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Properties

Label 20.2.33170570354...1216.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{10}\cdot 11^{16}\cdot 89^{3}$
Root discriminant $18.88$
Ramified primes $2, 11, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T432

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -5, -17, -38, -66, -114, -103, 37, 104, -65, -121, 37, 15, -19, 11, -16, 8, -5, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 4*x^18 - 5*x^17 + 8*x^16 - 16*x^15 + 11*x^14 - 19*x^13 + 15*x^12 + 37*x^11 - 121*x^10 - 65*x^9 + 104*x^8 + 37*x^7 - 103*x^6 - 114*x^5 - 66*x^4 - 38*x^3 - 17*x^2 - 5*x - 1)
 
gp: K = bnfinit(x^20 + 4*x^18 - 5*x^17 + 8*x^16 - 16*x^15 + 11*x^14 - 19*x^13 + 15*x^12 + 37*x^11 - 121*x^10 - 65*x^9 + 104*x^8 + 37*x^7 - 103*x^6 - 114*x^5 - 66*x^4 - 38*x^3 - 17*x^2 - 5*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} + 4 x^{18} - 5 x^{17} + 8 x^{16} - 16 x^{15} + 11 x^{14} - 19 x^{13} + 15 x^{12} + 37 x^{11} - 121 x^{10} - 65 x^{9} + 104 x^{8} + 37 x^{7} - 103 x^{6} - 114 x^{5} - 66 x^{4} - 38 x^{3} - 17 x^{2} - 5 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-33170570354885225234441216=-\,2^{10}\cdot 11^{16}\cdot 89^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.88$
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:  $2, 11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{19253706121059427337} a^{19} - \frac{6690134913691291721}{19253706121059427337} a^{18} + \frac{681579561500369846}{19253706121059427337} a^{17} + \frac{6690242188385324453}{19253706121059427337} a^{16} + \frac{5311290733243384109}{19253706121059427337} a^{15} + \frac{8305138882905554877}{19253706121059427337} a^{14} + \frac{1565780306890925170}{19253706121059427337} a^{13} - \frac{5740074596078439217}{19253706121059427337} a^{12} + \frac{1134879732628666616}{19253706121059427337} a^{11} - \frac{2206240720991512839}{19253706121059427337} a^{10} - \frac{4381008644316637942}{19253706121059427337} a^{9} + \frac{7254845166819697018}{19253706121059427337} a^{8} + \frac{3937610112493888490}{19253706121059427337} a^{7} + \frac{1243353149956484807}{19253706121059427337} a^{6} + \frac{5218539642324844653}{19253706121059427337} a^{5} - \frac{3178242445528201596}{19253706121059427337} a^{4} + \frac{9002907712750642348}{19253706121059427337} a^{3} - \frac{2238653621533908494}{19253706121059427337} a^{2} + \frac{5605559138114867796}{19253706121059427337} a - \frac{8752970009915643045}{19253706121059427337}$

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:  $10$
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:  \( 44393.4228485 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T432:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n432 are not computed
Character table for t20n432 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.2.19077940409.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
89Data not computed