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Properties

Label 20.20.7063888397...9408.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{30}\cdot 3^{15}\cdot 71^{9}$
Root discriminant $43.90$
Ramified primes $2, 3, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1144, 11874, -14680, -90361, 91516, 196554, -165550, -188890, 138594, 92520, -63492, -24119, 17018, 3138, -2666, -123, 226, -12, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 - 12*x^18 + 226*x^17 - 123*x^16 - 2666*x^15 + 3138*x^14 + 17018*x^13 - 24119*x^12 - 63492*x^11 + 92520*x^10 + 138594*x^9 - 188890*x^8 - 165550*x^7 + 196554*x^6 + 91516*x^5 - 90361*x^4 - 14680*x^3 + 11874*x^2 - 1144*x + 1)
 
gp: K = bnfinit(x^20 - 8*x^19 - 12*x^18 + 226*x^17 - 123*x^16 - 2666*x^15 + 3138*x^14 + 17018*x^13 - 24119*x^12 - 63492*x^11 + 92520*x^10 + 138594*x^9 - 188890*x^8 - 165550*x^7 + 196554*x^6 + 91516*x^5 - 90361*x^4 - 14680*x^3 + 11874*x^2 - 1144*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} - 12 x^{18} + 226 x^{17} - 123 x^{16} - 2666 x^{15} + 3138 x^{14} + 17018 x^{13} - 24119 x^{12} - 63492 x^{11} + 92520 x^{10} + 138594 x^{9} - 188890 x^{8} - 165550 x^{7} + 196554 x^{6} + 91516 x^{5} - 90361 x^{4} - 14680 x^{3} + 11874 x^{2} - 1144 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(706388839731582814105670315409408=2^{30}\cdot 3^{15}\cdot 71^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.90$
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:  $2, 3, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{1482568011698658110733988411759} a^{19} - \frac{285584972430803318719262513399}{1482568011698658110733988411759} a^{18} + \frac{314288759275483037679175795819}{1482568011698658110733988411759} a^{17} - \frac{84390615429592701803371920137}{1482568011698658110733988411759} a^{16} - \frac{223501630140857532159046970467}{1482568011698658110733988411759} a^{15} - \frac{15494035670863591923545095923}{36160195407284344164243619799} a^{14} - \frac{589795408957959443290368022514}{1482568011698658110733988411759} a^{13} - \frac{61117405357686432626968789193}{1482568011698658110733988411759} a^{12} + \frac{131980059453745839792180112449}{1482568011698658110733988411759} a^{11} + \frac{441289182549943709763332351169}{1482568011698658110733988411759} a^{10} + \frac{7169167704754833369746237592}{36160195407284344164243619799} a^{9} - \frac{594921287065328307404293098791}{1482568011698658110733988411759} a^{8} + \frac{618639688626627267713852869495}{1482568011698658110733988411759} a^{7} + \frac{56753463735416924288799240671}{1482568011698658110733988411759} a^{6} - \frac{97277652315050138457908205695}{1482568011698658110733988411759} a^{5} + \frac{691983637865149370495713206692}{1482568011698658110733988411759} a^{4} + \frac{438970259064672157984814142332}{1482568011698658110733988411759} a^{3} + \frac{15816286443123885576150110681}{36160195407284344164243619799} a^{2} - \frac{10752673665537582672725907815}{1482568011698658110733988411759} a - \frac{23383984029560059131753547846}{1482568011698658110733988411759}$

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:  $19$
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:  \( 7244239011.09 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.122688.1, 10.10.6323239406592.1

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 R R $20$ $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
3Data not computed
$71$71.5.0.1$x^{5} - x + 8$$1$$5$$0$$C_5$$[\ ]^{5}$
71.5.0.1$x^{5} - x + 8$$1$$5$$0$$C_5$$[\ ]^{5}$
71.10.9.5$x^{10} - 18176$$10$$1$$9$$C_{10}$$[\ ]_{10}$