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Properties

Label 20.0.10995116277...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 5^{15}$
Root discriminant $22.49$
Ramified primes $2, 5$
Class number $4$
Class group $[4]$
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![256, -2560, 12288, -36736, 75672, -112960, 126216, -107952, 71958, -37920, 16456, -6608, 3125, -1912, 1256, -728, 352, -136, 40, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 40*x^18 - 136*x^17 + 352*x^16 - 728*x^15 + 1256*x^14 - 1912*x^13 + 3125*x^12 - 6608*x^11 + 16456*x^10 - 37920*x^9 + 71958*x^8 - 107952*x^7 + 126216*x^6 - 112960*x^5 + 75672*x^4 - 36736*x^3 + 12288*x^2 - 2560*x + 256)
 
gp: K = bnfinit(x^20 - 8*x^19 + 40*x^18 - 136*x^17 + 352*x^16 - 728*x^15 + 1256*x^14 - 1912*x^13 + 3125*x^12 - 6608*x^11 + 16456*x^10 - 37920*x^9 + 71958*x^8 - 107952*x^7 + 126216*x^6 - 112960*x^5 + 75672*x^4 - 36736*x^3 + 12288*x^2 - 2560*x + 256, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 40 x^{18} - 136 x^{17} + 352 x^{16} - 728 x^{15} + 1256 x^{14} - 1912 x^{13} + 3125 x^{12} - 6608 x^{11} + 16456 x^{10} - 37920 x^{9} + 71958 x^{8} - 107952 x^{7} + 126216 x^{6} - 112960 x^{5} + 75672 x^{4} - 36736 x^{3} + 12288 x^{2} - 2560 x + 256 \)

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:  \(1099511627776000000000000000=2^{55}\cdot 5^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.49$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{20} a^{12} - \frac{1}{10} a^{11} + \frac{1}{20} a^{8} - \frac{1}{2} a^{7} + \frac{3}{10} a^{6} + \frac{3}{10} a^{4} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{20} a^{13} + \frac{1}{20} a^{11} + \frac{1}{20} a^{9} + \frac{1}{10} a^{8} - \frac{9}{20} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{10} a^{4} - \frac{1}{2} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5}$, $\frac{1}{20} a^{14} + \frac{1}{10} a^{11} + \frac{1}{20} a^{10} + \frac{1}{10} a^{9} + \frac{1}{10} a^{7} - \frac{1}{2} a^{6} + \frac{1}{10} a^{5} - \frac{3}{10} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5}$, $\frac{1}{40} a^{15} + \frac{1}{20} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{7} + \frac{1}{4} a^{6} + \frac{7}{20} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{5}$, $\frac{1}{40} a^{16} + \frac{1}{20} a^{11} + \frac{1}{8} a^{8} + \frac{1}{4} a^{7} - \frac{2}{5} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{5} a$, $\frac{1}{80} a^{17} + \frac{1}{20} a^{11} + \frac{1}{16} a^{9} + \frac{1}{10} a^{8} - \frac{9}{20} a^{7} - \frac{2}{5} a^{6} + \frac{3}{8} a^{5} + \frac{1}{10} a^{4} - \frac{2}{5} a^{2} - \frac{3}{10} a - \frac{2}{5}$, $\frac{1}{32800} a^{18} - \frac{87}{16400} a^{17} - \frac{71}{8200} a^{16} + \frac{1}{328} a^{15} + \frac{13}{4100} a^{14} + \frac{11}{4100} a^{13} - \frac{73}{4100} a^{12} + \frac{141}{4100} a^{11} - \frac{3891}{32800} a^{10} + \frac{1157}{16400} a^{9} - \frac{199}{1640} a^{8} + \frac{187}{8200} a^{7} - \frac{1}{3280} a^{6} - \frac{619}{8200} a^{5} + \frac{787}{2050} a^{4} + \frac{1337}{4100} a^{3} - \frac{121}{820} a^{2} + \frac{223}{1025} a + \frac{226}{1025}$, $\frac{1}{485190809929662400} a^{19} + \frac{1320035664149}{242595404964831200} a^{18} - \frac{112283427234367}{30324425620603900} a^{17} + \frac{225954231697957}{30324425620603900} a^{16} + \frac{152488294984383}{60648851241207800} a^{15} - \frac{935576225424763}{60648851241207800} a^{14} - \frac{917458367823051}{60648851241207800} a^{13} + \frac{177706530010909}{12129770248241560} a^{12} - \frac{1869933115581651}{19407632397186496} a^{11} - \frac{1873574473355699}{242595404964831200} a^{10} + \frac{743944947325697}{7581106405150975} a^{9} - \frac{13417517995308179}{60648851241207800} a^{8} + \frac{100792214764696943}{242595404964831200} a^{7} + \frac{52189249657793331}{121297702482415600} a^{6} + \frac{4253994527451961}{12129770248241560} a^{5} + \frac{350474981098093}{3032442562060390} a^{4} + \frac{4497731868264789}{60648851241207800} a^{3} - \frac{12892004196682259}{30324425620603900} a^{2} + \frac{1637171758705547}{15162212810301950} a - \frac{2300640966692029}{7581106405150975}$

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

Class group and class number

$C_{4}$, 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:  \( 1993746.99362 \)
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

\(\Q(\sqrt{10}) \), 4.0.256000.2, 5.1.256000.1 x5, 10.2.2621440000000.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.256000.1
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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$2.4.11.10$x^{4} + 8 x + 6$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.10$x^{4} + 8 x + 6$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.10$x^{4} + 8 x + 6$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.10$x^{4} + 8 x + 6$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.10$x^{4} + 8 x + 6$$4$$1$$11$$C_4$$[3, 4]$
$5$5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$