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Properties

Label 20.10.2196009914...1875.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,5^{30}\cdot 11^{9}$
Root discriminant $32.89$
Ramified primes $5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_5^2:C_4$ (as 20T94)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 5, 0, -250, 475, 994, -1705, -1150, 2195, 750, -1880, -245, 1075, 35, -375, -8, 90, 0, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 15*x^18 + 90*x^16 - 8*x^15 - 375*x^14 + 35*x^13 + 1075*x^12 - 245*x^11 - 1880*x^10 + 750*x^9 + 2195*x^8 - 1150*x^7 - 1705*x^6 + 994*x^5 + 475*x^4 - 250*x^3 + 5*x - 1)
 
gp: K = bnfinit(x^20 - 15*x^18 + 90*x^16 - 8*x^15 - 375*x^14 + 35*x^13 + 1075*x^12 - 245*x^11 - 1880*x^10 + 750*x^9 + 2195*x^8 - 1150*x^7 - 1705*x^6 + 994*x^5 + 475*x^4 - 250*x^3 + 5*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 15 x^{18} + 90 x^{16} - 8 x^{15} - 375 x^{14} + 35 x^{13} + 1075 x^{12} - 245 x^{11} - 1880 x^{10} + 750 x^{9} + 2195 x^{8} - 1150 x^{7} - 1705 x^{6} + 994 x^{5} + 475 x^{4} - 250 x^{3} + 5 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2196009914390742778778076171875=-\,5^{30}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.89$
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:  $5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{167373957049468626165557949} a^{19} - \frac{73826206441646341090665362}{167373957049468626165557949} a^{18} - \frac{79960393748794234143774653}{167373957049468626165557949} a^{17} - \frac{73019172809343761418711020}{167373957049468626165557949} a^{16} - \frac{81016478717704811442126803}{167373957049468626165557949} a^{15} + \frac{1389471888117099935955250}{5771515760326504350536481} a^{14} + \frac{12135624836419423271236619}{167373957049468626165557949} a^{13} - \frac{318998171848957688042434}{5771515760326504350536481} a^{12} + \frac{36686343541531318778926877}{167373957049468626165557949} a^{11} - \frac{3492943323545874125637889}{18597106338829847351728661} a^{10} + \frac{56545829695471161129049657}{167373957049468626165557949} a^{9} - \frac{28325288068146503973466976}{167373957049468626165557949} a^{8} + \frac{3428827889505089232192457}{18597106338829847351728661} a^{7} - \frac{7548646986606718284292087}{167373957049468626165557949} a^{6} - \frac{298709191850952852371075}{8809155634182559271871471} a^{5} - \frac{45660858044252253313069186}{167373957049468626165557949} a^{4} - \frac{12497557759213126400390416}{55791319016489542055185983} a^{3} + \frac{32677483376890600268172974}{167373957049468626165557949} a^{2} - \frac{65574325270669260702709471}{167373957049468626165557949} a - \frac{41039127356300384392547750}{167373957049468626165557949}$

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:  $14$
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:  \( 61759270.0621 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5^2:C_4$ (as 20T94):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $D_5^2:C_4$
Character table for $D_5^2:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 10.6.3692626953125.1

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 $20$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R $20$ R $20$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$