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Properties

Label 20.10.3101144877...4448.2
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{20}\cdot 11^{16}\cdot 23^{5}$
Root discriminant $29.82$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T749

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -6, 47, 392, 777, 380, 537, 486, -2924, -1638, 1386, 732, 145, -38, -12, 34, -42, -4, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 3*x^18 - 4*x^17 - 42*x^16 + 34*x^15 - 12*x^14 - 38*x^13 + 145*x^12 + 732*x^11 + 1386*x^10 - 1638*x^9 - 2924*x^8 + 486*x^7 + 537*x^6 + 380*x^5 + 777*x^4 + 392*x^3 + 47*x^2 - 6*x - 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 3*x^18 - 4*x^17 - 42*x^16 + 34*x^15 - 12*x^14 - 38*x^13 + 145*x^12 + 732*x^11 + 1386*x^10 - 1638*x^9 - 2924*x^8 + 486*x^7 + 537*x^6 + 380*x^5 + 777*x^4 + 392*x^3 + 47*x^2 - 6*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 3 x^{18} - 4 x^{17} - 42 x^{16} + 34 x^{15} - 12 x^{14} - 38 x^{13} + 145 x^{12} + 732 x^{11} + 1386 x^{10} - 1638 x^{9} - 2924 x^{8} + 486 x^{7} + 537 x^{6} + 380 x^{5} + 777 x^{4} + 392 x^{3} + 47 x^{2} - 6 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-310114487798903480985555304448=-\,2^{20}\cdot 11^{16}\cdot 23^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.82$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{4687} a^{18} + \frac{1552}{4687} a^{17} + \frac{1314}{4687} a^{16} + \frac{171}{4687} a^{15} + \frac{396}{4687} a^{14} - \frac{38}{4687} a^{13} - \frac{525}{4687} a^{12} + \frac{495}{4687} a^{11} - \frac{1803}{4687} a^{10} - \frac{1294}{4687} a^{9} - \frac{1227}{4687} a^{8} - \frac{2108}{4687} a^{7} + \frac{2170}{4687} a^{6} - \frac{982}{4687} a^{5} + \frac{347}{4687} a^{4} + \frac{248}{4687} a^{3} + \frac{1396}{4687} a^{2} - \frac{146}{4687} a - \frac{588}{4687}$, $\frac{1}{263739949623403725808566253} a^{19} + \frac{22131727275288587918336}{263739949623403725808566253} a^{18} + \frac{24997622900497921979111396}{263739949623403725808566253} a^{17} - \frac{20957672220256618195569712}{263739949623403725808566253} a^{16} - \frac{22443723863957782824813313}{263739949623403725808566253} a^{15} + \frac{125652115589312398332279982}{263739949623403725808566253} a^{14} - \frac{58092989650730199467137199}{263739949623403725808566253} a^{13} - \frac{572211249146569702200694}{2419632565352327759711617} a^{12} + \frac{112447467644192391590193688}{263739949623403725808566253} a^{11} - \frac{18775836327129809139344392}{263739949623403725808566253} a^{10} - \frac{54780816763439099089872745}{263739949623403725808566253} a^{9} - \frac{69045584265751843501412101}{263739949623403725808566253} a^{8} + \frac{697575173051362427068628}{6133487200544272693222471} a^{7} + \frac{97247352413808035250754178}{263739949623403725808566253} a^{6} + \frac{27274236571634762641282454}{263739949623403725808566253} a^{5} + \frac{26451350269513459910520837}{263739949623403725808566253} a^{4} - \frac{59411202002856072067395891}{263739949623403725808566253} a^{3} + \frac{7099133826970695521291223}{263739949623403725808566253} a^{2} - \frac{112987987589514358630670320}{263739949623403725808566253} a - \frac{47487147056531257186007304}{263739949623403725808566253}$

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:  $14$
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:  \( 25843851.6431 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T749:

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 t20n749 are not computed
Character table for t20n749 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.116117348402176.2

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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
$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}$$[\ ]_{5}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$