↑

Properties

Label 20.20.1753149859...5856.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{10}\cdot 11^{16}\cdot 43^{2}\cdot 67^{4}$
Root discriminant $32.52$
Ramified primes $2, 11, 43, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T751

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -5, 97, -155, -724, 1361, 2210, -4342, -3349, 7005, 2501, -6233, -678, 3080, -163, -802, 127, 96, -21, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 21*x^18 + 96*x^17 + 127*x^16 - 802*x^15 - 163*x^14 + 3080*x^13 - 678*x^12 - 6233*x^11 + 2501*x^10 + 7005*x^9 - 3349*x^8 - 4342*x^7 + 2210*x^6 + 1361*x^5 - 724*x^4 - 155*x^3 + 97*x^2 - 5*x - 1)
 
gp: K = bnfinit(x^20 - 4*x^19 - 21*x^18 + 96*x^17 + 127*x^16 - 802*x^15 - 163*x^14 + 3080*x^13 - 678*x^12 - 6233*x^11 + 2501*x^10 + 7005*x^9 - 3349*x^8 - 4342*x^7 + 2210*x^6 + 1361*x^5 - 724*x^4 - 155*x^3 + 97*x^2 - 5*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 127 x^{16} - 802 x^{15} - 163 x^{14} + 3080 x^{13} - 678 x^{12} - 6233 x^{11} + 2501 x^{10} + 7005 x^{9} - 3349 x^{8} - 4342 x^{7} + 2210 x^{6} + 1361 x^{5} - 724 x^{4} - 155 x^{3} + 97 x^{2} - 5 x - 1 \)

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:  \(1753149859092675220167728505856=2^{10}\cdot 11^{16}\cdot 43^{2}\cdot 67^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.52$
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:  $2, 11, 43, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{1136556250916791} a^{19} - \frac{166334597715347}{1136556250916791} a^{18} + \frac{398064690184518}{1136556250916791} a^{17} - \frac{125075598502816}{1136556250916791} a^{16} - \frac{441412997990347}{1136556250916791} a^{15} - \frac{193149866924}{1916620996487} a^{14} + \frac{440312557769655}{1136556250916791} a^{13} - \frac{9748209461462}{1136556250916791} a^{12} - \frac{510474543782876}{1136556250916791} a^{11} - \frac{512791749061641}{1136556250916791} a^{10} + \frac{502906089213075}{1136556250916791} a^{9} - \frac{349519080466351}{1136556250916791} a^{8} + \frac{222512715203053}{1136556250916791} a^{7} + \frac{147293853008566}{1136556250916791} a^{6} - \frac{243638233302982}{1136556250916791} a^{5} - \frac{504199762666477}{1136556250916791} a^{4} - \frac{565854312944198}{1136556250916791} a^{3} - \frac{561176830024954}{1136556250916791} a^{2} - \frac{192747894807312}{1136556250916791} a - \frac{336194961404967}{1136556250916791}$

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:  \( 388032792.777 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T751:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n751 are not computed
Character table for t20n751 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.617567936161.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.5$x^{10} - 9 x^{8} + 50 x^{6} - 50 x^{4} + 45 x^{2} - 5$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$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}$
43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
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}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$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$$[\ ]^{2}$
67.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.4.3.1$x^{4} + 268$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$