↑

Properties

Label 20.0.73811250000...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{23}$
Root discriminant $22.05$
Ramified primes $2, 3, 5$
Class number $4$
Class group $[2, 2]$
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![256, 640, 160, -880, -980, -172, 730, 1010, 155, -950, -261, 350, 20, 30, -25, -18, 20, -10, 5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 5*x^18 - 10*x^17 + 20*x^16 - 18*x^15 - 25*x^14 + 30*x^13 + 20*x^12 + 350*x^11 - 261*x^10 - 950*x^9 + 155*x^8 + 1010*x^7 + 730*x^6 - 172*x^5 - 980*x^4 - 880*x^3 + 160*x^2 + 640*x + 256)
 
gp: K = bnfinit(x^20 + 5*x^18 - 10*x^17 + 20*x^16 - 18*x^15 - 25*x^14 + 30*x^13 + 20*x^12 + 350*x^11 - 261*x^10 - 950*x^9 + 155*x^8 + 1010*x^7 + 730*x^6 - 172*x^5 - 980*x^4 - 880*x^3 + 160*x^2 + 640*x + 256, 1)
 

Normalized defining polynomial

\( x^{20} + 5 x^{18} - 10 x^{17} + 20 x^{16} - 18 x^{15} - 25 x^{14} + 30 x^{13} + 20 x^{12} + 350 x^{11} - 261 x^{10} - 950 x^{9} + 155 x^{8} + 1010 x^{7} + 730 x^{6} - 172 x^{5} - 980 x^{4} - 880 x^{3} + 160 x^{2} + 640 x + 256 \)

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:  \(738112500000000000000000000=2^{20}\cdot 3^{10}\cdot 5^{23}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.05$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{9} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{4} + \frac{1}{6} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{12} a^{12} + \frac{1}{6} a^{10} - \frac{1}{12} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{5} + \frac{1}{4} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a$, $\frac{1}{12} a^{13} + \frac{1}{12} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{36} a^{14} - \frac{1}{36} a^{13} + \frac{1}{36} a^{12} + \frac{1}{18} a^{11} + \frac{1}{12} a^{10} - \frac{1}{36} a^{9} - \frac{1}{36} a^{8} - \frac{1}{9} a^{7} - \frac{1}{4} a^{6} + \frac{17}{36} a^{5} - \frac{7}{36} a^{4} - \frac{1}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{9}$, $\frac{1}{72} a^{15} - \frac{1}{72} a^{14} + \frac{1}{72} a^{13} + \frac{1}{36} a^{12} - \frac{1}{24} a^{11} - \frac{1}{72} a^{10} - \frac{13}{72} a^{9} - \frac{1}{18} a^{8} - \frac{1}{24} a^{7} + \frac{5}{72} a^{6} + \frac{29}{72} a^{5} + \frac{5}{18} a^{4} + \frac{13}{36} a^{3} - \frac{1}{3} a^{2} - \frac{5}{18} a - \frac{1}{3}$, $\frac{1}{72} a^{16} - \frac{1}{24} a^{13} - \frac{1}{72} a^{12} - \frac{1}{18} a^{11} - \frac{7}{36} a^{10} + \frac{13}{72} a^{9} + \frac{17}{72} a^{8} - \frac{5}{36} a^{7} + \frac{5}{36} a^{6} + \frac{31}{72} a^{5} - \frac{13}{36} a^{4} - \frac{11}{36} a^{3} - \frac{5}{18} a^{2} + \frac{1}{18} a + \frac{1}{3}$, $\frac{1}{216} a^{17} + \frac{1}{216} a^{16} + \frac{1}{216} a^{15} - \frac{7}{216} a^{13} + \frac{7}{216} a^{12} - \frac{1}{216} a^{11} + \frac{11}{108} a^{10} + \frac{1}{216} a^{9} - \frac{7}{216} a^{8} - \frac{19}{216} a^{7} - \frac{1}{108} a^{6} - \frac{17}{36} a^{5} + \frac{47}{108} a^{4} - \frac{11}{54} a^{3} - \frac{1}{27} a^{2} + \frac{1}{27} a - \frac{10}{27}$, $\frac{1}{61344} a^{18} - \frac{17}{7668} a^{17} + \frac{389}{61344} a^{16} + \frac{5}{30672} a^{15} - \frac{1}{3834} a^{14} + \frac{419}{10224} a^{13} - \frac{113}{6816} a^{12} + \frac{35}{1136} a^{11} - \frac{401}{1917} a^{10} - \frac{329}{10224} a^{9} - \frac{1517}{61344} a^{8} + \frac{533}{10224} a^{7} - \frac{55}{864} a^{6} + \frac{2447}{30672} a^{5} + \frac{6385}{30672} a^{4} - \frac{493}{15336} a^{3} - \frac{1831}{5112} a^{2} + \frac{31}{71} a - \frac{790}{1917}$, $\frac{1}{6452962083897408} a^{19} + \frac{771893153}{201655065121794} a^{18} - \frac{1079832686507}{6452962083897408} a^{17} + \frac{309736062501}{119499297849952} a^{16} - \frac{529996788523}{1613240520974352} a^{15} + \frac{27938188425287}{3226481041948704} a^{14} + \frac{103604501673599}{6452962083897408} a^{13} - \frac{114056803760725}{3226481041948704} a^{12} + \frac{82773705266557}{1613240520974352} a^{11} + \frac{546560748295639}{3226481041948704} a^{10} - \frac{1155540229181101}{6452962083897408} a^{9} - \frac{167086134679255}{3226481041948704} a^{8} + \frac{100119430903481}{2150987361299136} a^{7} + \frac{673855921499537}{3226481041948704} a^{6} + \frac{800066767141481}{3226481041948704} a^{5} + \frac{452885102514013}{1613240520974352} a^{4} + \frac{157380949876535}{1613240520974352} a^{3} + \frac{181981863252757}{403310130243588} a^{2} - \frac{5006783334674}{33609177520299} a + \frac{16347210318049}{33609177520299}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:  \( 179913.802936 \)
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

\(\Q(\sqrt{5}) \), 4.0.18000.1, 5.1.450000.1 x5, 10.2.1012500000000.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.450000.1
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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$2$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5Data not computed