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Properties

Label 20.4.25000000000...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{24}\cdot 5^{26}$
Root discriminant $18.62$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $C_2\times A_5$ (as 20T31)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 15, 20, 30, -100, -195, 60, 290, 250, 88, -90, -180, -70, 55, 50, 5, -10, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^18 - 10*x^17 + 5*x^16 + 50*x^15 + 55*x^14 - 70*x^13 - 180*x^12 - 90*x^11 + 88*x^10 + 250*x^9 + 290*x^8 + 60*x^7 - 195*x^6 - 100*x^5 + 30*x^4 + 20*x^3 + 15*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^18 - 10*x^17 + 5*x^16 + 50*x^15 + 55*x^14 - 70*x^13 - 180*x^12 - 90*x^11 + 88*x^10 + 250*x^9 + 290*x^8 + 60*x^7 - 195*x^6 - 100*x^5 + 30*x^4 + 20*x^3 + 15*x^2 - 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{18} - 10 x^{17} + 5 x^{16} + 50 x^{15} + 55 x^{14} - 70 x^{13} - 180 x^{12} - 90 x^{11} + 88 x^{10} + 250 x^{9} + 290 x^{8} + 60 x^{7} - 195 x^{6} - 100 x^{5} + 30 x^{4} + 20 x^{3} + 15 x^{2} - 10 x + 1 \)

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:  \(25000000000000000000000000=2^{24}\cdot 5^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.62$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{1442493746323896469} a^{19} + \frac{411254159238905278}{1442493746323896469} a^{18} + \frac{69015115140141645}{1442493746323896469} a^{17} + \frac{48145433237616808}{1442493746323896469} a^{16} + \frac{187238968589504288}{1442493746323896469} a^{15} - \frac{656865971099294966}{1442493746323896469} a^{14} + \frac{391971903862878215}{1442493746323896469} a^{13} + \frac{522879621497093074}{1442493746323896469} a^{12} + \frac{566922030344488262}{1442493746323896469} a^{11} + \frac{96060249156705115}{206070535189128067} a^{10} - \frac{283226112551815877}{1442493746323896469} a^{9} - \frac{686900379002183067}{1442493746323896469} a^{8} - \frac{418997442700735205}{1442493746323896469} a^{7} - \frac{513865909306343455}{1442493746323896469} a^{6} + \frac{320785677145194365}{1442493746323896469} a^{5} - \frac{194027268856491851}{1442493746323896469} a^{4} + \frac{356356605090603881}{1442493746323896469} a^{3} + \frac{297478290855814370}{1442493746323896469} a^{2} - \frac{118162023445193445}{1442493746323896469} a - \frac{32005849040641396}{1442493746323896469}$

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:  \( 94707.1867115 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times A_5$ (as 20T31):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 120
The 10 conjugacy class representatives for $C_2\times A_5$
Character table for $C_2\times A_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 10.2.1000000000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 sibling: data not computed
Degree 12 siblings: data not computed
Degree 20 sibling: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.8.12.16$x^{8} + 24 x^{2} + 4$$4$$2$$12$$A_4\times C_2$$[2, 2]^{6}$
2.12.12.11$x^{12} - 6 x^{10} - 73 x^{8} + 140 x^{6} + 79 x^{4} - 6 x^{2} + 57$$2$$6$$12$$A_4 \times C_2$$[2, 2]^{6}$
$5$5.10.13.2$x^{10} + 10 x^{4} + 5$$10$$1$$13$$D_{10}$$[3/2]_{2}^{2}$
5.10.13.2$x^{10} + 10 x^{4} + 5$$10$$1$$13$$D_{10}$$[3/2]_{2}^{2}$