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Properties

Label 20.16.2005405060...3125.1
Degree $20$
Signature $[16, 2]$
Discriminant $5^{10}\cdot 11^{16}\cdot 446909$
Root discriminant $29.18$
Ramified primes $5, 11, 446909$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T409

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 34, -57, -160, 446, 98, -1142, 682, 1052, -1459, 110, 987, -722, -72, 344, -145, -21, 34, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 34*x^18 - 21*x^17 - 145*x^16 + 344*x^15 - 72*x^14 - 722*x^13 + 987*x^12 + 110*x^11 - 1459*x^10 + 1052*x^9 + 682*x^8 - 1142*x^7 + 98*x^6 + 446*x^5 - 160*x^4 - 57*x^3 + 34*x^2 - 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 34*x^18 - 21*x^17 - 145*x^16 + 344*x^15 - 72*x^14 - 722*x^13 + 987*x^12 + 110*x^11 - 1459*x^10 + 1052*x^9 + 682*x^8 - 1142*x^7 + 98*x^6 + 446*x^5 - 160*x^4 - 57*x^3 + 34*x^2 - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 34 x^{18} - 21 x^{17} - 145 x^{16} + 344 x^{15} - 72 x^{14} - 722 x^{13} + 987 x^{12} + 110 x^{11} - 1459 x^{10} + 1052 x^{9} + 682 x^{8} - 1142 x^{7} + 98 x^{6} + 446 x^{5} - 160 x^{4} - 57 x^{3} + 34 x^{2} - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(200540506089835653323720703125=5^{10}\cdot 11^{16}\cdot 446909\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.18$
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:  $5, 11, 446909$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{331} a^{18} - \frac{9}{331} a^{17} + \frac{127}{331} a^{16} - \frac{150}{331} a^{15} + \frac{81}{331} a^{14} + \frac{20}{331} a^{13} - \frac{65}{331} a^{12} - \frac{71}{331} a^{11} - \frac{87}{331} a^{10} + \frac{63}{331} a^{9} - \frac{9}{331} a^{8} - \frac{144}{331} a^{7} - \frac{49}{331} a^{6} + \frac{9}{331} a^{5} + \frac{74}{331} a^{4} + \frac{114}{331} a^{3} - \frac{111}{331} a^{2} - \frac{125}{331} a - \frac{159}{331}$, $\frac{1}{331} a^{19} + \frac{46}{331} a^{17} + \frac{55}{331} a^{15} + \frac{87}{331} a^{14} + \frac{115}{331} a^{13} + \frac{6}{331} a^{12} - \frac{64}{331} a^{11} - \frac{58}{331} a^{10} - \frac{104}{331} a^{9} + \frac{106}{331} a^{8} - \frac{21}{331} a^{7} - \frac{101}{331} a^{6} + \frac{155}{331} a^{5} + \frac{118}{331} a^{4} - \frac{78}{331} a^{3} - \frac{131}{331} a^{2} + \frac{40}{331} a - \frac{107}{331}$

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:  $17$
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:  \( 47906378.6045 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T409:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n409 are not computed
Character table for t20n409 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1

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 $20$ $20$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
5Data not computed
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
446909Data not computed