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Properties

Label 20.2.65045103627...1875.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,5^{10}\cdot 7^{10}\cdot 11^{9}$
Root discriminant $17.40$
Ramified primes $5, 7, 11$
Class number $1$
Class group Trivial
Galois group $C_{10}^2:C_2^2$ (as 20T106)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -12, 63, -194, 396, -575, 602, -395, 30, 247, -271, 136, -5, -79, 115, -95, 57, -28, 11, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 11*x^18 - 28*x^17 + 57*x^16 - 95*x^15 + 115*x^14 - 79*x^13 - 5*x^12 + 136*x^11 - 271*x^10 + 247*x^9 + 30*x^8 - 395*x^7 + 602*x^6 - 575*x^5 + 396*x^4 - 194*x^3 + 63*x^2 - 12*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 11*x^18 - 28*x^17 + 57*x^16 - 95*x^15 + 115*x^14 - 79*x^13 - 5*x^12 + 136*x^11 - 271*x^10 + 247*x^9 + 30*x^8 - 395*x^7 + 602*x^6 - 575*x^5 + 396*x^4 - 194*x^3 + 63*x^2 - 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 11 x^{18} - 28 x^{17} + 57 x^{16} - 95 x^{15} + 115 x^{14} - 79 x^{13} - 5 x^{12} + 136 x^{11} - 271 x^{10} + 247 x^{9} + 30 x^{8} - 395 x^{7} + 602 x^{6} - 575 x^{5} + 396 x^{4} - 194 x^{3} + 63 x^{2} - 12 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-6504510362736328701171875=-\,5^{10}\cdot 7^{10}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.40$
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:  $5, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{15} a^{15} + \frac{2}{15} a^{14} + \frac{2}{15} a^{11} + \frac{2}{15} a^{10} - \frac{7}{15} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{4}{15} a^{5} + \frac{2}{15} a^{4} - \frac{1}{15} a^{3} - \frac{7}{15} a^{2} - \frac{1}{15} a + \frac{4}{15}$, $\frac{1}{285} a^{16} + \frac{2}{285} a^{15} - \frac{1}{19} a^{14} + \frac{3}{19} a^{13} + \frac{14}{95} a^{12} - \frac{11}{95} a^{11} - \frac{37}{285} a^{10} - \frac{23}{95} a^{9} + \frac{4}{285} a^{8} - \frac{22}{57} a^{7} - \frac{7}{95} a^{6} + \frac{29}{95} a^{5} + \frac{14}{285} a^{4} + \frac{73}{285} a^{3} + \frac{28}{95} a^{2} - \frac{26}{285} a + \frac{14}{57}$, $\frac{1}{285} a^{17} + \frac{6}{95} a^{14} + \frac{47}{285} a^{13} - \frac{22}{285} a^{12} - \frac{28}{285} a^{11} + \frac{43}{285} a^{10} - \frac{86}{285} a^{9} + \frac{91}{285} a^{8} - \frac{7}{19} a^{7} + \frac{34}{285} a^{6} + \frac{106}{285} a^{5} + \frac{83}{285} a^{4} + \frac{109}{285} a^{3} + \frac{53}{285} a^{2} + \frac{8}{285} a - \frac{64}{285}$, $\frac{1}{285} a^{18} - \frac{1}{285} a^{15} + \frac{3}{95} a^{14} - \frac{22}{285} a^{13} - \frac{28}{285} a^{12} + \frac{1}{57} a^{11} - \frac{29}{285} a^{10} + \frac{34}{285} a^{9} - \frac{29}{285} a^{8} + \frac{53}{285} a^{7} + \frac{11}{285} a^{6} - \frac{88}{285} a^{5} + \frac{71}{285} a^{4} - \frac{23}{285} a^{3} + \frac{47}{95} a^{2} - \frac{3}{19} a + \frac{1}{15}$, $\frac{1}{285} a^{19} - \frac{8}{285} a^{15} + \frac{4}{57} a^{14} + \frac{17}{285} a^{13} + \frac{47}{285} a^{12} - \frac{1}{57} a^{11} - \frac{41}{285} a^{10} - \frac{4}{19} a^{9} - \frac{1}{5} a^{8} + \frac{22}{57} a^{7} + \frac{27}{95} a^{6} - \frac{36}{95} a^{5} - \frac{142}{285} a^{4} + \frac{46}{95} a^{3} + \frac{77}{285} a^{2} - \frac{83}{285} a - \frac{2}{95}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $10$
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:  \( 27264.0677779 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{10}^2:C_2^2$ (as 20T106):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 46 conjugacy class representatives for $C_{10}^2:C_2^2$
Character table for $C_{10}^2:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.13475.1, 10.2.109853253125.1

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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R R R $20$ $20$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$