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Properties

Label 20.0.28846728944...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 29^{8}$
Root discriminant $14.89$
Ramified primes $3, 5, 29$
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, 1, 5, 4, 23, 26, -12, 44, 19, -96, 68, 26, -34, 7, -6, -4, 12, -6, 3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 3*x^18 - 6*x^17 + 12*x^16 - 4*x^15 - 6*x^14 + 7*x^13 - 34*x^12 + 26*x^11 + 68*x^10 - 96*x^9 + 19*x^8 + 44*x^7 - 12*x^6 + 26*x^5 + 23*x^4 + 4*x^3 + 5*x^2 + x + 1)
 
gp: K = bnfinit(x^20 - 3*x^19 + 3*x^18 - 6*x^17 + 12*x^16 - 4*x^15 - 6*x^14 + 7*x^13 - 34*x^12 + 26*x^11 + 68*x^10 - 96*x^9 + 19*x^8 + 44*x^7 - 12*x^6 + 26*x^5 + 23*x^4 + 4*x^3 + 5*x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 3 x^{18} - 6 x^{17} + 12 x^{16} - 4 x^{15} - 6 x^{14} + 7 x^{13} - 34 x^{12} + 26 x^{11} + 68 x^{10} - 96 x^{9} + 19 x^{8} + 44 x^{7} - 12 x^{6} + 26 x^{5} + 23 x^{4} + 4 x^{3} + 5 x^{2} + x + 1 \)

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:  \(288467289442715712890625=3^{10}\cdot 5^{10}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.89$
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:  $3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{15} a^{14} + \frac{1}{15} a^{13} - \frac{1}{15} a^{12} + \frac{1}{15} a^{11} - \frac{1}{15} a^{10} - \frac{1}{5} a^{9} + \frac{7}{15} a^{8} - \frac{7}{15} a^{7} + \frac{2}{15} a^{6} + \frac{2}{15} a^{5} - \frac{7}{15} a^{4} - \frac{1}{15} a^{3} + \frac{2}{15} a + \frac{1}{5}$, $\frac{1}{15} a^{15} - \frac{2}{15} a^{13} + \frac{2}{15} a^{12} - \frac{2}{15} a^{11} - \frac{2}{15} a^{10} - \frac{1}{3} a^{9} + \frac{1}{15} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{15} a^{3} + \frac{2}{15} a^{2} + \frac{1}{15} a - \frac{1}{5}$, $\frac{1}{45} a^{16} - \frac{1}{45} a^{15} + \frac{1}{45} a^{14} + \frac{7}{45} a^{13} - \frac{7}{45} a^{12} + \frac{1}{15} a^{11} + \frac{4}{45} a^{10} - \frac{2}{5} a^{9} - \frac{7}{15} a^{8} - \frac{4}{9} a^{7} - \frac{1}{15} a^{6} + \frac{7}{15} a^{5} + \frac{1}{5} a^{4} - \frac{17}{45} a^{3} - \frac{1}{45} a^{2} - \frac{2}{5} a - \frac{8}{45}$, $\frac{1}{45} a^{17} - \frac{1}{45} a^{14} + \frac{2}{15} a^{13} + \frac{1}{9} a^{12} - \frac{2}{45} a^{11} - \frac{1}{9} a^{10} + \frac{2}{5} a^{9} + \frac{1}{45} a^{8} + \frac{2}{9} a^{7} - \frac{2}{5} a^{5} - \frac{1}{9} a^{4} + \frac{2}{15} a^{3} + \frac{11}{45} a^{2} + \frac{16}{45} a - \frac{1}{9}$, $\frac{1}{2025} a^{18} + \frac{19}{2025} a^{17} - \frac{4}{675} a^{16} + \frac{8}{2025} a^{15} - \frac{16}{2025} a^{14} - \frac{7}{81} a^{13} + \frac{34}{675} a^{12} - \frac{274}{2025} a^{11} + \frac{127}{2025} a^{10} - \frac{578}{2025} a^{9} + \frac{536}{2025} a^{8} - \frac{94}{405} a^{7} - \frac{163}{675} a^{6} + \frac{556}{2025} a^{5} + \frac{127}{2025} a^{4} + \frac{347}{2025} a^{3} + \frac{197}{675} a^{2} + \frac{199}{405} a + \frac{487}{2025}$, $\frac{1}{3719925} a^{19} - \frac{64}{3719925} a^{18} + \frac{25951}{3719925} a^{17} - \frac{6871}{3719925} a^{16} - \frac{18676}{743985} a^{15} - \frac{79352}{3719925} a^{14} - \frac{389473}{3719925} a^{13} + \frac{22262}{148797} a^{12} + \frac{4604}{45925} a^{11} - \frac{599629}{3719925} a^{10} - \frac{5017}{16533} a^{9} + \frac{2119}{112725} a^{8} + \frac{771526}{3719925} a^{7} + \frac{772438}{3719925} a^{6} - \frac{859396}{3719925} a^{5} - \frac{594338}{1239975} a^{4} - \frac{221786}{743985} a^{3} - \frac{1832443}{3719925} a^{2} - \frac{121307}{413325} a + \frac{410434}{3719925}$

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:  \( -\frac{156544}{743985} a^{19} + \frac{628072}{743985} a^{18} - \frac{910432}{743985} a^{17} + \frac{1316017}{743985} a^{16} - \frac{2764099}{743985} a^{15} + \frac{2407223}{743985} a^{14} + \frac{619618}{743985} a^{13} - \frac{1968806}{743985} a^{12} + \frac{654008}{82665} a^{11} - \frac{9126962}{743985} a^{10} - \frac{281333}{27555} a^{9} + \frac{801974}{22545} a^{8} - \frac{2979599}{148797} a^{7} - \frac{7546483}{743985} a^{6} + \frac{7589557}{743985} a^{5} - \frac{847999}{247995} a^{4} - \frac{29816}{743985} a^{3} + \frac{3297802}{743985} a^{2} + \frac{11239}{82665} a + \frac{721814}{743985} \) (order $6$)
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:  \( 9642.05211515 \)
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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 10.2.179030503125.1 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$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}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.10.8.1$x^{10} - 899 x^{5} + 204363$$5$$2$$8$$D_5$$[\ ]_{5}^{2}$