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Properties

Label 20.0.59131912388...5625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 7^{10}\cdot 11^{8}$
Root discriminant $15.44$
Ramified primes $5, 7, 11$
Class number $1$
Class group Trivial
Galois group $D_5^2$ (as 20T28)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 5, -9, 18, -29, 37, -24, 26, -13, 31, -13, 26, -24, 37, -29, 18, -9, 5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 5*x^18 - 9*x^17 + 18*x^16 - 29*x^15 + 37*x^14 - 24*x^13 + 26*x^12 - 13*x^11 + 31*x^10 - 13*x^9 + 26*x^8 - 24*x^7 + 37*x^6 - 29*x^5 + 18*x^4 - 9*x^3 + 5*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^20 - 3*x^19 + 5*x^18 - 9*x^17 + 18*x^16 - 29*x^15 + 37*x^14 - 24*x^13 + 26*x^12 - 13*x^11 + 31*x^10 - 13*x^9 + 26*x^8 - 24*x^7 + 37*x^6 - 29*x^5 + 18*x^4 - 9*x^3 + 5*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 5 x^{18} - 9 x^{17} + 18 x^{16} - 29 x^{15} + 37 x^{14} - 24 x^{13} + 26 x^{12} - 13 x^{11} + 31 x^{10} - 13 x^{9} + 26 x^{8} - 24 x^{7} + 37 x^{6} - 29 x^{5} + 18 x^{4} - 9 x^{3} + 5 x^{2} - 3 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(591319123885120791015625=5^{10}\cdot 7^{10}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.44$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{15} - \frac{2}{5} a^{14} + \frac{2}{5} a^{13} + \frac{1}{5} a^{12} + \frac{2}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{17} - \frac{1}{5} a^{15} + \frac{1}{5} a^{14} + \frac{2}{5} a^{13} - \frac{2}{5} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{10955} a^{18} - \frac{456}{10955} a^{17} + \frac{618}{10955} a^{16} - \frac{250}{2191} a^{15} + \frac{2563}{10955} a^{14} - \frac{5161}{10955} a^{13} + \frac{4183}{10955} a^{12} + \frac{779}{1565} a^{11} - \frac{5109}{10955} a^{10} - \frac{957}{2191} a^{9} - \frac{5109}{10955} a^{8} + \frac{779}{1565} a^{7} + \frac{4183}{10955} a^{6} - \frac{5161}{10955} a^{5} + \frac{2563}{10955} a^{4} - \frac{250}{2191} a^{3} + \frac{618}{10955} a^{2} - \frac{456}{10955} a + \frac{1}{10955}$, $\frac{1}{208145} a^{19} - \frac{4}{208145} a^{18} + \frac{9224}{208145} a^{17} - \frac{6744}{208145} a^{16} - \frac{8634}{41629} a^{15} + \frac{97253}{208145} a^{14} + \frac{83702}{208145} a^{13} + \frac{64493}{208145} a^{12} - \frac{1313}{10955} a^{11} - \frac{1929}{29735} a^{10} - \frac{13352}{41629} a^{9} - \frac{3516}{10955} a^{8} - \frac{13464}{208145} a^{7} - \frac{3571}{29735} a^{6} + \frac{9222}{29735} a^{5} + \frac{11948}{29735} a^{4} + \frac{2780}{5947} a^{3} - \frac{6171}{29735} a^{2} - \frac{1346}{41629} a + \frac{9216}{208145}$

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:  $9$
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:  \( 4090.28443721 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5^2$ (as 20T28):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 100
The 16 conjugacy class representatives for $D_5^2$
Character table for $D_5^2$

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{5}, \sqrt{-7})\), 10.2.109853253125.1 x5

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 sibling: data not computed
Degree 25 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$5.4.2.1$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}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$