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Properties

Label 20.20.3724513239...5625.1
Degree $20$
Signature $[20, 0]$
Discriminant $5^{10}\cdot 11^{16}\cdot 43^{2}\cdot 67^{2}$
Root discriminant $33.77$
Ramified primes $5, 11, 43, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2\times C_2^4:C_5$ (as 20T86)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -25, -71, 1710, -5900, 3122, 16581, -23057, -12282, 33661, -1727, -20965, 5177, 6605, -2213, -1076, 414, 82, -35, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 35*x^18 + 82*x^17 + 414*x^16 - 1076*x^15 - 2213*x^14 + 6605*x^13 + 5177*x^12 - 20965*x^11 - 1727*x^10 + 33661*x^9 - 12282*x^8 - 23057*x^7 + 16581*x^6 + 3122*x^5 - 5900*x^4 + 1710*x^3 - 71*x^2 - 25*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 - 35*x^18 + 82*x^17 + 414*x^16 - 1076*x^15 - 2213*x^14 + 6605*x^13 + 5177*x^12 - 20965*x^11 - 1727*x^10 + 33661*x^9 - 12282*x^8 - 23057*x^7 + 16581*x^6 + 3122*x^5 - 5900*x^4 + 1710*x^3 - 71*x^2 - 25*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 35 x^{18} + 82 x^{17} + 414 x^{16} - 1076 x^{15} - 2213 x^{14} + 6605 x^{13} + 5177 x^{12} - 20965 x^{11} - 1727 x^{10} + 33661 x^{9} - 12282 x^{8} - 23057 x^{7} + 16581 x^{6} + 3122 x^{5} - 5900 x^{4} + 1710 x^{3} - 71 x^{2} - 25 x + 1 \)

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:  \(3724513239982001674003134765625=5^{10}\cdot 11^{16}\cdot 43^{2}\cdot 67^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.77$
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:  $5, 11, 43, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{22388783382264152896681} a^{19} + \frac{5613864710298985133538}{22388783382264152896681} a^{18} + \frac{1428262582051521899835}{22388783382264152896681} a^{17} + \frac{7355223291983591103318}{22388783382264152896681} a^{16} + \frac{767570674255297608867}{22388783382264152896681} a^{15} - \frac{11045627859753841456727}{22388783382264152896681} a^{14} - \frac{938713284272468972129}{22388783382264152896681} a^{13} - \frac{6492483142925681909228}{22388783382264152896681} a^{12} + \frac{5655706054990638181833}{22388783382264152896681} a^{11} + \frac{9234214025400747493291}{22388783382264152896681} a^{10} + \frac{8878277698572293331286}{22388783382264152896681} a^{9} - \frac{1857956040481100126090}{22388783382264152896681} a^{8} + \frac{10704455105731284801406}{22388783382264152896681} a^{7} - \frac{1279841506598711619586}{22388783382264152896681} a^{6} + \frac{10945247565348011649352}{22388783382264152896681} a^{5} - \frac{10200549671977427304145}{22388783382264152896681} a^{4} + \frac{527913352842534041200}{22388783382264152896681} a^{3} - \frac{36141770293761940089}{22388783382264152896681} a^{2} + \frac{11053051837111912345830}{22388783382264152896681} a + \frac{6457768381441429098672}{22388783382264152896681}$

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:  \( 518758634.936 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_2^4:C_5$ (as 20T86):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 32 conjugacy class representatives for $C_2^2\times C_2^4:C_5$
Character table for $C_2^2\times C_2^4:C_5$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{11})^+\), 10.10.617567936161.1, 10.10.1929899800503125.1, 10.10.669871503125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$43$43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$67$67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$