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Properties

Label 20.12.2103876049...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{16}\cdot 3^{2}\cdot 5^{15}\cdot 43^{8}$
Root discriminant $29.25$
Ramified primes $2, 3, 5, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T135

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 9, -36, -54, 378, -335, -890, 2212, -1105, -2098, 3846, -2440, -252, 1770, -1572, 720, -141, -39, 36, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 36*x^18 - 39*x^17 - 141*x^16 + 720*x^15 - 1572*x^14 + 1770*x^13 - 252*x^12 - 2440*x^11 + 3846*x^10 - 2098*x^9 - 1105*x^8 + 2212*x^7 - 890*x^6 - 335*x^5 + 378*x^4 - 54*x^3 - 36*x^2 + 9*x + 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 36*x^18 - 39*x^17 - 141*x^16 + 720*x^15 - 1572*x^14 + 1770*x^13 - 252*x^12 - 2440*x^11 + 3846*x^10 - 2098*x^9 - 1105*x^8 + 2212*x^7 - 890*x^6 - 335*x^5 + 378*x^4 - 54*x^3 - 36*x^2 + 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 36 x^{18} - 39 x^{17} - 141 x^{16} + 720 x^{15} - 1572 x^{14} + 1770 x^{13} - 252 x^{12} - 2440 x^{11} + 3846 x^{10} - 2098 x^{9} - 1105 x^{8} + 2212 x^{7} - 890 x^{6} - 335 x^{5} + 378 x^{4} - 54 x^{3} - 36 x^{2} + 9 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(210387604996818000000000000000=2^{16}\cdot 3^{2}\cdot 5^{15}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.25$
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:  $2, 3, 5, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{30} a^{12} + \frac{2}{15} a^{11} + \frac{1}{15} a^{10} - \frac{1}{2} a^{9} + \frac{3}{10} a^{8} - \frac{1}{15} a^{7} + \frac{11}{30} a^{6} + \frac{7}{15} a^{5} - \frac{3}{10} a^{4} - \frac{3}{10} a^{3} + \frac{1}{30} a^{2} + \frac{1}{10} a + \frac{1}{30}$, $\frac{1}{30} a^{13} - \frac{2}{15} a^{11} - \frac{1}{10} a^{10} - \frac{1}{30} a^{9} - \frac{4}{15} a^{8} - \frac{1}{30} a^{7} + \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{10} a^{4} + \frac{7}{30} a^{3} - \frac{1}{30} a^{2} - \frac{1}{30} a + \frac{1}{5}$, $\frac{1}{150} a^{14} - \frac{1}{75} a^{13} - \frac{1}{150} a^{12} - \frac{23}{150} a^{11} + \frac{1}{150} a^{10} + \frac{59}{150} a^{9} + \frac{11}{75} a^{8} + \frac{8}{75} a^{7} - \frac{11}{75} a^{6} - \frac{7}{50} a^{5} - \frac{37}{75} a^{4} - \frac{12}{25} a^{3} + \frac{32}{75} a^{2} + \frac{7}{150} a - \frac{3}{50}$, $\frac{1}{150} a^{15} - \frac{1}{10} a^{11} - \frac{2}{75} a^{10} - \frac{4}{15} a^{9} - \frac{11}{30} a^{8} + \frac{11}{30} a^{7} + \frac{2}{5} a^{6} + \frac{59}{150} a^{5} - \frac{1}{15} a^{4} + \frac{1}{5} a^{3} + \frac{1}{30} a^{2} + \frac{1}{6} a - \frac{13}{150}$, $\frac{1}{150} a^{16} + \frac{1}{25} a^{11} - \frac{1}{15} a^{10} + \frac{2}{15} a^{9} - \frac{1}{15} a^{8} + \frac{1}{5} a^{7} + \frac{37}{75} a^{6} + \frac{3}{10} a^{4} + \frac{2}{15} a^{3} + \frac{4}{15} a^{2} - \frac{3}{25} a + \frac{13}{30}$, $\frac{1}{150} a^{17} + \frac{1}{150} a^{12} + \frac{2}{15} a^{11} + \frac{1}{15} a^{10} + \frac{13}{30} a^{9} + \frac{7}{30} a^{8} - \frac{11}{25} a^{7} - \frac{11}{30} a^{6} + \frac{1}{6} a^{5} + \frac{13}{30} a^{4} - \frac{13}{30} a^{3} - \frac{23}{150} a^{2} - \frac{1}{3} a - \frac{11}{30}$, $\frac{1}{150} a^{18} + \frac{1}{150} a^{13} - \frac{2}{15} a^{11} - \frac{1}{6} a^{10} - \frac{1}{10} a^{9} + \frac{9}{25} a^{8} + \frac{7}{30} a^{7} + \frac{1}{30} a^{6} + \frac{7}{30} a^{5} - \frac{7}{30} a^{4} + \frac{7}{150} a^{3} - \frac{7}{15} a^{2} - \frac{13}{30} a + \frac{1}{5}$, $\frac{1}{16350} a^{19} + \frac{3}{1090} a^{18} + \frac{2}{8175} a^{17} - \frac{37}{16350} a^{16} + \frac{2}{8175} a^{15} - \frac{41}{16350} a^{14} - \frac{121}{16350} a^{13} - \frac{89}{16350} a^{12} + \frac{1829}{16350} a^{11} + \frac{191}{8175} a^{10} - \frac{107}{8175} a^{9} + \frac{341}{2725} a^{8} + \frac{3992}{8175} a^{7} - \frac{2183}{5450} a^{6} + \frac{4049}{8175} a^{5} - \frac{457}{1635} a^{4} + \frac{754}{2725} a^{3} - \frac{79}{545} a^{2} + \frac{3041}{8175} a - \frac{3707}{8175}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 33467788.43 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T135:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 22 conjugacy class representatives for t20n135
Character table for t20n135 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.3698000.1, 10.10.68376020000000.1

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 siblings: data not computed
Degree 32 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 R R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$43$43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$