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Properties

Label 20.4.61250000000...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{20}\cdot 5^{23}\cdot 7^{2}$
Root discriminant $15.47$
Ramified primes $2, 5, 7$
Class number $1$
Class group Trivial
Galois group 20T135

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 45, -120, 215, -282, 280, -210, 115, -20, -41, 50, -45, 10, 20, -8, 5, 0, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^18 + 5*x^16 - 8*x^15 + 20*x^14 + 10*x^13 - 45*x^12 + 50*x^11 - 41*x^10 - 20*x^9 + 115*x^8 - 210*x^7 + 280*x^6 - 282*x^5 + 215*x^4 - 120*x^3 + 45*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^18 + 5*x^16 - 8*x^15 + 20*x^14 + 10*x^13 - 45*x^12 + 50*x^11 - 41*x^10 - 20*x^9 + 115*x^8 - 210*x^7 + 280*x^6 - 282*x^5 + 215*x^4 - 120*x^3 + 45*x^2 - 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{18} + 5 x^{16} - 8 x^{15} + 20 x^{14} + 10 x^{13} - 45 x^{12} + 50 x^{11} - 41 x^{10} - 20 x^{9} + 115 x^{8} - 210 x^{7} + 280 x^{6} - 282 x^{5} + 215 x^{4} - 120 x^{3} + 45 x^{2} - 10 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(612500000000000000000000=2^{20}\cdot 5^{23}\cdot 7^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.47$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{33042200669} a^{19} - \frac{2282223241}{33042200669} a^{18} - \frac{14490367712}{33042200669} a^{17} + \frac{3003604226}{33042200669} a^{16} - \frac{5108677923}{33042200669} a^{15} + \frac{2358848768}{33042200669} a^{14} - \frac{3156008731}{33042200669} a^{13} + \frac{4072101975}{33042200669} a^{12} - \frac{8503933941}{33042200669} a^{11} + \frac{595657427}{33042200669} a^{10} - \frac{12383642878}{33042200669} a^{9} + \frac{7251053777}{33042200669} a^{8} - \frac{7002599944}{33042200669} a^{7} - \frac{10218228618}{33042200669} a^{6} + \frac{15319463226}{33042200669} a^{5} - \frac{3938967010}{33042200669} a^{4} - \frac{16281914329}{33042200669} a^{3} + \frac{12784505043}{33042200669} a^{2} - \frac{1320472278}{33042200669} a - \frac{2295903914}{33042200669}$

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:  \( 10062.6245548 \)
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

\(\Q(\sqrt{5}) \), 5.1.50000.1, 10.2.12500000000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.0.1$x^{8} - x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
7.8.0.1$x^{8} - x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$