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Properties

Label 20.16.1460932248...8784.1
Degree $20$
Signature $[16, 2]$
Discriminant $2^{20}\cdot 11^{18}\cdot 1583^{2}$
Root discriminant $36.16$
Ramified primes $2, 11, 1583$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T340

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -30, -3, 2232, -13860, 36048, -40389, -1620, 52117, -51150, 12365, 13244, -14059, 6564, -1559, 78, 22, -12, 19, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 19*x^18 - 12*x^17 + 22*x^16 + 78*x^15 - 1559*x^14 + 6564*x^13 - 14059*x^12 + 13244*x^11 + 12365*x^10 - 51150*x^9 + 52117*x^8 - 1620*x^7 - 40389*x^6 + 36048*x^5 - 13860*x^4 + 2232*x^3 - 3*x^2 - 30*x + 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 19*x^18 - 12*x^17 + 22*x^16 + 78*x^15 - 1559*x^14 + 6564*x^13 - 14059*x^12 + 13244*x^11 + 12365*x^10 - 51150*x^9 + 52117*x^8 - 1620*x^7 - 40389*x^6 + 36048*x^5 - 13860*x^4 + 2232*x^3 - 3*x^2 - 30*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 19 x^{18} - 12 x^{17} + 22 x^{16} + 78 x^{15} - 1559 x^{14} + 6564 x^{13} - 14059 x^{12} + 13244 x^{11} + 12365 x^{10} - 51150 x^{9} + 52117 x^{8} - 1620 x^{7} - 40389 x^{6} + 36048 x^{5} - 13860 x^{4} + 2232 x^{3} - 3 x^{2} - 30 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14609322487882432594514132598784=2^{20}\cdot 11^{18}\cdot 1583^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.16$
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:  $2, 11, 1583$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{270657065159570251614008911069} a^{19} + \frac{63408957432930851344258481009}{270657065159570251614008911069} a^{18} + \frac{73560392387794223117748842744}{270657065159570251614008911069} a^{17} + \frac{296954519326261268442133010}{3041090619770452265325942821} a^{16} - \frac{45265113158705372460408868497}{270657065159570251614008911069} a^{15} + \frac{99014527305839458013330388972}{270657065159570251614008911069} a^{14} - \frac{97615883245129963344640884714}{270657065159570251614008911069} a^{13} - \frac{51896703874572883831764089233}{270657065159570251614008911069} a^{12} - \frac{98317021689600103839410109789}{270657065159570251614008911069} a^{11} + \frac{99126134752218858214808404440}{270657065159570251614008911069} a^{10} + \frac{23490214354169933754977562459}{270657065159570251614008911069} a^{9} - \frac{106846194199326051576498621239}{270657065159570251614008911069} a^{8} - \frac{54359285806359421766248193241}{270657065159570251614008911069} a^{7} - \frac{87776421109013590231379492668}{270657065159570251614008911069} a^{6} - \frac{93102642084541281266127226860}{270657065159570251614008911069} a^{5} - \frac{97818666825597248077937455343}{270657065159570251614008911069} a^{4} - \frac{20520048719435954116972312545}{270657065159570251614008911069} a^{3} + \frac{46545793888734861415072900736}{270657065159570251614008911069} a^{2} + \frac{56453753192111494939650831375}{270657065159570251614008911069} a + \frac{9085920856656211676534626756}{270657065159570251614008911069}$

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:  $17$
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:  \( 570345011.017 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T340:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 80 conjugacy class representatives for t20n340 are not computed
Character table for t20n340 is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{44})^+\)

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 ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
1583Data not computed