243969266768604318503043198392749} a^{17} - \frac{1673009388479757324000700431910413211758102732105333}{4646471356354126959243969266768604318503043198392749} a^{15} - \frac{2318456618591067998719941090330163358215000335030507}{4646471356354126959243969266768604318503043198392749} a^{13} - \frac{244556699919204390289749701205178605116570401502547}{4646471356354126959243969266768604318503043198392749} a^{11} + \frac{126460369284492567016828786789760402498048625840401}{4646471356354126959243969266768604318503043198392749} a^{9} + \frac{2067644785572527378453565052060978079792281257171593}{4646471356354126959243969266768604318503043198392749} a^{7} + \frac{2172896608249019945294755786521922707548752510328325}{4646471356354126959243969266768604318503043198392749} a^{5} + \frac{621997707069666239309328280661745337649397281968126}{4646471356354126959243969266768604318503043198392749} a^{3} + \frac{78282060386881851185706893421431167044449954}{4569695895611950577589881664916344809360969549} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{12}\times C_{399660}$, which has order $38367360$ (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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 161406.837641 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times C_2^4:C_5$ (as 20T75):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 32 conjugacy class representatives for $C_4\times C_2^4:C_5$
Character table for $C_4\times C_2^4:C_5$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.390625.1, \(\Q(\zeta_{25})^+\)

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

Sibling fields

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 $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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

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
5Data not computed
251Data not computed
4051Data not computed