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Properties

Label 20.4.23928882935...3125.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{15}\cdot 601^{5}$
Root discriminant $16.56$
Ramified primes $5, 601$
Class number $1$
Class group Trivial
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, 0, 14, 28, 81, 215, 168, -213, -330, -2, 112, 0, -1, 1, -26, 10, 24, -4, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^18 - 4*x^17 + 24*x^16 + 10*x^15 - 26*x^14 + x^13 - x^12 + 112*x^10 - 2*x^9 - 330*x^8 - 213*x^7 + 168*x^6 + 215*x^5 + 81*x^4 + 28*x^3 + 14*x^2 + 1)
 
gp: K = bnfinit(x^20 - 8*x^18 - 4*x^17 + 24*x^16 + 10*x^15 - 26*x^14 + x^13 - x^12 + 112*x^10 - 2*x^9 - 330*x^8 - 213*x^7 + 168*x^6 + 215*x^5 + 81*x^4 + 28*x^3 + 14*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{18} - 4 x^{17} + 24 x^{16} + 10 x^{15} - 26 x^{14} + x^{13} - x^{12} + 112 x^{10} - 2 x^{9} - 330 x^{8} - 213 x^{7} + 168 x^{6} + 215 x^{5} + 81 x^{4} + 28 x^{3} + 14 x^{2} + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2392888293548614501953125=5^{15}\cdot 601^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.56$
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:  $5, 601$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5609} a^{18} + \frac{2731}{5609} a^{17} + \frac{1326}{5609} a^{16} - \frac{2476}{5609} a^{15} + \frac{1076}{5609} a^{14} - \frac{1327}{5609} a^{13} + \frac{2543}{5609} a^{12} - \frac{505}{5609} a^{11} + \frac{2301}{5609} a^{10} + \frac{2121}{5609} a^{9} + \frac{246}{5609} a^{8} - \frac{1970}{5609} a^{7} - \frac{952}{5609} a^{6} - \frac{1162}{5609} a^{5} - \frac{1405}{5609} a^{4} + \frac{1440}{5609} a^{3} - \frac{270}{5609} a^{2} + \frac{562}{5609} a - \frac{162}{5609}$, $\frac{1}{21903351982527131} a^{19} + \frac{597038739611}{21903351982527131} a^{18} + \frac{157065646444628}{21903351982527131} a^{17} - \frac{10454764870778167}{21903351982527131} a^{16} - \frac{3069711377357402}{21903351982527131} a^{15} + \frac{7176182168066818}{21903351982527131} a^{14} - \frac{7002501043877290}{21903351982527131} a^{13} + \frac{2754357591685707}{21903351982527131} a^{12} + \frac{8873555534990310}{21903351982527131} a^{11} + \frac{10314693825468026}{21903351982527131} a^{10} + \frac{8589223330512014}{21903351982527131} a^{9} - \frac{5821513244429892}{21903351982527131} a^{8} + \frac{10168480824286714}{21903351982527131} a^{7} - \frac{39037706816528}{21903351982527131} a^{6} + \frac{6184829011749592}{21903351982527131} a^{5} - \frac{3555749724044548}{21903351982527131} a^{4} - \frac{6884908989364872}{21903351982527131} a^{3} - \frac{1290483324979173}{21903351982527131} a^{2} + \frac{9440541523729023}{21903351982527131} a + \frac{10290231345309694}{21903351982527131}$

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:  $11$
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:  \( 14441.4999597 \)
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

\(\Q(\sqrt{5}) \), 4.4.75125.1, 10.2.1128753125.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}$ $20$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
601Data not computed